Gr 12 Mathematics: November Medium P(1)
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Questions and Answers

What happens to the function when $x = -6$?

  • The function equals zero.
  • The function is undefined. (correct)
  • The function approaches positive infinity.
  • The function reaches its maximum.

    • What is the limit of the function as $x$ approaches -6?

  • 6
  • 0
  • -8 (correct)
  • -6

    • Which of the following represents the derivative of the function $f(x) = x^3$?

  • 3x^3
  • 2x^3
  • 3x^2 (correct)
  • x^2

    • What is the derivative of a constant $k$?

    <p>0</p> Signup and view all the answers

    When should you use first principles to find the derivative instead of the rules for differentiation?

    <p>If specified in the question.</p> Signup and view all the answers

    What does the notation $rac{dy}{dx}$ represent?

    <p>The derivative of $y$ with respect to $x$.</p> Signup and view all the answers

    Which of the following statements about limits is true?

    <p>Limits can exist even if the function is undefined at that point.</p> Signup and view all the answers

    What is the term for the procedure of determining the derivative from first principles?

    <p>Differentiation</p> Signup and view all the answers

    If $f(x) = 2x^2 + 3x$, what is $f'(x)$ using the rules of differentiation?

    <p>$4x + 3$</p> Signup and view all the answers

    What is the derivative of the function $f(x) = x^n$ where $n$ is a non-zero constant?

    <p>$nx^{n-1}$</p> Signup and view all the answers

    What is the asymptote of the exponential function f(x) = 10^x?

    <p>y = 0</p> Signup and view all the answers

    What is the domain of the logarithmic function f^{-1}(x) = log x?

    <p>x &gt; 0</p> Signup and view all the answers

    What is the formula used to calculate the accumulated amount in simple interest?

    <p>A = P(1 + in)</p> Signup and view all the answers

    What is the purpose of a present value annuity?

    <p>To pay off a loan or debt over time</p> Signup and view all the answers

    What is the formula used to solve for the time period n in compound interest calculations?

    <p>n = log(A/P) / log(1 + i)</p> Signup and view all the answers

    What is the type of interest applied to the accumulating amount in a future value annuity?

    <p>Compound interest</p> Signup and view all the answers

    What is the purpose of using logarithms in population growth calculations?

    <p>To solve for the time period</p> Signup and view all the answers

    What is the formula used to calculate the pH level of a solution?

    <p>pH = -log10[H+]</p> Signup and view all the answers

    What is the difference between nominal and effective interest rates?

    <p>Nominal rate is the stated rate, while effective rate is the actual rate received</p> Signup and view all the answers

    What is the purpose of using annuities?

    <p>To pay off a loan or debt over time</p> Signup and view all the answers

    What is the general formula for a finite arithmetic series?

    <p>$S_n = rac{n}{2} (a + l)$</p> Signup and view all the answers

    What is a relation in mathematics?

    <p>A rule that associates each element of one set with at least one element of another set</p> Signup and view all the answers

    What is the condition for a function to have an inverse that is also a function?

    <p>The function must be one-to-one</p> Signup and view all the answers

    What is the graphical representation of a one-to-one function?

    <p>Every vertical line intersects the graph at most once</p> Signup and view all the answers

    What is the formula for the sum of a finite arithmetic series when the last term is unknown?

    <p>$S_n = rac{n}{2} (2a + (n - 1) d)$</p> Signup and view all the answers

    What is the definition of an inverse function?

    <p>A function that reverses the operation of a given function</p> Signup and view all the answers

    What is the graphical symmetry of an inverse function?

    <p>The graph of the inverse function is reflected across the line y = x</p> Signup and view all the answers

    What is the horizontal line test used for?

    <p>To determine if a function has an inverse that is also a function</p> Signup and view all the answers

    What are the steps to find the inverse of a linear function?

    <p>Interchange x and y, solve for y, and swap the domain and range</p> Signup and view all the answers

    What is the notation $f^{-1}(x)$ used to represent?

    <p>The inverse function of f(x)</p> Signup and view all the answers

    What is the formula for the sum of the first (n) terms of a finite geometric series?

    <p>[ S_n = rac{a(1 - r^n)}{1 - r} ]</p> Signup and view all the answers

    Which of the following conditions must be met for an infinite geometric series to converge?

    <p>The common ratio (r) must be between -1 and 1.</p> Signup and view all the answers

    What is the sum of the first 100 positive integers?

    <p>5050</p> Signup and view all the answers

    What is the common difference of the arithmetic sequence (2, 5, 8, 11, ...)?

    <p>3</p> Signup and view all the answers

    What is the 10th term of the geometric sequence (2, 6, 18, 54, ...)?

    <p>59049</p> Signup and view all the answers

    Which of the following is a finite arithmetic series?

    <p>1 + 2 + 3 + ... + 100</p> Signup and view all the answers

    What is the sum of the infinite geometric series (1 + 1/2 + 1/4 + 1/8 + ...)?

    <p>2</p> Signup and view all the answers

    Which of the following infinite geometric series is divergent?

    <p>1 - 2 + 4 - 8 + ...</p> Signup and view all the answers

    What is the general formula for the nth term of an arithmetic sequence?

    <p>[ T_n = a + (n - 1) d ]</p> Signup and view all the answers

    What is the difference between a finite geometric series and an infinite geometric series?

    <p>A finite geometric series has a finite number of terms, while an infinite geometric series has an infinite number of terms.</p> Signup and view all the answers

    What is the common ratio in a geometric sequence?

    <p>The constant value multiplied to each term</p> Signup and view all the answers

    How do you check if a sequence is geometric?

    <p>The ratios between consecutive terms must be constant</p> Signup and view all the answers

    What does the variable ( FV ) represent in the context of annuities?

    <p>Future Value of the annuity</p> Signup and view all the answers

    If the first term of a geometric sequence is 4 and the common ratio is 3, what is the third term?

    <p>36</p> Signup and view all the answers

    What does the notation $ ext{S}_n$ represent in series?

    <p>The sum of the first n terms</p> Signup and view all the answers

    Which formula is used to calculate the Present Value of an annuity?

    <p>PV = P rac{1 - (1 + i)^{-n}}{i}</p> Signup and view all the answers

    What is the value of the geometric mean between 9 and 16?

    <p>12</p> Signup and view all the answers

    In the formula ( FV = P rac{(1 + i)^n - 1}{i} ), what does the variable ( P ) represent?

    <p>Payment amount per period</p> Signup and view all the answers

    If a geometric sequence has a common ratio $r$ of 0.5, what does this indicate about the sequence?

    <p>The sequence will decay exponentially</p> Signup and view all the answers

    Which of the following best describes future value annuities?

    <p>They involve making regular payments into an account that earns compound interest.</p> Signup and view all the answers

    What is the purpose of calculating the present value annuity?

    <p>To find the required initial amount to achieve a series of future payments.</p> Signup and view all the answers

    What does the summation symbol $ ext{Σ}$ represent in mathematical notation?

    <p>Sum of the terms in a sequence</p> Signup and view all the answers

    What does the variable ( n ) signify in the annuity formulas?

    <p>The total duration of the investment or loan</p> Signup and view all the answers

    Which formula correctly expresses the sum of the first n terms of a geometric series?

    <p>$S_n = a + ar + ar^2 + ext{...} + ar^{n-1}$</p> Signup and view all the answers

    What happens to a geometric series with a common ratio greater than 1?

    <p>It grows exponentially</p> Signup and view all the answers

    In the formula for the $n$-th term of a geometric sequence, which symbol represents the position of the term?

    <p>$n$</p> Signup and view all the answers

    What does the variable 'n' represent in the Present Value of an Annuity formula?

    <p>Total number of payments</p> Signup and view all the answers

    Which formula is used to calculate the Future Value of Annuities?

    <p>F = x [ (1 + i)^n - 1 ] / i</p> Signup and view all the answers

    In the formula for calculating the Effective Annual Rate (EAR), what does 'm' represent?

    <p>The number of compounding periods per year</p> Signup and view all the answers

    What is the main purpose of calculating the Present Value of an Annuity?

    <p>To calculate the current worth of a series of future payments</p> Signup and view all the answers

    Which of the following is a key aspect of the Achilles and the Tortoise paradox?

    <p>It highlights the concept of limits in calculus</p> Signup and view all the answers

    The formula for Compound Interest includes which of the following parameters?

    <p>Principal amount</p> Signup and view all the answers

    In calculating the Outstanding Loan Balance, which variable represents the monthly payment amount?

    <p>x</p> Signup and view all the answers

    Which of the following represents the formula to calculate Total Interest Paid?

    <p>I = T - P</p> Signup and view all the answers

    In the Nominal and Effective Interest Rate formula, what does 'i' stand for?

    <p>The nominal interest rate</p> Signup and view all the answers

    What does the logarithm conversion formula allow for?

    <p>Changing the base of logarithms</p> Signup and view all the answers

    What defines an arithmetic sequence?

    <p>Each term is calculated by adding a constant value to the previous term.</p> Signup and view all the answers

    Given the first term of an arithmetic sequence is 5 and the common difference is 3, what is the 4th term?

    <p>17</p> Signup and view all the answers

    How can you determine if a sequence is arithmetic?

    <p>Verify if the differences between consecutive terms are constant.</p> Signup and view all the answers

    In the sequence 2, 5, 8, 11, what is the common difference?

    <p>3</p> Signup and view all the answers

    What is the arithmetic mean of the numbers 4 and 10?

    <p>7</p> Signup and view all the answers

    What shape does the graph of an arithmetic sequence typically represent?

    <p>A straight line</p> Signup and view all the answers

    If the 5th term of an arithmetic sequence is 20 and the common difference is 4, what is the first term?

    <p>8</p> Signup and view all the answers

    Which of the following statements is incorrect regarding arithmetic sequences?

    <p>Consecutive terms can differ by varying amounts.</p> Signup and view all the answers

    What is the definition of a stationary point of a function?

    <p>A point where the derivative of the function is zero</p> Signup and view all the answers

    What is the purpose of finding the concavity of a function?

    <p>To determine the shape of the graph</p> Signup and view all the answers

    What is the formula for synthetic division?

    <p>q2 = a3, q1 = a2 + q2 * d/c, q0 = a1 + q1 * d/c</p> Signup and view all the answers

    What is the relationship between the stationary points and the local maximum and minimum points?

    <p>A local maximum or minimum point is always a stationary point</p> Signup and view all the answers

    What is the purpose of finding the y-intercept of a cubic polynomial?

    <p>To find the shape of the graph</p> Signup and view all the answers

    What is the method used to solve cubic equations?

    <p>All of the above</p> Signup and view all the answers

    What is the definition of a point of inflection?

    <p>A point where the function changes from concave up to concave down</p> Signup and view all the answers

    What is the purpose of finding the x-intercepts of a cubic polynomial?

    <p>To find the shape of the graph</p> Signup and view all the answers

    What is the formula for the division rule for polynomials?

    <p>a(x) = b(x) * Q(x) + R(x)</p> Signup and view all the answers

    What is the application of differential calculus in optimisation problems?

    <p>To determine the stationary points</p> Signup and view all the answers

    What is the remainder when a polynomial p(x) is divided by cx - d?

    <p>p(d/c)</p> Signup and view all the answers

    What is the condition for cx - d to be a factor of p(x)?

    <p>p(d/c) = 0</p> Signup and view all the answers

    What is the general form of a polynomial p(x) divided by cx - d?

    <p>p(x) = (cx - d)Q(x) + R</p> Signup and view all the answers

    What is the relationship between the degree of the polynomial p(x) and the degree of the quotient Q(x)?

    <p>The degree of Q(x) is one degree lower than p(x)</p> Signup and view all the answers

    What is the result of substituting x = d/c into p(x) if cx - d is a factor of p(x)?

    <p>p(d/c) = 0</p> Signup and view all the answers

    What is the purpose of the Factor Theorem in solving cubic equations?

    <p>To factorize the polynomial into linear factors</p> Signup and view all the answers

    What is the formula for the addition rule in probability?

    <p>P(A or B) = P(A) + P(B) - P(A and B)</p> Signup and view all the answers

    What is the condition for two events A and B to be mutually exclusive?

    <p>P(A and B) = 0</p> Signup and view all the answers

    What is the result of the addition rule for mutually exclusive events A and B?

    <p>P(A or B) = P(A) + P(B)</p> Signup and view all the answers

    What is the purpose of the Quadratic Formula in solving cubic equations?

    <p>To solve the quadratic polynomial obtained after division</p> Signup and view all the answers

    What is the relationship between the gradients of the tangent and the normal to a curve at a given point?

    <p>The product of the gradients is -1.</p> Signup and view all the answers

    What does the second derivative of a function tell us about the original function?

    <p>The second derivative tells us the concavity of the function.</p> Signup and view all the answers

    Which of the following is NOT a valid notation for the second derivative of a function ( f(x) )?

    <p>( Df(x) )</p> Signup and view all the answers

    What is the y-intercept of the cubic function ( f(x) = 2x^3 - 5x^2 + 3x - 1 )?

    <p>( -1 )</p> Signup and view all the answers

    What is the slope of the tangent line to the graph of ( f(x) = x^3 - 2x^2 + 1 ) at the point ( x = 2 )?

    <p>( 4 )</p> Signup and view all the answers

    How does the coefficient ( a ) affect the shape and orientation of the cubic function ( y = ax^3 + bx^2 + cx + d )?

    <p>The coefficient ( a ) determines the shape and orientation of the function.</p> Signup and view all the answers

    What is the first step in finding the equation of a tangent line to a curve at a given point?

    <p>Find the derivative of the function.</p> Signup and view all the answers

    Which of the following is a valid notation for the derivative of ( y ) with respect to ( x )?

    <p>( y' )</p> Signup and view all the answers

    What is the derivative of the function ( f(x) = 3x^2 - 4x + 2 )?

    <p>( 6x - 4 )</p> Signup and view all the answers

    What is the second derivative of the function ( y = x^4 + 2x^3 - 5x + 1 )?

    <p>( 12x^2 + 12x )</p> Signup and view all the answers

    What is the formula for the probability of the complement of event A?

    <p>$P(A') = 1 - P(A)$</p> Signup and view all the answers

    Which relationship holds true for mutually exclusive events A and B?

    <p>$P(A ext{ and } B) = 0$</p> Signup and view all the answers

    For independent events A and B, what is the correct expression for their joint probability?

    <p>$P(A ext{ and } B) = P(A) imes P(B)$</p> Signup and view all the answers

    How is the addition rule for two events A and B expressed mathematically?

    <p>$P(A ext{ or } B) = P(A) + P(B) - P(A ext{ and } B)$</p> Signup and view all the answers

    What does it mean for two events A and B to be dependent?

    <p>$P(A ext{ and } B) eq P(A) imes P(B)$</p> Signup and view all the answers

    Which statement about mutually exclusive and independent events is true?

    <p>Mutually exclusive events cannot be independent events.</p> Signup and view all the answers

    What is the condition for two events A and B to be mutually exclusive?

    <p>$P(A ext{ and } B) = 0$</p> Signup and view all the answers

    Which of the following is not a characteristic of complementary events?

    <p>They can occur at the same time.</p> Signup and view all the answers

    What can be depicted using a Venn diagram in probability?

    <p>The relationship between events and their probabilities.</p> Signup and view all the answers

    What is the intersection of two mutually exclusive events A and B represented as?

    <p>$A ext{ and } B$</p> Signup and view all the answers

    Which of the following events are mutually exclusive?

    <p>Rolling a 3 and a 6 on a single die</p> Signup and view all the answers

    What is the probability of event (A) given that event (B) has already occurred?

    <p>(P(A | B) )</p> Signup and view all the answers

    What is the probability of drawing a red card from a standard deck of cards, given that the first card drawn was a heart?

    <p>(13/51 )</p> Signup and view all the answers

    If two events are independent, what is the probability of both events occurring?

    <p>The product of the individual probabilities</p> Signup and view all the answers

    What is the probability of rolling a 6 on a standard die, followed by rolling a 2 on the same die?

    <p>(1/36 )</p> Signup and view all the answers

    A restaurant offers a choice of 3 appetizers, 5 main courses, and 2 desserts. How many different meals can be ordered?

    <p>30</p> Signup and view all the answers

    How many ways can 5 people be arranged in a line?

    <p>720</p> Signup and view all the answers

    What is the value of (6!)?

    <p>720</p> Signup and view all the answers

    How many ways can you choose 3 letters from the word 'MATH'?

    <p>12</p> Signup and view all the answers

    What is the probability of getting at least one head when flipping a coin twice?

    <p>(3/4 )</p> Signup and view all the answers

    What is the inverse function of y = ax + q expressed in function notation?

    <p>f^{-1}(x) = \frac{1}{a}x - \frac{q}{a}</p> Signup and view all the answers

    For the quadratic function y = ax^2, what is the appropriate restriction to ensure the inverse is a function when a > 0?

    <p>x ≥ 0</p> Signup and view all the answers

    What is the inverse of the function y = b^x expressed in function notation?

    <p>y = \log_b x</p> Signup and view all the answers

    What is the domain of the logarithmic function y = \log_b x?

    <p>x &gt; 0</p> Signup and view all the answers

    Which of the following statements about the graph of the exponential function y = b^x is true when b > 1?

    <p>The function approaches an asymptote at y = 0.</p> Signup and view all the answers

    What does the Product Rule of logarithms state?

    <p>log_a(xy) = log_a x + log_a y</p> Signup and view all the answers

    In the function y = ax^2, if a < 0, how should the domain typically be restricted to allow the inverse to be a function?

    <p>x ≤ 0</p> Signup and view all the answers

    What is the vertical asymptote of the logarithmic function y = \log_b x?

    <p>x = 0</p> Signup and view all the answers

    Which of the following statements is true for the inverse of the function f(x) = -x^2?

    <p>The graph reflects about the line y = x.</p> Signup and view all the answers

    What is the range of the exponential function y = b^x when b > 1?

    <p>y &gt; 0</p> Signup and view all the answers

    What is the general formula for the nth term of an arithmetic sequence?

    <p>$T_n = a + (n - 1)d$</p> Signup and view all the answers

    What is the common difference of an arithmetic sequence used to find?

    <p>The difference between consecutive terms</p> Signup and view all the answers

    What is the graphical representation of an arithmetic sequence?

    <p>A straight line</p> Signup and view all the answers

    What is the formula used to find the arithmetic mean between two numbers?

    <p>($a + b$)/2</p> Signup and view all the answers

    What is tested to determine if a sequence is arithmetic?

    <p>The difference between consecutive terms</p> Signup and view all the answers

    What is the characteristic of an arithmetic sequence when the common difference is positive?

    <p>The sequence increases</p> Signup and view all the answers

    What is the purpose of finding the common difference in an arithmetic sequence?

    <p>To find the nth term of the sequence</p> Signup and view all the answers

    What is the significance of the gradient of the line in the graphical representation of an arithmetic sequence?

    <p>It represents the common difference</p> Signup and view all the answers

    What is the formula used to calculate the Future Value of an annuity?

    <p>[ FV = P \frac{(1 + i)^n - 1}{i} ]</p> Signup and view all the answers

    In the formula for the Present Value of an annuity, what does the variable ( i ) represent?

    <p>The interest rate per period</p> Signup and view all the answers

    Which of the following scenarios would be best represented by a Future Value Annuity?

    <p>A person saving for retirement by making regular monthly contributions</p> Signup and view all the answers

    What is the primary difference between a Future Value Annuity and a Present Value Annuity?

    <p>The direction of the cash flows</p> Signup and view all the answers

    If you want to calculate the amount you need to deposit today to receive a specific amount of money in the future, which formula would you use?

    <p>Present Value Annuity</p> Signup and view all the answers

    Which of the following statements is TRUE about the formulas for Future Value and Present Value Annuities?

    <p>They are similar formulas, but with different time orientations</p> Signup and view all the answers

    What is the inverse of the function y = ax^2?

    <p>y = ±√(x/a)</p> Signup and view all the answers

    What is the domain of the inverse function f⁻¹(x) = log_b x?

    <p>x &gt; 0</p> Signup and view all the answers

    What is the graph of the exponential function y = b^x like?

    <p>Rising rapidly</p> Signup and view all the answers

    What is the logarithmic function y = log_b x equivalent to?

    <p>x = b^y</p> Signup and view all the answers

    What is the product rule of logarithms?

    <p>log_a(xy) = log_a x + log_a y</p> Signup and view all the answers

    What is the change of base formula for logarithms?

    <p>log_a x = log_b x / log_b a</p> Signup and view all the answers

    What is the inverse of the function y = b^x?

    <p>y = log_b x</p> Signup and view all the answers

    What is the property of the exponential function y = b^x?

    <p>It is always one-to-one</p> Signup and view all the answers

    What is the asymptote of the logarithmic function y = log_b x?

    <p>Vertical at x = 0</p> Signup and view all the answers

    What is the domain of the exponential function y = b^x?

    <p>x ∈ ℝ</p> Signup and view all the answers

    What is the simplified value of $S_{100}$ given that $2S_{100} = 10100$?

    <p>5050</p> Signup and view all the answers

    Which of the following formulas can be used to find the sum of a finite arithmetic series?

    <p>$S_n = rac{n}{2}(2a + (n - 1)d)$</p> Signup and view all the answers

    In a relation, how many elements from the first set can a single element map to in the second set?

    <p>One or more</p> Signup and view all the answers

    What is a required property for a function to have an inverse that is also a function?

    <p>It must be a one-to-one relation.</p> Signup and view all the answers

    What does the notation $f^{-1}(x)$ indicate in mathematics?

    <p>The inverse function of $f(x)$</p> Signup and view all the answers

    How can the inverse of a linear function be determined graphically?

    <p>By reflecting the graph across the line $y = x$.</p> Signup and view all the answers

    What result comes from applying the Horizontal Line Test to a function?

    <p>The function has an inverse.</p> Signup and view all the answers

    Which of the following describes a many-to-one function?

    <p>Some inputs share the same output.</p> Signup and view all the answers

    What step is NOT part of finding the inverse of a linear function?

    <p>Adding a constant to both sides</p> Signup and view all the answers

    In the formula $S_n = rac{n}{2} (a + l)$, what do the variables represent?

    <p>The first and last term of an arithmetic series</p> Signup and view all the answers

    What is the formula used to calculate the present value of an annuity?

    <p>$P = x \left[rac{1 - (1 + i)^{-n}}{i} ight]$</p> Signup and view all the answers

    Which of the following formulas correctly represents the outstanding loan balance?

    <p>$P_{balance} = x \left[rac{1 - (1 + i)^{-n_{remaining}}}{i} ight]$</p> Signup and view all the answers

    What is the formula used to calculate the future value of a series of payments?

    <p>$F = x \left[rac{(1 + i)^n - 1}{i} ight]$</p> Signup and view all the answers

    Which of the following formulas correctly represents the effective annual rate (EAR)?

    <p>$\text{EAR} = \left(1 + \frac{i_{nominal}}{m}\right)^m - 1$</p> Signup and view all the answers

    What is the formula used to calculate the payment amount for a present value annuity?

    <p>$x = \frac{P \cdot i}{1 - (1 + i)^{-n}}$</p> Signup and view all the answers

    Which of the following formulas correctly represents the change of base for logarithms?

    <p>$\log_a x = \frac{\log_b x}{\log_b a}$</p> Signup and view all the answers

    Which of the following formulas correctly represents the simple interest formula?

    <p>$A = P(1 + in)$</p> Signup and view all the answers

    What is the formula used to calculate the time period (n) in compound interest calculations?

    <p>$n = \frac{\log(\frac{A}{P})}{\log(1 + i)}$</p> Signup and view all the answers

    What is the formula used to calculate the total interest paid on a loan?

    <p>$I = T - P$</p> Signup and view all the answers

    Which of the following formulas correctly represents the compound depreciation formula?

    <p>$A = P(1 - i)^n$</p> Signup and view all the answers

    What is the process of determining the derivative of a function from its definition called?

    <p>Differentiation from first principles</p> Signup and view all the answers

    What is the derivative of the function f(x) = x, using the general rule for differentiation?

    <p>1</p> Signup and view all the answers

    What is the notation for the derivative of a function f(x), with respect to x?

    <p>f'(x)</p> Signup and view all the answers

    What is the derivative of a constant k, using the rules for differentiation?

    <p>0</p> Signup and view all the answers

    What is the purpose of using the rules for differentiation?

    <p>To make finding the derivative of a function much simpler</p> Signup and view all the answers

    What does the symbol D represent in the notation Dy or Df(x)?

    <p>Differential operator</p> Signup and view all the answers

    What is the derivative of the function f(x) = 2x^2 + 3x, using the rules of differentiation?

    <p>4x + 3</p> Signup and view all the answers

    What is the purpose of differentiation?

    <p>To find the gradient of the tangent to a curve</p> Signup and view all the answers

    What is the general rule for differentiation of x^n, where n is a non-zero constant?

    <p>nx^(n-1)</p> Signup and view all the answers

    When should you use first principles to find the derivative, instead of the rules for differentiation?

    <p>When the question specifies to use first principles</p> Signup and view all the answers

    What is the formula for the geometric mean between two numbers (a) and (b)?

    <p>$\sqrt{ab}$</p> Signup and view all the answers

    If the common ratio of a geometric sequence is greater than 1, how does the sequence behave?

    <p>The sequence increases exponentially.</p> Signup and view all the answers

    What is the general formula for the (n)-th term of a geometric sequence?

    <p>$T_n = ar^{n-1}$</p> Signup and view all the answers

    How do you determine the common ratio (r) of a geometric sequence?

    <p>Divide any term by the term before it.</p> Signup and view all the answers

    What is the meaning of the symbol (\frac{dy}{dx}) in differentiation?

    <p>It represents the rate of change of y with respect to x.</p> Signup and view all the answers

    What does Sigma notation ((\Sigma)) represent?

    <p>The sum of terms in a sequence</p> Signup and view all the answers

    What is the purpose of finding the second derivative of a function?

    <p>To determine the change in gradient of the original function.</p> Signup and view all the answers

    What is the difference between a finite series and an infinite series?

    <p>A finite series has a limited number of terms, while an infinite series has an unlimited number of terms.</p> Signup and view all the answers

    What is the equation of a tangent to a curve at a given point?

    <p>y - y1 = m(x - x1)</p> Signup and view all the answers

    What is the formula for the sum of a finite geometric series?

    <p>$S_n = \frac{a(1 - r^n)}{1 - r}$</p> Signup and view all the answers

    Which of the following represents the general form of Sigma notation?

    <p>$\sum_{i=m}^{n} T_i = T_m + T_{m+1} + ... + T_{n-1} + T_n$</p> Signup and view all the answers

    How does the coefficient a affect the shape and orientation of a cubic graph?

    <p>It affects the shape and orientation of the graph.</p> Signup and view all the answers

    How would you graphically represent a geometric sequence?

    <p>As an exponential curve</p> Signup and view all the answers

    What is the relationship between the gradients of the tangent and normal to a curve at a given point?

    <p>Their product is -1.</p> Signup and view all the answers

    What is the condition for an infinite geometric series to converge?

    <p>$-1 &lt; r &lt; 1$</p> Signup and view all the answers

    What is the derivative of a function used to describe?

    <p>The rate of change of the function.</p> Signup and view all the answers

    In a geometric sequence, if the common ratio (r) is negative, what happens to the terms?

    <p>They alternate in sign.</p> Signup and view all the answers

    What is the sum of the first 100 positive integers?

    <p>5050</p> Signup and view all the answers

    If a sequence is geometric, what is true about the ratio between consecutive terms?

    <p>It is always constant.</p> Signup and view all the answers

    What is the purpose of finding the equation of a tangent line to a curve?

    <p>To find the gradient of the curve at a given point.</p> Signup and view all the answers

    What is the formula for the nth term of an arithmetic sequence?

    <p>$T_n = a + (n - 1) d$</p> Signup and view all the answers

    What is the difference between a finite geometric series and an infinite geometric series?

    <p>A finite geometric series has a fixed number of terms, while an infinite geometric series has an infinite number of terms.</p> Signup and view all the answers

    What is the purpose of finding the stationary points of a cubic function?

    <p>To solve optimisation problems</p> Signup and view all the answers

    What is the notation for the second derivative of a function?

    <p>f''(x)</p> Signup and view all the answers

    What is the common ratio in a geometric sequence?

    <p>The ratio of consecutive terms</p> Signup and view all the answers

    What is the importance of the coefficient a in a cubic function?

    <p>It affects the shape and orientation of the graph.</p> Signup and view all the answers

    What is the formula used to determine the concavity of a curve?

    <p>f''(x) &gt; 0</p> Signup and view all the answers

    How do you check if a sequence is geometric?

    <p>Check if the ratio of consecutive terms is constant</p> Signup and view all the answers

    What is the purpose of finding the intercepts of a cubic function?

    <p>To graph the function.</p> Signup and view all the answers

    What is the method used to classify a stationary point as a local maximum or local minimum?

    <p>Determining the concavity of the curve</p> Signup and view all the answers

    What is the purpose of using synthetic division in factoring cubic polynomials?

    <p>To factorise the polynomial</p> Signup and view all the answers

    What is the formula for the sum of an infinite geometric series?

    <p>$S_ \infty = \frac{a}{1 - r}$</p> Signup and view all the answers

    What is the formula used to find the y-intercept of a cubic polynomial?

    <p>f(0) = d</p> Signup and view all the answers

    What is the purpose of Gauss's method for summing an arithmetic series?

    <p>To find the sum of a finite arithmetic series</p> Signup and view all the answers

    What is the purpose of finding the points of inflection of a curve?

    <p>To sketch the graph of the function</p> Signup and view all the answers

    What is the sum of the infinite geometric series $1 + 1/2 + 1/4 + 1/8 + \cdots$?

    <p>2</p> Signup and view all the answers

    What is the method used to solve cubic polynomial equations?

    <p>All of the above</p> Signup and view all the answers

    What is the purpose of finding the x-intercepts of a cubic polynomial?

    <p>To sketch the graph of the function</p> Signup and view all the answers

    What is the formula used to determine the end behavior of a cubic polynomial?

    <p>Analyzing the behavior as x approaches positive and negative infinity</p> Signup and view all the answers

    What is the purpose of using the Rational Root Theorem in factoring cubic polynomials?

    <p>To find possible rational roots of the polynomial</p> Signup and view all the answers

    What is the probability of the union of two mutually exclusive events A and B?

    <p>P(A) + P(B)</p> Signup and view all the answers

    What is the product rule for independent events A and B?

    <p>P(A and B) = P(A) × P(B)</p> Signup and view all the answers

    What is the complementary rule in probability?

    <p>P(A) = 1 - P(A')</p> Signup and view all the answers

    What is the addition rule for any two events A and B?

    <p>P(A or B) = P(A) + P(B) - P(A and B)</p> Signup and view all the answers

    What is the purpose of a Venn diagram in probability?

    <p>To show how events are related to one another</p> Signup and view all the answers

    What is the term for events that cannot happen at the same time?

    <p>Mutually exclusive events</p> Signup and view all the answers

    What is the symbol used to represent the intersection of two sets A and B?

    <p>A ∩ B</p> Signup and view all the answers

    What is the probability of the complement of an event A?

    <p>P(A') = 1 - P(A)</p> Signup and view all the answers

    What is the purpose of a tree diagram in probability?

    <p>To organize and visualize the different possible outcomes of a sequence of events</p> Signup and view all the answers

    What is the term for the procedure of determining the probability of an event from the sample space?

    <p>Experiment</p> Signup and view all the answers

    When calculating the probability of a sequence of outcomes from two events, what operation is used between the probabilities of each outcome?

    <p>Multiplication</p> Signup and view all the answers

    Which of the following correctly describes mutually exclusive events?

    <p>They cannot occur together.</p> Signup and view all the answers

    Which formula represents the Complementary Rule in probability?

    <p>$P( ext{not } A) = 1 - P(A)$</p> Signup and view all the answers

    What does the notation $n!$ represent in factorial notation?

    <p>The product of the first n positive integers</p> Signup and view all the answers

    In the context of the Fundamental Counting Principle, how do you determine the total number of outcomes for two combined events?

    <p>Multiply the outcomes of both events</p> Signup and view all the answers

    Which statement correctly describes dependent events?

    <p>The occurrence of one event affects the probability of the other.</p> Signup and view all the answers

    What is the formula for calculating the probability of the union of two mutually exclusive events?

    <p>$P(A ext{ or } B) = P(A) + P(B)$</p> Signup and view all the answers

    Which of the following statements best describes the Fundamental Counting Principle?

    <p>It allows for the calculation of outcomes for independent events by multiplication.</p> Signup and view all the answers

    What is the implication of having complementary events in probability?

    <p>They represent the complete sample space together.</p> Signup and view all the answers

    What does the Remainder Theorem state about the remainder when dividing a polynomial by a linear polynomial?

    <p>The remainder equals the value of the polynomial at the root of the divisor.</p> Signup and view all the answers

    If a polynomial p(x) is divided by cx - d and p(d/c) = 0, what can be concluded?

    <p>cx - d is a factor of p(x).</p> Signup and view all the answers

    When using the Factor Theorem, substituting which value helps identify if cx - d is a factor of p(x)?

    <p>p(d/c)</p> Signup and view all the answers

    What is the main purpose of the Quadratic Formula when solving cubic equations?

    <p>To solve the resulting quadratic equation after finding a factor.</p> Signup and view all the answers

    Which of the following statements accurately describes the relationship between the Remainder Theorem and the Factor Theorem?

    <p>The Factor Theorem can be applied when the remainder is zero using the Remainder Theorem.</p> Signup and view all the answers

    In the context of probability, what does the Addition Rule help to determine regarding events A and B?

    <p>The probability of at least one of the events A or B occurring.</p> Signup and view all the answers

    Which of the following equations represents the addition rule for two mutually exclusive events?

    <p>P(A or B) = P(A) + P(B)</p> Signup and view all the answers

    What indicates that the polynomial p(x) can be expressed as p(x) = (cx - d)Q(x)?

    <p>If cx - d is a divisor of p(x) without a remainder.</p> Signup and view all the answers

    Which step is NOT involved when solving cubic equations using the Factor Theorem?

    <p>Finding the derivative of the cubic equation.</p> Signup and view all the answers

    What characterizes the reflection of an inverse function in relation to its original function?

    <p>It reflects about the line $y = x$.</p> Signup and view all the answers

    Which statement correctly describes the domain of the logarithmic function?

    <p>Greater than 0.</p> Signup and view all the answers

    What is the formula used to calculate the accumulated amount in simple interest?

    <p>$A = P(1 + in)$</p> Signup and view all the answers

    Which of the following applications of logarithms is NOT mentioned in the context provided?

    <p>Analyzing market trends.</p> Signup and view all the answers

    In the context of compound interest, what is the correct formula for solving for the time period $n$?

    <p>$n = rac{ ext{log}(A/P)}{ ext{log}(1 + i)}$</p> Signup and view all the answers

    What type of interest is used in calculating the future value of an annuity?

    <p>Compound interest</p> Signup and view all the answers

    Which of the following represents a key property of the exponential function $f(x) = 10^x$?

    <p>It has an asymptote at $y=0$.</p> Signup and view all the answers

    What is the correct interpretation of the formula $A = P(1 - i)^n$?

    <p>Formula for simple depreciation.</p> Signup and view all the answers

    What does the variable $P$ represent in the formula $A = P(1 + i)^n$?

    <p>Principal amount.</p> Signup and view all the answers

    Which of the following correctly defines a Present Value Annuity (PVA)?

    <p>Equal payments made to pay off a loan over time.</p> Signup and view all the answers

    Given an arithmetic sequence with the first term, (a), and common difference, (d), which formula correctly represents the (n)-th term, (T_n)?

    <p>(T_n = a + (n - 1)d)</p> Signup and view all the answers

    In an arithmetic sequence, the difference between consecutive terms is known as the...

    <p>Common difference</p> Signup and view all the answers

    What is the arithmetic mean between the numbers 10 and 22?

    <p>16</p> Signup and view all the answers

    If the common difference, (d), of an arithmetic sequence is positive, then the sequence will...

    <p>Increase</p> Signup and view all the answers

    Which of the following sequences is an arithmetic sequence?

    <p>3, 7, 11, 15, ...</p> Signup and view all the answers

    What is the 15th term of the arithmetic sequence 5, 8, 11, 14, ...?

    <p>44</p> Signup and view all the answers

    If the first term of an arithmetic sequence is 2 and the common difference is 3, what is the sum of the first 10 terms?

    <p>155</p> Signup and view all the answers

    Which of the following is a graphical representation of an arithmetic sequence?

    <p>A straight line</p> Signup and view all the answers

    Which of the following statements correctly describes a one-to-one function?

    <p>Each element in the domain maps to a unique element in the range.</p> Signup and view all the answers

    In the general formula for a finite arithmetic series, (S_n = \frac{n}{2}(a + l)), what does the variable (l) represent?

    <p>The last term of the arithmetic series</p> Signup and view all the answers

    What is the condition for a function (f(x)) to have an inverse that is also a function?

    <p>The function must be one-to-one.</p> Signup and view all the answers

    Which of the following is NOT a key property of inverse functions?

    <p>The inverse function is always a linear function.</p> Signup and view all the answers

    What is the purpose of the horizontal line test when analyzing a function?

    <p>To determine if the function is a one-to-one function.</p> Signup and view all the answers

    Given a function (f(x) = 2x + 3), what is its inverse function, (f^{-1}(x))?

    <p>(f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} )</p> Signup and view all the answers

    What is the sum of the first 100 positive integers?

    <p>5050</p> Signup and view all the answers

    Which of the following is a valid formula for the sum of a finite arithmetic series when the last term is unknown?

    <p>(S_n = \frac{n}{2}(2a + (n-1)d) )</p> Signup and view all the answers

    What is the relationship between the graphs of a function and its inverse function?

    <p>They are reflections of each other across the line (y = x).</p> Signup and view all the answers

    In the formula (S_n = \frac{n}{2}(2a + (n - 1)d)), what does the variable (d) represent?

    <p>The common difference of the arithmetic series</p> Signup and view all the answers

    What is the formula to derive the sum of a finite geometric series?

    <p>$S_n = \frac{a(1 - r^n)}{1 - r}$</p> Signup and view all the answers

    What is the condition for an infinite geometric series to converge?

    <p>$-1 &lt; r &lt; 1$</p> Signup and view all the answers

    What is the formula for the sum of an infinite geometric series?

    <p>$S_\infty = \frac{a}{1 - r}$</p> Signup and view all the answers

    What is the definition of a finite geometric series?

    <p>A series with a finite number of terms where each term is found by multiplying the previous term by a constant value.</p> Signup and view all the answers

    What is the formula for the nth term of a geometric sequence?

    <p>$T_n = ar^{n-1}$</p> Signup and view all the answers

    What is the method used to find the sum of the first 100 positive integers?

    <p>Karl Friedrich Gauss's Method</p> Signup and view all the answers

    What is the common ratio in a geometric sequence?

    <p>The constant value multiplied by the previous term to get the next term.</p> Signup and view all the answers

    What is the purpose of using annuities?

    <p>To calculate the future value of a present amount.</p> Signup and view all the answers

    What is the formula used to calculate the sum of a finite arithmetic series?

    <p>$S_n = \frac{n(a + l)}{2}$</p> Signup and view all the answers

    What is the difference between a finite arithmetic series and a finite geometric series?

    <p>A finite arithmetic series has a common difference, whereas a finite geometric series has a common ratio.</p> Signup and view all the answers

    What happens to the terms of a geometric sequence if the common ratio is negative?

    <p>The terms alternate in sign.</p> Signup and view all the answers

    Which of the following formulas correctly describes the geometric mean of two numbers, $a$ and $b$?

    <p>$\sqrt{ab}$</p> Signup and view all the answers

    How can one confirm a sequence is geometric?

    <p>The ratio between consecutive terms must be constant.</p> Signup and view all the answers

    If the first term of a geometric sequence is 5 and the common ratio is 2, what is the fifth term?

    <p>40</p> Signup and view all the answers

    What is the notation used to represent the sum of the first $n$ terms in a finite series?

    <p>$S_n$</p> Signup and view all the answers

    In the formula for the $n$-th term of a geometric sequence, what does the symbol $r$ represent?

    <p>The common ratio.</p> Signup and view all the answers

    What is the behavior of a geometric sequence in which the common ratio $r$ is greater than 1?

    <p>The terms grow exponentially.</p> Signup and view all the answers

    Which notation is used to designate the sum in sigma notation?

    <p>$\Sigma$</p> Signup and view all the answers

    What is a characteristic of a finite series?

    <p>It only includes terms up to a specified limit.</p> Signup and view all the answers

    What should be done to find the number of terms in a geometric sequence if the $n$-th term is set equal to a specific value?

    <p>Use the formula $T_n = ar^{n-1}$ and solve for $n$.</p> Signup and view all the answers

    What is the formula used to calculate the present value of an annuity?

    <p>[ PV = P \frac{1 - (1 + i)^{-n}}{i} ]</p> Signup and view all the answers

    Which of the following best describes the process involved in a future value annuity?

    <p>A series of payments is made over a specific time period, with interest being earned on the accumulated balance.</p> Signup and view all the answers

    In the formula for the future value of an annuity, what does the variable 'i' represent?

    <p>The interest rate per period</p> Signup and view all the answers

    What is the purpose of the present value formula for an annuity?

    <p>To calculate the initial amount required to achieve a series of future payments</p> Signup and view all the answers

    Which of the following is a correct interpretation of the term 'annuity' in the context of finance?

    <p>A series of equal payments made over a specific time period.</p> Signup and view all the answers

    What is the formula used to calculate the accumulated amount in compound interest?

    <p>$A = P(1 + i)^n$</p> Signup and view all the answers

    What is the key difference between a future value annuity and a present value annuity?

    <p>Future value annuities involve making payments, while present value annuities involve receiving payments.</p> Signup and view all the answers

    What is the formula used to solve for the time period (n) in compound interest calculations?

    <p>$n = rac{log(rac{A}{P})}{log(1 + i)}$</p> Signup and view all the answers

    Which type of annuity is used to accumulate a sum of money in the future by making regular deposits?

    <p>Future Value Annuity (FVA)</p> Signup and view all the answers

    Which of the following financial concepts is used to calculate the reduction in value of an asset over time, considering the reduced value in each period?

    <p>Compound Depreciation</p> Signup and view all the answers

    What is the formula used to calculate the value of an asset after depreciation using the simple depreciation method?

    <p>$A = P(1 - in)$</p> Signup and view all the answers

    What is the purpose of a present value annuity (PVA)?

    <p>To pay off a loan or debt over time with regular payments</p> Signup and view all the answers

    What does the variable 'i' represent in the compound interest formula $A = P(1 + i)^n$?

    <p>The interest rate per period</p> Signup and view all the answers

    Which of the following is a common application of logarithms in financial calculations?

    <p>Calculating loan repayments and interest rates</p> Signup and view all the answers

    What is the difference between nominal and effective interest rates?

    <p>Nominal interest rate is the stated rate, while effective interest rate considers the compounding frequency.</p> Signup and view all the answers

    In the context of annuities, what does the variable 'FV' represent?

    <p>The future value of the annuity</p> Signup and view all the answers

    What is the inverse of the linear function expressed as $y = ax + q$?

    <p>$f^{-1}(x) = rac{1}{a}x - rac{q}{a}$</p> Signup and view all the answers

    What restriction must be applied to the quadratic function $y = ax^2$ for its inverse to also be a function?

    <p>$x &lt; 0$</p> Signup and view all the answers

    For the function $y = b^x$, what is the correct expression for its inverse?

    <p>$y = ext{log}_b x$</p> Signup and view all the answers

    What is the range of the logarithmic function $y = ext{log}_b x$?

    <p>$y ext{ is any real number}$</p> Signup and view all the answers

    For the exponential function $y = b^x$ with base $b > 1$, how does the graph behave?

    <p>The graph rises rapidly as x increases.</p> Signup and view all the answers

    What transformation is necessary to find the inverse of the quadratic function $y = ax^2$?

    <p>Interchange $x$ and $y$.</p> Signup and view all the answers

    Which of the following statements best describes the domain of the logarithmic function $y = ext{log}_b x$?

    <p>$x ext{ must be greater than 0}$</p> Signup and view all the answers

    What is the relationship between the domain of a function and the range of its inverse?

    <p>The domain of the function is the range of its inverse.</p> Signup and view all the answers

    Which of the following is a property of both exponential and logarithmic functions?

    <p>Both are always increasing.</p> Signup and view all the answers

    What does the symbol ( D ) represent in the notation ( Df(x) )?

    <p>The differential operator</p> Signup and view all the answers

    What is the relationship between the gradients of the tangent and the normal to a curve at a given point?

    <p>m_tangent m_normal = -1</p> Signup and view all the answers

    What is the purpose of finding the second derivative of a function?

    <p>To describe the change in gradient of the original function</p> Signup and view all the answers

    What is the equation of the tangent line to the curve y = f(x) at x = a?

    <p>y - f(a) = f'(a)(x - a)</p> Signup and view all the answers

    What is the effect of the coefficient a on the shape of the cubic graph y = ax^3 + bx^2 + cx + d?

    <p>It affects the shape and orientation of the graph</p> Signup and view all the answers

    What does the variable 'n' represent in the formulas associated with annuities?

    <p>The number of periods or payments</p> Signup and view all the answers

    How do you find the x-intercepts of a cubic function f(x) = ax^3 + bx^2 + cx + d?

    <p>Set y = 0 and solve for x</p> Signup and view all the answers

    What is the notation for the second derivative of a function f(x)?

    <p>f''(x)</p> Signup and view all the answers

    Which formula can be used to calculate the effective annual rate (EAR)?

    <p>EAR = (1 + i_nominal/m)^m - 1</p> Signup and view all the answers

    What is the purpose of finding the derivative of a function?

    <p>To describe the rate of change of the function</p> Signup and view all the answers

    In the context of a loan balance calculation, what does the term 'P_balance' represent?

    <p>Remaining loan balance</p> Signup and view all the answers

    What is the equation of the normal to a curve at a given point?

    <p>y - y1 = -1/m(x - x1)</p> Signup and view all the answers

    What is the purpose of the 'Change of Base for Logarithms' formula?

    <p>To redefine logarithms to different bases</p> Signup and view all the answers

    What is the derivative of the function f(x) = ax^3 + bx^2 + cx + d?

    <p>f'(x) = 3ax^2 + 2bx + c</p> Signup and view all the answers

    In the present value of an annuity formula, what does 'x' represent?

    <p>The payment amount per period</p> Signup and view all the answers

    How is 'Total Interest Paid' calculated in relation to loans?

    <p>I = T - P</p> Signup and view all the answers

    What does the variable 'A' represent in the simple interest formula?

    <p>Accumulated amount after interest</p> Signup and view all the answers

    In the formula for calculating the period of an investment using compound interest, what is the variable 'i'?

    <p>Interest rate per period</p> Signup and view all the answers

    What does the compound depreciation formula calculate?

    <p>Current value after depreciation</p> Signup and view all the answers

    What does the formula for future value of annuities calculate?

    <p>The value of payments made in the future</p> Signup and view all the answers

    What is the purpose of finding the stationary points of a function?

    <p>To determine the local maximum and minimum values.</p> Signup and view all the answers

    What is the result of synthetic division in polynomial division?

    <p>Quotient and remainder.</p> Signup and view all the answers

    What is the condition for a point of inflection?

    <p>f''(x) = 0</p> Signup and view all the answers

    What is the formula for long division in polynomial division?

    <p>a(x) = b(x) * Q(x) + R(x)</p> Signup and view all the answers

    What is the purpose of finding the concavity of a function?

    <p>To determine the shape of the graph.</p> Signup and view all the answers

    What is the formula for synthetic division?

    <p>q_2 = a_3, q_1 = a_2 + q_2 * d/c, q_0 = a_1 + q_1 * d/c</p> Signup and view all the answers

    What is the condition for a local maximum or local minimum?

    <p>f'(x) = 0</p> Signup and view all the answers

    What is the purpose of finding the x-intercepts of a function?

    <p>To determine the shape of the graph.</p> Signup and view all the answers

    What is the formula for the remainder in synthetic division?

    <p>R = a_0 + q_0 * d/c</p> Signup and view all the answers

    What is the purpose of drawing the graph of a cubic polynomial?

    <p>To identify the end behavior of the function.</p> Signup and view all the answers

    What is the value of the limit of the function as x approaches -6?

    <p>-8</p> Signup and view all the answers

    What does the derivative of a function represent?

    <p>The rate of change of the function</p> Signup and view all the answers

    What is the probability of a sequence of outcomes in a tree diagram?

    <p>The product of the probabilities along the branches of the sequence</p> Signup and view all the answers

    What is the purpose of a two-way contingency table?

    <p>To keep a record of the counts or percentages in a probability problem</p> Signup and view all the answers

    Which of the following statements about the function y = (x + 6)(x - 2)/(x + 6) is true?

    <p>It has a hole at x = -6.</p> Signup and view all the answers

    What is the formula for the probability of mutually exclusive events A and B?

    <p>P(A or B) = P(A) + P(B)</p> Signup and view all the answers

    Which rule is applied when determining the derivative of a constant multiplied by a function?

    <p>Constant remains, apply the derivative to the function</p> Signup and view all the answers

    What is the notation for the sample space in probability?

    <p>S</p> Signup and view all the answers

    How is the gradient of the tangent to a curve at a specific point defined?

    <p>Using the limit as h approaches 0 of the difference quotient</p> Signup and view all the answers

    From what principle is the process of finding a derivative derived?

    <p>First principles or the definition of a derivative</p> Signup and view all the answers

    What is the formula for the total number of outcomes for k events?

    <p>n_1 × n_2 × ... × n_k</p> Signup and view all the answers

    What is the definition of n! (read as 'n factorial')?

    <p>The product of all positive integers up to n</p> Signup and view all the answers

    Which of the following notations is NOT associated with the derivative?

    <p>D_x(x^n)</p> Signup and view all the answers

    What is the total number of possible arrangements of n different objects?

    <p>n!</p> Signup and view all the answers

    What happens to the function value when x equals -6?

    <p>The function value is undefined.</p> Signup and view all the answers

    What is the general rule for differentiating a function of the form x^n?

    <p>rac{d}{dx}[x^n] = nx^{n-1}</p> Signup and view all the answers

    What is the formula for the total number of possibilities if there are n objects to choose from and you choose from them r times?

    <p>n × n × ... × n (r times) = n^r</p> Signup and view all the answers

    What is the purpose of the complementary rule?

    <p>To find the probability of the complement of an event</p> Signup and view all the answers

    Which operation does the notation rac{dy}{dx} signify?

    <p>The instantaneous rate of change of y with respect to x</p> Signup and view all the answers

    What is the definition of the union of sets A and B?

    <p>All elements in A or B</p> Signup and view all the answers

    What does the Remainder Theorem indicate about the remainder when dividing a polynomial by a linear divisor?

    <p>The remainder is obtained by substituting the value $\frac{d}{c}$ into the polynomial.</p> Signup and view all the answers

    Under what circumstance does the Factor Theorem affirm that a polynomial has a certain factor?

    <p>If the polynomial evaluated at $\frac{d}{c}$ equals zero.</p> Signup and view all the answers

    Which of the following statements about cubic equations is true?

    <p>Finding one factor allows us to reduce the cubic to solving a quadratic equation.</p> Signup and view all the answers

    What is the result when applying the addition rule for two mutually exclusive events A and B?

    <p>P(A or B) is equal to the sum of P(A) and P(B).</p> Signup and view all the answers

    Which of the following equations correctly represents the relationship described by the Remainder Theorem?

    <p>p(x) = (cx - d) \cdot Q(x) + R</p> Signup and view all the answers

    How would you express a polynomial if $cx - d$ is found to be a factor of $p(x)$?

    <p>p(x) = (cx - d) \cdot Q(x) + R</p> Signup and view all the answers

    When determining the solutions of a cubic polynomial, what is typically the first step?

    <p>Identify a factor using the Factor Theorem.</p> Signup and view all the answers

    What does the addition rule for probabilities account for that makes it different from simply adding individual probabilities?

    <p>The intersection of events.</p> Signup and view all the answers

    Which of the following is true regarding the remainder when dividing a polynomial by a linear polynomial?

    <p>The remainder is a constant value when divided by a linear polynomial.</p> Signup and view all the answers

    Which of the following statements accurately describes the relationship between mutually exclusive events and independent events?

    <p>Mutually exclusive events and independent events are distinct concepts, and one does not imply the other.</p> Signup and view all the answers

    The addition rule for the probability of the union of two events, ( A ) and ( B ), is given by ( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) ). What is the justification for subtracting ( P(A \text{ and } B) ) from the sum of ( P(A) ) and ( P(B) )?

    <p>To avoid counting the probability of the intersection of ( A ) and ( B ) twice.</p> Signup and view all the answers

    Two events, ( A ) and ( B ), are said to be complementary if they satisfy which of the following conditions?

    <p>They are mutually exclusive and their union forms the entire sample space.</p> Signup and view all the answers

    What does the symbol ( A' ) represent in the context of probability?

    <p>The complement of event ( A ), containing all outcomes not in ( A ).</p> Signup and view all the answers

    A tree diagram is a visual representation used in probability to:

    <p>Organize and visualize the different possible outcomes of a sequence of events.</p> Signup and view all the answers

    The product rule for independent events states that ( P(A \text{ and } B) = P(A) \times P(B) ). Which of the following scenarios demonstrates this rule?

    <p>Flipping a coin twice and getting heads on both flips.</p> Signup and view all the answers

    Which of the following statements is TRUE about the complementary rule in probability?

    <p>It states that the probability of an event and its complement always add up to 1.</p> Signup and view all the answers

    The symbol ( \emptyset ) in probability represents:

    <p>The empty set, containing no elements.</p> Signup and view all the answers

    Venn diagrams are graphical representations used in probability to:

    <p>Show the relationship between events and their probabilities.</p> Signup and view all the answers

    Which of the following statements correctly describes the difference between the addition rule and the product rule in probability?

    <p>The addition rule calculates the probability of the union of two events, while the product rule calculates the probability of the intersection of two events.</p> Signup and view all the answers

    What is the formula for the nth term of an arithmetic sequence?

    <p>T_n = a + (n - 1)d</p> Signup and view all the answers

    What is the common difference of an arithmetic sequence?

    <p>The constant value added to each term to get the next term</p> Signup and view all the answers

    What is the graphical representation of an arithmetic sequence?

    <p>A straight line</p> Signup and view all the answers

    What is the arithmetic mean between two numbers?

    <p>The average of the two numbers</p> Signup and view all the answers

    How do you test if a sequence is arithmetic?

    <p>By checking if the differences between consecutive terms are equal</p> Signup and view all the answers

    What is the characteristic of an arithmetic sequence with a positive common difference?

    <p>The sequence increases</p> Signup and view all the answers

    What is the purpose of finding the common difference in an arithmetic sequence?

    <p>To determine if the sequence is arithmetic</p> Signup and view all the answers

    What is the importance of the first term and the common difference in an arithmetic sequence?

    <p>They are used to find the nth term of the sequence</p> Signup and view all the answers

    What is the formula for the (n)-th term of a geometric sequence?

    <p>$T_n = ar^{n-1}$</p> Signup and view all the answers

    What is the geometric mean between 4 and 9?

    <p>6</p> Signup and view all the answers

    Which of the following statements is true about a geometric sequence with a common ratio (r) between 0 and 1?

    <p>The sequence decays exponentially.</p> Signup and view all the answers

    What is the sum of the first five terms of the geometric sequence 2, 6, 18, 54, ...?

    <p>242</p> Signup and view all the answers

    What is the general form for sigma notation?

    <p><em>{i=m}^n T_i = T_m + T</em>{m+1} + ... + T_{n-1} + T_n</p> Signup and view all the answers

    Which of the following is a finite arithmetic series?

    <p>2 + 5 + 8 + 11 + 14</p> Signup and view all the answers

    What is the common ratio of the geometric sequence 1, 3, 9, 27, ...?

    <p>3</p> Signup and view all the answers

    Which of the following conditions must be met for an infinite geometric series to converge?

    <p>The common ratio must be less than 1.</p> Signup and view all the answers

    What is the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ...?

    <p>2</p> Signup and view all the answers

    If the first term of a geometric sequence is 4 and the common ratio is 3, what is the third term?

    <p>36</p> Signup and view all the answers

    What is the common ratio in the geometric sequence defined by the formula $T_n = 5 imes 2^{n-1}$?

    <p>2</p> Signup and view all the answers

    For which values of the common ratio $r$ would an infinite geometric series diverge?

    <p>1</p> Signup and view all the answers

    How is the sum for the first 100 integers calculated using Karl Friedrich Gauss's method?

    <p>By pairing numbers from each end of the sequence</p> Signup and view all the answers

    What is the formula for the sum of the first $n$ terms of a finite geometric series when the common ratio $r$ is less than 1?

    <p>$S_n = \frac{a(1 - r^n)}{1 - r}$</p> Signup and view all the answers

    Which of the following correctly represents the formula for the $n$-th term of an arithmetic sequence?

    <p>$T_n = a + (n - 1)d$</p> Signup and view all the answers

    What must be true for a geometric series to be convergent?

    <p>-1 &lt; r &lt; 1</p> Signup and view all the answers

    What is the sum of the infinite geometric series $1 + rac{1}{2} + rac{1}{4} + rac{1}{8} + ...$?

    <p>2</p> Signup and view all the answers

    If the first term of a geometric sequence is 3 and the common ratio is 0.5, what is the fifth term of this sequence?

    <p>0.375</p> Signup and view all the answers

    Which of the following statements about the sum of a finite arithmetic series is accurate?

    <p>The sum can be computed using the formula $S_n = \frac{n}{2}(a + l)$</p> Signup and view all the answers

    What is the formula for the sum of the first (n) terms of an arithmetic series when the last term, (l), is unknown?

    <p>$S_n = \frac{n}{2}(2a + (n - 1)d)$</p> Signup and view all the answers

    If a function is not one-to-one, what can be said about its inverse?

    <p>It may not be a function.</p> Signup and view all the answers

    Which of the following correctly describes the graphical symmetry of a function and its inverse?

    <p>The graph of the inverse is a reflection of the original function across the line (y = x).</p> Signup and view all the answers

    What is the purpose of the horizontal line test?

    <p>To determine if a function is one-to-one.</p> Signup and view all the answers

    What is the correct notation for the inverse function of (f(x))?

    <p>(f^{-1}(x))</p> Signup and view all the answers

    What is the inverse of the linear function (y = 3x - 2)?

    <p>$y = \frac{1}{3}x + \frac{2}{3}$</p> Signup and view all the answers

    Which of the following is NOT a key property of inverse functions?

    <p>Inverse functions always have the same domain and range.</p> Signup and view all the answers

    Which of the following statements about the sum of an arithmetic series is FALSE?

    <p>The sum of an arithmetic series is always a positive number.</p> Signup and view all the answers

    What is the sum of the first 100 positive integers?

    <p>5050</p> Signup and view all the answers

    What is the common difference of the arithmetic sequence (2, 5, 8, 11, ...)?

    <p>3</p> Signup and view all the answers

    What is the main purpose of using the present value of an annuity formula?

    <p>To calculate the current value of future payment streams</p> Signup and view all the answers

    In the formula for future value of an annuity, what does the variable 'n' represent?

    <p>The number of total payment periods</p> Signup and view all the answers

    Which of the following components is added to the payments in the calculation of future value annuities?

    <p>Compound interest</p> Signup and view all the answers

    What is represented by 'i' in the annuity formulas?

    <p>The interest rate per period</p> Signup and view all the answers

    What distinguishes present value annuities from future value annuities?

    <p>They involve repaying debts with equal payments.</p> Signup and view all the answers

    If an individual wants to find out how much they need to invest now to achieve a specific annuity in the future, which formula should they use?

    <p>Present Value of an Annuity formula</p> Signup and view all the answers

    What is the formula for the accumulated amount (A) after (n) periods of compound depreciation?

    <p>$A = P(1 - i)^n$</p> Signup and view all the answers

    Which of the following statements is true about the relationship between the graph of a function and its inverse?

    <p>The graph of the inverse is a reflection of the original function about the line (y = x).</p> Signup and view all the answers

    If a population triples in size, and the growth rate is (i), what is the formula for the number of periods (n) it takes for the population to triple?

    <p>$n = rac{\log(3)}{\log(1 + i)}$</p> Signup and view all the answers

    What is the formula for calculating the effective annual interest rate (i), given the nominal interest rate (i_{(m)}) compounded (m) times per year?

    <p>$1 + i = \left(1 + rac{i_{(m)}}{m} ight)^m$</p> Signup and view all the answers

    What is the purpose of a future value annuity (FVA)?

    <p>To accumulate a sum of money in the future by making regular deposits.</p> Signup and view all the answers

    What is the formula for calculating the future value (FV) of an annuity, given the regular payment (P), the interest rate (i) per period, and the number of periods (n)?

    <p>$FV = P\left(rac{(1 + i)^n - 1}{i} ight)$</p> Signup and view all the answers

    What is the main advantage of using compound interest over simple interest?

    <p>Compound interest earns interest on both the principal and previously accumulated interest.</p> Signup and view all the answers

    Which of the following is NOT a common application of logarithms?

    <p>Calculating the area of a triangle.</p> Signup and view all the answers

    What is the main difference between a future value annuity (FVA) and a present value annuity (PVA)?

    <p>FVA accumulates funds over time, while PVA repays funds over time.</p> Signup and view all the answers

    Which of the following best describes simple interest?

    <p>Interest calculated on the principal only.</p> Signup and view all the answers

    What is the limit of the function y = (x^2 + 4x - 12)/(x + 6) as x approaches -6?

    <p>-8</p> Signup and view all the answers

    What is the derivative of the function f(x) = x^2 using the definition of a derivative?

    <p>2x</p> Signup and view all the answers

    What is the notation for the derivative of a function f(x) with respect to x?

    <p>All of the above</p> Signup and view all the answers

    What is the rule for differentiating a constant function f(x) = k?

    <p>0</p> Signup and view all the answers

    What is the derivative of the function f(x) = x^n where n is a non-zero constant?

    <p>nx^(n-1)</p> Signup and view all the answers

    What is the general rule for differentiating a function f(x) = x^n where n is a real number?

    <p>nx^(n-1)</p> Signup and view all the answers

    What is the derivative of the function f(x) = k * f(x) where k is a constant?

    <p>k * f'(x)</p> Signup and view all the answers

    What is the derivative of the function f(x) = f(x) + g(x)?

    <p>f'(x) + g'(x)</p> Signup and view all the answers

    What is the derivative of the function f(x) = f(x) - g(x)?

    <p>f'(x) - g'(x)</p> Signup and view all the answers

    When should you use first principles to find the derivative of a function?

    <p>Only when specified by the question</p> Signup and view all the answers

    What is the correct expression for the inverse of the function defined by $y = ax + q$?

    <p>$f^{-1}(x) = rac{1}{a}x - rac{q}{a}$</p> Signup and view all the answers

    What is the restricted domain of the inverse function for $y = ax^2$ when $a > 0$?

    <p>$x ext{ can be any real number}$</p> Signup and view all the answers

    Which inequality describes the domain of the function $f(x) = b^x$ when $b > 1$?

    <p>$x ext{ is any real number}$</p> Signup and view all the answers

    What is the range of the logarithmic function $y = ext{log}_b x$ where $b > 1$?

    <p>$y ext{ can be any real number}$</p> Signup and view all the answers

    What is the inverse of the exponential function $y = b^x$?

    <p>$y = ext{log}_b x$</p> Signup and view all the answers

    Which of the following is true about the graph of the function $f(x) = b^x$?

    <p>It has a horizontal asymptote at $y = 0$.</p> Signup and view all the answers

    What happens to the function $y = ext{log}_b x$ near its vertical asymptote?

    <p>The function is undefined at $x = 0$.</p> Signup and view all the answers

    What is derived from the property $ ext{log}_a(xy)$ according to the laws of logarithms?

    <p>$ ext{log}_a x + ext{log}_a y$</p> Signup and view all the answers

    For which of the following would the base of a logarithm not be defined?

    <p>$b &lt; 0$</p> Signup and view all the answers

    If $y = ax^2$ and $a < 0$, what is the typical restriction on the domain for its inverse to be a function?

    <p>$x ext{ must be non-positive}$</p> Signup and view all the answers

    What is the purpose of the first derivative at a point on a curve?

    <p>To find the gradient of the tangent line at that point.</p> Signup and view all the answers

    Which statement about the second derivative is true?

    <p>It indicates the rate of acceleration of a function.</p> Signup and view all the answers

    What does the notation $m_{ ext{tangent}} imes m_{ ext{normal}} = -1$ signify?

    <p>The tangent line is perpendicular to the normal line.</p> Signup and view all the answers

    What happens to the cubic function when the coefficient $a$ is negative?

    <p>The graph will fall to the right and rise to the left.</p> Signup and view all the answers

    What does the Remainder Theorem allow you to find when dividing a polynomial by a linear polynomial?

    <p>The remainder of the division</p> Signup and view all the answers

    When does a polynomial have a factor according to the Factor Theorem?

    <p>When substituting a certain value gives a remainder of zero</p> Signup and view all the answers

    Which of the following describes how to find the y-intercept of a cubic function?

    <p>Evaluate the function at $x = 0$.</p> Signup and view all the answers

    How do you derive the equation of a tangent line at a point on a curve?

    <p>By substituting the x-coordinate into the derivative to find the slope.</p> Signup and view all the answers

    Which method is NOT commonly used to solve cubic equations?

    <p>Finding the first derivative</p> Signup and view all the answers

    What is the correct notation for the second derivative of $y$ with respect to $x$?

    <p>$y'' = rac{d^2y}{dx^2}$</p> Signup and view all the answers

    How is the polynomial expressed after finding a factor using the Factor Theorem?

    <p>As a product of a linear factor and a quadratic polynomial</p> Signup and view all the answers

    What is the key condition for the addition rule in probability for two mutually exclusive events?

    <p>The intersection probability must be zero</p> Signup and view all the answers

    Which of the following best describes a gradient function?

    <p>It indicates how steep a curve is at a specific point.</p> Signup and view all the answers

    What step is NOT included in finding the equation of a tangent line?

    <p>Evaluate the second derivative to find concavity.</p> Signup and view all the answers

    In the context of a polynomial, what causes the quotient polynomial Q(x) to differ in degree from p(x)?

    <p>The divisor is a linear polynomial</p> Signup and view all the answers

    What is the first step in solving cubic equations using the Factor Theorem?

    <p>Identify a potential root by trial and error</p> Signup and view all the answers

    What is the complementary rule in probability?

    <p>P(not A) = 1 - P(A)</p> Signup and view all the answers

    What results when substituting a certain value into a polynomial confirms that it is a root?

    <p>The polynomial evaluates to zero</p> Signup and view all the answers

    What is the product rule for independent events?

    <p>P(A and B) = P(A) × P(B)</p> Signup and view all the answers

    What simplification occurs in the addition rule for mutually exclusive events?

    <p>The intersection probability can be ignored.</p> Signup and view all the answers

    What is the addition rule for mutually exclusive events?

    <p>P(A or B) = P(A) + P(B)</p> Signup and view all the answers

    What is the symbol for the sample space in probability?

    <p>S</p> Signup and view all the answers

    What is the purpose of Venn diagrams in probability?

    <p>To show the relationship between events</p> Signup and view all the answers

    What is the term for events that cannot happen at the same time?

    <p>Mutually exclusive events</p> Signup and view all the answers

    What is the formula for the probability of the complement of an event A?

    <p>P(A') = 1 - P(A)</p> Signup and view all the answers

    What is the term for the procedure of organizing and visualizing the different possible outcomes of a sequence of events?

    <p>Tree diagrams</p> Signup and view all the answers

    What is the symbol for the number of elements in a set A?

    <p>n(A)</p> Signup and view all the answers

    What is the term for events that are mutually exclusive and make up the whole sample space?

    <p>Complementary events</p> Signup and view all the answers

    What is the relationship between the second derivative ( f''(x) ) of a cubic function and its concavity?

    <p>If ( f''(x) &gt; 0 ), the graph is concave up.</p> Signup and view all the answers

    What is the first step in finding the x-intercepts of a cubic polynomial of the form ( f(x) = ax^3 + bx^2 + cx + d )?

    <p>Solve the equation ( f(x) = 0 ).</p> Signup and view all the answers

    Which of the following describes a local maximum in a cubic function?

    <p>A point where the function changes from increasing to decreasing.</p> Signup and view all the answers

    What is the remainder when the polynomial ( a(x) = x^3 - 3x^2 + 2x - 1 ) is divided by ( b(x) = x - 1 )?

    <p>-1</p> Signup and view all the answers

    In the context of long division of polynomials, what does the quotient ( Q(x) ) represent?

    <p>The result of dividing the dividend by the divisor.</p> Signup and view all the answers

    What is the relationship between the stationary points of a cubic function and its turning points?

    <p>The stationary points are the x-coordinates of the turning points.</p> Signup and view all the answers

    What is the purpose of synthetic division in the context of factoring cubic polynomials?

    <p>To find the remainder after dividing the polynomial by a linear factor.</p> Signup and view all the answers

    Which of the following is a factor of the cubic polynomial ( x^3 - 2x^2 - 5x + 6 )?

    <p>x - 2</p> Signup and view all the answers

    What is the general formula for finding the quotient ( Q(x) ) in synthetic division given the dividend polynomial ( a(x) = a_3x^3 + a_2x^2 + a_1x + a_0 ) and the divisor ( b(x) = cx - d )?

    <p>[ Q(x) = q_2x^2 + q_1x + q_0 ] where ( q_2 = a_3 ), ( q_1 = a_2 + q_2 \cdot rac{d}{c} ), and ( q_0 = a_1 + q_1 \cdot rac{d}{c} )</p> Signup and view all the answers

    Which of the following describes the end behavior of a cubic polynomial ( f(x) = ax^3 + bx^2 + cx + d ) when the leading coefficient ( a ) is positive?

    <p>As ( x ) approaches negative infinity, ( f(x) ) approaches negative infinity, and as ( x ) approaches positive infinity, ( f(x) ) approaches positive infinity.</p> Signup and view all the answers

    What is the formula for calculating the probability of either event A or event B, assuming they are mutually exclusive?

    <p>P(A or B) = P(A) + P(B)</p> Signup and view all the answers

    What does the complementary rule state about the probability of an event and its complement?

    <p>P(not A) = 1 - P(A)</p> Signup and view all the answers

    What is the total number of outcomes when two independent events A and B occur, with n(A) and n(B) outcomes respectively?

    <p>n(A) n(B)</p> Signup and view all the answers

    Which notation represents the factorial of n, the product of all positive integers up to n?

    <p>n!</p> Signup and view all the answers

    For what type of events can the Product Rule for probabilities be applied?

    <p>Independent events</p> Signup and view all the answers

    How is the total number of arrangements of n different objects represented?

    <p>n!</p> Signup and view all the answers

    Which of the following statements correctly defines mutually exclusive events?

    <p>Events that cannot occur at the same time</p> Signup and view all the answers

    What is the method for calculating the total number of outcomes for k events when repetitions are allowed?

    <p>n^k</p> Signup and view all the answers

    In a two-way contingency table, what is primarily used to assess the relationship between two events?

    <p>Count or percentage</p> Signup and view all the answers

    What is the correct product rule for two independent events A and B?

    <p>P(A and B) = P(A) P(B)</p> Signup and view all the answers

    What is the formula used to calculate the present value (P) of an annuity?

    <p>$P = x \left[\frac{1 - (1 + i)^{-n}}{i}\right]$</p> Signup and view all the answers

    Which formula is used to calculate the future value (F) of an annuity?

    <p>$F = x \left[\frac{(1 + i)^n - 1}{i}\right]$</p> Signup and view all the answers

    In the context of an annuity, what does the variable 'x' represent?

    <p>The payment amount per period</p> Signup and view all the answers

    What is the formula used to calculate the effective annual rate (EAR) given a nominal interest rate (i) and the number of compounding periods per year (m)?

    <p>$\text{EAR} = \left(1 + \frac{i}{m}\right)^m - 1$</p> Signup and view all the answers

    What is the formula used to calculate the outstanding loan balance (P_balance) after a certain number of payments?

    <p>$P_\text{balance} = x \left[\frac{1 - (1 + i)^{-n_\text{remaining}}}{i}\right]$</p> Signup and view all the answers

    Which formula is used to calculate the total amount paid (T) on a loan or annuity, given the monthly payment amount (x) and the total number of payments (n)?

    <p>$T = n \times x$</p> Signup and view all the answers

    What is the formula used to calculate the total interest paid (I) on a loan or annuity, given the total amount paid (T) and the principal amount (P)?

    <p>$I = T - P$</p> Signup and view all the answers

    What is the formula used to calculate the time period (n) of an investment, using compound interest, given the accumulated amount (A), the principal amount (P), and the interest rate per period (i)?

    <p>$n = \frac{\log(\frac{A}{P})}{\log(1 + i)}$</p> Signup and view all the answers

    Which of the following formulas correctly represents the change of base for logarithms?

    <p>$\log_a x = \frac{\log_b x}{\log_b a}$</p> Signup and view all the answers

    What is the formula used to calculate the payment amount (x) for a future value annuity, given the future value (F), the interest rate per period (i), and the total number of payments (n)?

    <p>$x = \frac{F \cdot i}{(1 + i)^n - 1}$</p> Signup and view all the answers

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