Gr12 Mathematics: June Exam Easy P(1)
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Questions and Answers

What is the definition of a limit in the context of calculus?

  • The value of a function at a specific point.
  • The maximum value of a function.
  • The derivative of a function at a point.
  • The value a function approaches as the input approaches a certain point. (correct)
  • In the function $y = \frac{x^2 + 4x - 12}{x + 6}$, what happens when $x$ equals -6?

  • The function is undefined. (correct)
  • The function has a maximum value.
  • The function value is 0.
  • The function approaches positive infinity.
  • What is the simplified form of the function $y$ when $x \neq -6$?

  • $y = \frac{x^2 + 4x - 12}{x - 6}$
  • $y = x - 2$ (correct)
  • $y = x + 2$
  • $y = x^2 - 2$
  • According to the Achilles and the Tortoise paradox, what does this illustrate about limits?

    <p>A competitor can approach a goal without ever overtaking it.</p> Signup and view all the answers

    What does the graphical representation of the function $y = \frac{x^2 + 4x - 12}{x + 6}$ show at $x = -6$?

    <p>There is a hole in the graph at that point.</p> Signup and view all the answers

    As $x$ approaches -6 in the function $y = \frac{x^2 + 4x - 12}{x + 6}$, what does $y$ approach?

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

    What is the derivative of $x^4$?

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

    What is the derivative of $5x^2$?

    <p>$10x$</p> Signup and view all the answers

    What is the derivative of $3x^3 + 2x$?

    <p>$9x^2 + 2$</p> Signup and view all the answers

    What is the derivative of $2x^2 - 3x$?

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

    What is the derivative of $7$?

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

    Which of the following is a correct notation for the derivative of a function $f(x)$?

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

    What is the derivative of $f(x) = x^2$ using the definition of the derivative?

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

    What is the gradient of the tangent to the curve $y = x^3$ at the point $x = 2$?

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

    What is the equation of the tangent to the curve $y = x^2 + 1$ at the point $(1, 2)$?

    <p>$y = 2x + 1$</p> Signup and view all the answers

    What is the derivative of $f(x) = \frac{1}{x}$?

    <p>$-\frac{1}{x^2}$</p> Signup and view all the answers

    What does the second derivative indicate about a function?

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

    If the coefficient 'a' of a cubic function is negative, what can be inferred about the graph?

    <p>The graph falls to the left and rises to the right.</p> Signup and view all the answers

    Which of the following denotes the second derivative of a function?

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

    What is the first step in finding the equation of a tangent line to a function?

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

    What relationship exists between the gradients of the tangent and the normal at a point on a curve?

    <p>Their product equals negative one.</p> Signup and view all the answers

    How do you find the y-intercept of a cubic function?

    <p>Set x equal to 0.</p> Signup and view all the answers

    What does a stationary point signify on a graph?

    <p>The function stops changing.</p> Signup and view all the answers

    What is required to determine the normal line to a curve at a given point?

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

    What type of stationary point occurs when a function changes from increasing to decreasing?

    <p>Local maximum.</p> Signup and view all the answers

    To find the x-coordinates of stationary points for a function, what must be solved?

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

    What does the symbol (\Sigma) represent?

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

    What is the general formula for a finite geometric series?

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

    What is the condition for the sum of an infinite geometric series to exist?

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

    What is the definition of a finite series?

    <p>The sum of a specific number of terms of a sequence</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 definition of a geometric sequence?

    <p>A sequence of numbers with a constant ratio</p> Signup and view all the answers

    What is the general form of the summation notation?

    <p>(\sum_{i=m}^{n} T_i = T_m + T_{m+1} + \cdots + T_{n-1} + T_n)</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 fixed number of terms, while an infinite series has an infinite number of terms</p> Signup and view all the answers

    What happens to an infinite geometric series when r = 1?

    <p>The series diverges</p> Signup and view all the answers

    Which of the following is the correct expression for the remainder (R) when a polynomial (p(x)) is divided by (cx - d)?

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

    According to the Factor Theorem, if (cx - d) is a factor of polynomial (p(x)), what is the value of (p(d/c)) ?

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

    What is the degree of the quotient (Q(x)) when a polynomial (p(x)) of degree (n) is divided by a linear polynomial (cx - d)?

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

    Which of the following is NOT a key formula used in solving cubic equations?

    <p>Remainder Theorem</p> Signup and view all the answers

    In the process of solving a cubic equation, after finding a factor, how do you obtain the remaining polynomial factors?

    <p>Polynomial Long Division</p> Signup and view all the answers

    What is the common difference in an arithmetic sequence if the first term is 5 and the fourth term is 17?

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

    Which of the following is the correct formula for the (n)-th term of an arithmetic sequence?

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

    Given an arithmetic sequence with the first term 3 and a common difference of 2, what is the 10th term?

    <p>21</p> Signup and view all the answers

    If the 5th term of an arithmetic sequence is 12 and the common difference is 3, what is the first term?

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

    Which of the following sequences is NOT an arithmetic sequence?

    <p>1, 3, 6, 10, 15</p> Signup and view all the answers

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

    <p>Interchange x and y, Solve for y, Express in function notation</p> Signup and view all the answers

    What is the general form of the inverse of a linear function f(x) = ax + q?

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

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

    <p>Interchange x and y, Solve for y, Express in function notation</p> Signup and view all the answers

    What is the domain and range of the inverse function f^-1(x) = 1/a x - q/a?

    <p>Domain: R, Range: R, assuming a ≠ 0</p> Signup and view all the answers

    What is the definition of the logarithm?

    <p>If x = b^y, then y = log_b(x)</p> Signup and view all the answers

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

    <p>Increasing or decreasing</p> Signup and view all the answers

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

    <p>(1, 0)</p> Signup and view all the answers

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

    <p>Vertical asymptote at x = 0</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

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

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

    What is 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 is the common ratio of the geometric sequence 2, 6, 18, 54, ...?

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

    Which of the following sequences is an arithmetic sequence?

    <p>1, 3, 5, 7, ...</p> Signup and view all the answers

    What is the sum of the first 10 terms of the arithmetic sequence 3, 7, 11, 15, ...?

    <p>210</p> Signup and view all the answers

    Which of the following is a characteristic of an arithmetic sequence?

    <p>The terms form a linear pattern when plotted on a graph.</p> Signup and view all the answers

    What is the formula for the sum of the first (n) terms of an arithmetic sequence?

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

    What does the gradient (slope) of the line representing an arithmetic sequence represent?

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

    What is the sum of the first 5 terms of the geometric sequence 2, 4, 8, 16, ...?

    <p>62</p> Signup and view all the answers

    Which of the following is true about the graphical representation of a geometric sequence?

    <p>It is an exponential curve.</p> Signup and view all the answers

    What does it indicate when a graph is concave up?

    <p>The gradient is increasing.</p> Signup and view all the answers

    Which equation indicates a point of inflection?

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

    When performing synthetic division, what should you use as the divisor's root?

    <p>The opposite sign of the divisor's root</p> Signup and view all the answers

    What does the Remainder Theorem state about polynomial division?

    <p>The remainder can be found by evaluating at the divisor's root.</p> Signup and view all the answers

    Which method is not used for factorizing cubic polynomials?

    <p>Polynomial Interpolation</p> Signup and view all the answers

    Which statement correctly describes a stationary point?

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

    What is meant by the term 'optimisation problem' in calculus?

    <p>Determining maximum or minimum values of functions.</p> Signup and view all the answers

    How can the y-intercept of a cubic polynomial be found?

    <p>By setting x to zero in the polynomial equation.</p> Signup and view all the answers

    What do the signs of the coefficients in a cubic polynomial indicate?

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

    Which of the following correctly defines a cubic polynomial?

    <p>A polynomial of the form $ax^3 + bx^2 + cx + d$.</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 NOT a property of an inverse function?

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

    What is the value of ( S_{100} ) in the context of Gauss's method for finding the sum of the first 100 integers?

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

    What is the general formula for the sum of a finite arithmetic series, where ( a ) is the first term, ( d ) is the common difference, and ( n ) is the number of terms?

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

    Which type of function has an inverse that is also a function?

    <p>One-to-one function</p> Signup and view all the answers

    What is the process for finding the inverse of a function?

    <p>Interchange ( x ) and ( y ) in the equation ( y = f(x) ) and then solve for ( y ).</p> Signup and view all the answers

    In the context of inverse functions, what does the notation ( f^{-1}(x) ) represent?

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

    Which of the following is NOT a valid method for determining if a function has an inverse that is also a function?

    <p>Checking if the function is continuous</p> Signup and view all the answers

    In a one-to-one function, how many elements in the range are associated with each element in the domain?

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

    What is the difference between a relation and a function?

    <p>A relation can have multiple outputs for a single input, while a function can only have one output for a single input.</p> Signup and view all the answers

    What is the value of $log_a 1$?

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

    What does the Product Rule for logarithms state?

    <p>$log_a(xy) = log_a x + log_a y$</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 correct formula for calculating pH levels?

    <p>$pH = -log_{10}[ ext{H}^+]$</p> Signup and view all the answers

    Which of the following correctly describes the characteristics of the exponential function $f(x) = 10^x$?

    <p>Intercept is $(0, 1)$, asymptote is $y = 0$</p> Signup and view all the answers

    Using the Change of Base formula, how can $log_a x$ be expressed?

    <p>$rac{log x}{log a}$</p> Signup and view all the answers

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

    <p>The derivative of the function at x = a</p> Signup and view all the answers

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

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

    What is the rule for differentiating a constant?

    <p>The derivative of a constant is 0</p> Signup and view all the answers

    What is the general rule for differentiating x^n?

    <p>The derivative of x^n is nx^(n-1)</p> Signup and view all the answers

    What is the derivative of a sum of two functions?

    <p>The sum of the derivatives of the two functions</p> Signup and view all the answers

    What is the derivative of a difference of two functions?

    <p>The difference of the derivatives of the two functions</p> Signup and view all the answers

    When should you use the rules for differentiation?

    <p>When the question does not specify how to determine the derivative</p> Signup and view all the answers

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

    <p>To find the equation of the tangent to the function</p> Signup and view all the answers

    What is the notation for the differential operator?

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

    What does the derivative of a function describe?

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

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

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

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

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

    What is the first step in finding the equation of a tangent line to a function?

    <p>Find the derivative using the rules of differentiation</p> Signup and view all the answers

    What is the effect of a negative coefficient 'a' on a cubic graph?

    <p>The graph falls to the right and rises to the left</p> Signup and view all the answers

    How do you find the y-intercept of a cubic function?

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

    What does a stationary point signify on a graph?

    <p>A point where the function changes from increasing to decreasing or vice versa</p> Signup and view all the answers

    What is required to determine the normal line to a curve at a given point?

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

    What type of stationary point occurs when a function changes from increasing to decreasing?

    <p>Local maximum</p> Signup and view all the answers

    To find the x-coordinates of stationary points for a function, what must be solved?

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

    What notation is used to denote the second derivative of a function?

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

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

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

    What is the degree of the quotient Q(x) when a polynomial p(x) of degree n is divided by a linear polynomial cx - d?

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

    According to the Factor Theorem, what is the value of p(d/c) if cx - d is a factor of polynomial p(x)?

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

    What is the general form of an arithmetic sequence?

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

    What is the formula to solve a quadratic equation?

    <p>x = -b ± √(b^2 - 4ac) / 2a</p> Signup and view all the answers

    What is the first step to solve a cubic equation?

    <p>Find a factor by trial and error</p> Signup and view all the answers

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

    <p>They are perpendicular</p> Signup and view all the answers

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

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

    What is the condition for the sum of an infinite geometric series to exist?

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

    What is the definition of an arithmetic sequence?

    <p>A sequence of numbers in which each consecutive term is calculated by adding a constant value</p> Signup and view all the answers

    What is the condition for a point to be a point of inflection on a curve?

    <p>The second derivative of the function changes sign.</p> Signup and view all the answers

    When finding the x-intercepts of a cubic polynomial, what equation needs to be solved?

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

    Which of the following methods can be used to factorize a cubic polynomial?

    <p>Long Division</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

    In the division rule for polynomials, what does the term (R(x)) represent?

    <p>The remainder polynomial</p> Signup and view all the answers

    What is the first step in sketching the graph of a cubic polynomial?

    <p>Finding the y-intercept</p> Signup and view all the answers

    Which of the following is a correct notation for the second derivative of a function f(x)?

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

    What does the sign of the second derivative of a function tell us about the concavity of the graph?

    <p>Positive second derivative indicates concave up, negative second derivative indicates concave down.</p> Signup and view all the answers

    Which of the following is NOT a method for finding the x-intercepts of a cubic polynomial?

    <p>Using the quadratic formula</p> Signup and view all the answers

    What does a stationary point indicate on the graph of a function?

    <p>A point where the graph has a maximum or minimum value.</p> Signup and view all the answers

    What happens to the function when the variable $x$ approaches -6?

    <p>The function approaches a value of -8</p> Signup and view all the answers

    How can the numerator of the function $y = rac{x^2 + 4x - 12}{x + 6}$ be simplified?

    <p>It can be simplified to $(x + 6)(x - 2)$</p> Signup and view all the answers

    In the context of Zeno's paradox, what does Achilles' perceived failure to overtake the tortoise demonstrate?

    <p>The idea of limits in mathematics</p> Signup and view all the answers

    What is the graphical representation of the function $y = rac{x^2 + 4x - 12}{x + 6}$ at $x = -6$?

    <p>It shows a discontinuity with a hole at the point</p> Signup and view all the answers

    What condition must be met for the terms of the function $y$ to be canceled?

    <p>It is valid only when $x eq -6$</p> Signup and view all the answers

    Which of the following statements correctly describes the limit of the function as it approaches -6?

    <p>The limit approaches -8 from both sides</p> Signup and view all the answers

    What is the purpose of interchanging x and y when finding the inverse of a linear function?

    <p>To switch the roles of the dependent and independent variables</p> Signup and view all the answers

    What is the correct inverse of the function y = 2x + 3?

    <p>y = x/2 - 3/2</p> Signup and view all the answers

    What is the domain and range of the inverse function f^-1(x) = 1/a x - q/a, assuming a ≠ 0?

    <p>Domain: ℝ, Range: ℝ</p> Signup and view all the answers

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

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

    What is the purpose of restricting the domain of a quadratic function when finding its inverse?

    <p>To ensure the function is one-to-one</p> Signup and view all the answers

    What is the definition of a logarithm?

    <p>The exponent to which a base must be raised to yield a number</p> Signup and view all the answers

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

    <p>A curve that rises or falls rapidly</p> Signup and view all the answers

    What is the relationship between the graphs of exponential and logarithmic functions?

    <p>They are reflections about the line y = x</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 an exponential function f(x) = b^x when b > 1?

    <p>The function is increasing</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 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 'r' of a geometric sequence is greater than 1, what happens to the sequence?

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

    Which of the following represents the sum of the first 'n' terms of a sequence?

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

    What is the formula for the arithmetic mean between two numbers?

    <p>$rac{a + b}{2}$</p> Signup and view all the answers

    What does the gradient of a line represent when plotting the terms of an arithmetic sequence against their positions?

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

    What is the condition for a sequence to be considered geometric?

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

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

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

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

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

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

    <p>S = a / (1 - r)</p> Signup and view all the answers

    Which of the following statements about the logarithmic function is true?

    <p>The log function approaches negative infinity as x approaches 0.</p> Signup and view all the answers

    What is the product rule for logarithms?

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

    What can be inferred about the graph of an exponential function?

    <p>The domain includes all real numbers.</p> Signup and view all the answers

    What does the change of base formula for logarithms express?

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

    Which application of logarithms could be used to model population growth?

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

    Which of the following is the correct logarithmic identity?

    <p>log_a(a) = 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

    Which of the following is NOT a characteristic of a function?

    <p>A single element in the domain can map to multiple elements in the range.</p> Signup and view all the answers

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

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

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

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

    What is the first step in finding the inverse of a function?

    <p>Interchange the independent and dependent variables (x and y).</p> Signup and view all the answers

    What does the notation (f^{-1}(x)) represent?

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

    Which of the following is the general formula for the sum of a finite arithmetic series?

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

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

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

    What is the condition for the sum of an infinite geometric series to exist?

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

    What is the general form of the summation notation?

    <p>(\sum_{i=1}^{n} a_i)</p> Signup and view all the answers

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

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

    What is the general form of the summation notation?

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

    What is the condition for the sum of an infinite geometric series to exist?

    <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 series?

    <p>A series that sums a specific number of terms of a sequence.</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 definition of a geometric sequence?

    <p>A sequence where each term is found by multiplying the previous term by a constant value.</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 fixed number of terms, while an infinite series has an unlimited number of terms.</p> Signup and view all the answers

    What happens to an infinite geometric series when r = 1?

    <p>The series diverges to infinity.</p> Signup and view all the answers

    Which of the following is NOT a key formula used in solving cubic equations?

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

    What is the derivative of x^n?

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

    What is the derivative of a constant?

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

    What is the derivative of kf(x)?

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

    What is the derivative of f(x) + g(x)?

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

    What is the derivative of f(x) - g(x)?

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

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

    <p>f'(x), y', Dy/Dx, Df(x) and D_xy</p> Signup and view all the answers

    What is the derivative of a function f(x) represents?

    <p>The gradient of the tangent to the curve at a point.</p> Signup and view all the answers

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

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

    What does concavity indicate about a curve?

    <p>Whether the gradient of a curve is increasing or decreasing</p> Signup and view all the answers

    When should you use the rules for differentiation?

    <p>When the question does not specify how to determine the derivative.</p> Signup and view all the answers

    What is the condition for a point of inflection to occur?

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

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

    <p>To analyze the behavior of the function at a point.</p> Signup and view all the answers

    What is the purpose of synthetic division in factorising cubic polynomials?

    <p>To find the quotient and remainder</p> Signup and view all the answers

    How do we find the x-intercepts of a cubic polynomial?

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

    What does the first derivative of a function indicate?

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

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

    <p>To find the equation of a tangent line</p> Signup and view all the answers

    What is the general form of a cubic polynomial?

    <p>ax^3 + bx^2 + cx + d</p> Signup and view all the answers

    What does the second derivative of a function indicate?

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

    What is the purpose of the Remainder Theorem?

    <p>To find the remainder when a polynomial is divided by a linear polynomial</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 point?

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

    How do you find the y-intercept of a cubic function?

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

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

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

    What is the significance of a stationary point on a graph?

    <p>It indicates where the gradient of the curve is zero</p> Signup and view all the answers

    What does a stationary point on a graph signify?

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

    How do you find the x-coordinates of stationary points for a function?

    <p>Find the derivative of the function and set it to 0</p> Signup and view all the answers

    What is the last step in finding the equation of a tangent line to a function?

    <p>Write the equation in the form y = mx + c</p> Signup and view all the answers

    What type of stationary point occurs when a function changes from increasing to decreasing?

    <p>Local maximum</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 effect of the coefficient 'a' on a cubic function?

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

    What is the first step in finding the equation of a tangent line to a function?

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

    In the function (y = \frac{x^2 + 4x - 12}{x + 6}), what happens to the function when (x = -6)?

    <p>The function is undefined because the denominator becomes zero.</p> Signup and view all the answers

    What does the graph of the function (y = \frac{x^2 + 4x - 12}{x + 6}) show at (x = -6)?

    <p>A hole.</p> Signup and view all the answers

    Which of the following is a correct notation for the limit of a function (f(x)) as (x) approaches (a)?

    <p>(lim_{x \to a} f(x))</p> Signup and view all the answers

    In the context of the Achilles and the Tortoise paradox, what does this illustrate about limits?

    <p>That the distance between Achilles and the tortoise will eventually become infinitely small.</p> Signup and view all the answers

    What is the limit of the function (y = \frac{x^2 + 4x - 12}{x + 6}) as (x) approaches -6?

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

    What is the simplified form of the function (y = \frac{x^2 + 4x - 12}{x + 6}) when (x \neq -6)?

    <p>(y = x - 2)</p> Signup and view all the answers

    What is the formula to find the nth term of an arithmetic sequence?

    <p>T_n = a + (n - 1)d</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 to find the arithmetic mean between two numbers?

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

    What is the condition for the sum of an infinite geometric series to exist?

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

    What is the general 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 graphical representation of a geometric sequence?

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

    What is the formula to find the geometric mean between two numbers?

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

    What is the definition of a series?

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

    What is the symbol used to represent the sum of a series?

    <p>Σ</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 fixed number of terms, while an infinite series has an infinite number of terms</p> Signup and view all the answers

    What is the symbol Σ used to denote in mathematics?

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

    What is the general form of the summation notation?

    <p>∑(i=m to n) Ti = Tm + Tm+1 + … + Tn-1 + Tn</p> Signup and view all the answers

    What is the definition of a geometric sequence?

    <p>A sequence of numbers where each term is found by multiplying the previous term by a constant value</p> Signup and view all the answers

    What is the general formula for a finite geometric series?

    <p>Sn = a(1 - rn) / (1 - r)</p> Signup and view all the answers

    What is the condition for the sum of an infinite geometric series to exist?

    <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∞ = a / (1 - r)</p> Signup and view all the answers

    What is the definition of a finite series?

    <p>The sum of the terms of a sequence where the number of terms is finite</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 fixed number of terms, while an infinite series has an infinite number of terms</p> Signup and view all the answers

    What happens to an infinite geometric series when r = 1?

    <p>The series diverges to infinity</p> Signup and view all the answers

    What is the definition of an arithmetic sequence?

    <p>A sequence of numbers where each term is found by adding a constant value to the previous term</p> Signup and view all the answers

    What is the value of loga 1?

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

    What is the product rule of logarithms?

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

    What is the graph of the inverse of an exponential function?

    <p>Reflection of the original function about the line y = x</p> Signup and view all the answers

    What is the application of logarithms in pH levels?

    <p>To use the formula pH = -log10[H+] to calculate pH levels</p> Signup and view all the answers

    What is the range of the logarithmic function?

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

    What is the formula for population growth?

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

    What is the general form of the inverse of a linear function (f(x) = ax + q)?

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

    What is the inverse of the function (y = ax^2) when (a > 0)?

    <p>(y = \sqrt{rac{x}{a}})</p> Signup and view all the answers

    If a quadratic function is not naturally one-to-one, what needs to be done to ensure its inverse is a function?

    <p>Restrict the domain of the original function.</p> Signup and view all the answers

    Which of the following is the correct definition of a logarithm?

    <p>If (x = b^y), then (y = \log_b(x)).</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 domain of the function (y = \log_b x)?

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

    Which of the following functions is increasing?

    <p>(f(x) = 2^x)</p> Signup and view all the answers

    What is the shape of the graph of the exponential function (y = b^x) where (b > 1)?

    <p>Increasing and concave up</p> Signup and view all the answers

    What is the horizontal asymptote of the exponential function (y = b^x)?

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

    Which of the following is a correct conversion from exponential to logarithmic form?

    <p>(4^3 = 64) to (\log_4 64 = 3)</p> Signup and view all the answers

    Which of the following is a correct formula for the sum of an arithmetic series?

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

    Which of the following describes a function where each element of the domain maps to a unique element of the range?

    <p>One-to-one function</p> Signup and view all the answers

    Which of the following is a key property of inverse functions?

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

    Which of the following best describes the relationship between a function and its inverse?

    <p>The inverse function is a reflection of the original function across the line (y = x).</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 NOT a correct statement about relations and functions?

    <p>All relations are functions.</p> Signup and view all the answers

    Which of the following graphical representations represents a function?

    <p>A graph where every vertical line intersects the graph at most once.</p> Signup and view all the answers

    Which of the following best describes the horizontal line test for determining if a function has an inverse that is also a function?

    <p>If a horizontal line intersects the graph at most once, then the function has an inverse that is also a function.</p> Signup and view all the answers

    What is the first step in finding the inverse of a function?

    <p>Interchange (x) and (y) in the equation (y = f(x)).</p> Signup and view all the answers

    Which of the following is NOT a characteristic of a one-to-one function?

    <p>It can have multiple outputs for a single input.</p> Signup and view all the answers

    What is the remainder when the polynomial (p(x) = x^3 - 2x^2 + 5x - 1) is divided by (x - 2)?

    <p>11</p> Signup and view all the answers

    According to the Factor Theorem, which of the following is a factor of (p(x) = x^3 - 6x^2 + 11x - 6)?

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

    Which of the following is the correct general form of the polynomial (p(x)) after division by a linear factor (cx - d)?

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

    What is the degree of the quotient polynomial (Q(x)) when a polynomial (p(x)) of degree (n) is divided by a linear polynomial (cx - d)?

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

    What is the first step in solving a cubic equation using factorization methods?

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

    Which of the following is the correct formula for the (n)-th term of an arithmetic sequence?

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

    If the first term of an arithmetic sequence is 5 and the fourth term is 17, what is the common difference?

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

    Which of the following is a factor of the polynomial (p(x) = x^3 - 3x^2 - 4x + 12)?

    <p>(x - 2)</p> Signup and view all the answers

    What is the remainder when the polynomial (p(x) = 2x^3 - 5x^2 + 3x + 1) is divided by (2x - 1)?

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

    Given an arithmetic sequence with the first term 3 and a common difference of 2, what is the 10th term?

    <p>21</p> Signup and view all the answers

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

    <p>The function is undefined but has a limit.</p> Signup and view all the answers

    How does the Achilles and the Tortoise paradox relate to limits?

    <p>It illustrates that limits can exist even if a function is undefined.</p> Signup and view all the answers

    What graphical feature does the function $y = rac{x^2 + 4x - 12}{x + 6}$ exhibit at $x = -6$?

    <p>A hole at $y = -8$.</p> Signup and view all the answers

    What does the factorization of the function $y = rac{x^2 + 4x - 12}{x + 6}$ reveal?

    <p>When factored, it reveals a removable discontinuity.</p> Signup and view all the answers

    What is the role of limits in calculus?

    <p>To analyze the behavior of functions as they approach specific points.</p> Signup and view all the answers

    In the context of the function $y = rac{x^2 + 4x - 12}{x + 6}$, what limit does $y$ approach as $x$ nears -6?

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

    What is the correct definition of the derivative using first principles?

    <p>$f'(x) = rac{f(x + h) - f(x)}{h}$</p> Signup and view all the answers

    Which of the following notations represents the derivative of a function correctly?

    <p>$rac{df}{dx}$</p> Signup and view all the answers

    What does the derivative of a constant equal?

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

    Which rule allows you to differentiate a sum of two functions?

    <p>Sum Rule</p> Signup and view all the answers

    What is the general formula for differentiating a power function?

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

    In the context of calculus, what does the symbol $D$ represent?

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

    When would you choose to apply differentiation from first principles instead of rules of differentiation?

    <p>When asked specifically to do so</p> Signup and view all the answers

    What is the meaning of $f'(x)$?

    <p>The derivative of $f(x)$ with respect to $x$</p> Signup and view all the answers

    What happens to the gradient of the tangent to a curve at a point?

    <p>It changes with the function.</p> Signup and view all the answers

    Which rule is applied when differentiating a product of two functions?

    <p>Product Rule</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

    If a polynomial p(x) is divided by cx - d and the remainder is zero, what can be concluded?

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

    What is the degree of the quotient Q(x) when a polynomial p(x) of degree n is divided by a linear polynomial cx - d?

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

    What is the first step in solving a cubic equation of the form ax^3 + bx^2 + cx + d = 0?

    <p>Find a factor using the factor theorem</p> Signup and view all the answers

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

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

    What is the common difference in an arithmetic sequence if the first term is 5 and the fourth term is 17?

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

    Which of the following is NOT a step in solving a cubic equation?

    <p>Taking the derivative of the polynomial</p> Signup and view all the answers

    What is the degree of the polynomial Q(x) when a polynomial p(x) of degree n is divided by a linear polynomial cx - d?

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

    What is the formula for the sum of an infinite geometric series with first term a and common ratio r?

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

    What is the condition for the sum of an infinite geometric series to exist?

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

    What is the condition for a point to be a point of inflection on a graph?

    <p>The second derivative of the function changes sign.</p> Signup and view all the answers

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

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

    Which of the following is a correct step in determining the concavity of a cubic function?

    <p>Calculate the second derivative and determine its sign.</p> Signup and view all the answers

    When dividing a polynomial (a(x)) by (b(x)), what does (R(x)) represent?

    <p>The remainder of the division.</p> Signup and view all the answers

    What is the first step in finding the equation of a tangent line to a function?

    <p>Determine the slope of the tangent line.</p> Signup and view all the answers

    Which of the following is NOT a key formula used in solving cubic equations?

    <p>Quadratic Formula</p> Signup and view all the answers

    What is the correct formula for the (n)-th term of an arithmetic sequence?

    <p>a_n = a_1 + (n - 1)d</p> Signup and view all the answers

    Which of the following is a correct step in finding the x-intercepts of a cubic function?

    <p>Solve the equation f(x) = 0 to find the roots of the function.</p> Signup and view all the answers

    Which of the following is NOT a method for factorising cubic polynomials?

    <p>Quadratic Formula</p> Signup and view all the answers

    What does the symbol (\Sigma) represent?

    <p>The sum of a sequence</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 their gradients is -1.</p> Signup and view all the answers

    What is the second derivative of a function, and what does it indicate?

    <p>The second derivative is the derivative of the first derivative and indicates the change in gradient of the original function.</p> Signup and view all the answers

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

    <p>The coefficient 'a' controls the direction of the graph's rise and fall.</p> Signup and view all the answers

    What are the steps involved in finding the equation of a tangent line to a function (f(x)) at a point (x = a)?

    <p>Find (f'(x)), calculate (f'(a)), use the point-slope form.</p> Signup and view all the answers

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

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

    What does the summation symbol (\Sigma) represent?

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

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

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

    What is the first step in finding the equation of a tangent line to a function?

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

    What does a stationary point signify on a graph of a function?

    <p>A point where the function changes from increasing to decreasing or vice versa.</p> Signup and view all the answers

    What is the condition for the sum of an infinite geometric series to exist?

    <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 = rac{a}{1 - r})</p> Signup and view all the answers

    What type of stationary point occurs when a function changes from increasing to decreasing?

    <p>Local Maximum</p> Signup and view all the answers

    What are the steps involved in finding the normal line to a curve at a given point?

    <p>Find the derivative, find the negative reciprocal of the slope, use the point-slope form.</p> Signup and view all the answers

    What is the definition of a finite series?

    <p>The sum of the first n terms of a sequence</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

    Which of the following notations correctly represents the second derivative of a function (f(x))?

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

    What is the definition of a geometric sequence?

    <p>A sequence with a common ratio</p> Signup and view all the answers

    What is the general form of the summation notation?

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

    What happens to an infinite geometric series when r = 1?

    <p>The series diverges to infinity</p> Signup and view all the answers

    What is the sum of the first 100 integers?

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

    What is the general formula for a finite arithmetic series?

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

    What is a relation?

    <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 a function?

    <p>A relation where each element in set A maps to exactly one element in set B</p> Signup and view all the answers

    What is an inverse function?

    <p>A function that reverses the operation of another function</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 horizontal line test?

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

    What is the formula to find the inverse of a function?

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

    What is the notation for the inverse function?

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

    What is the graph of the inverse function symmetrical to?

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

    What is the common difference in an arithmetic sequence if the first term is 4 and the second term is 10?

    <p>5</p> Signup and view all the answers

    Which formula correctly defines the arithmetic mean of two numbers?

    <p>Mean = $\frac{a + b}{2}$</p> Signup and view all the answers

    How can you identify if a sequence is geometric?

    <p>By calculating the ratios between consecutive terms.</p> Signup and view all the answers

    What occurs when the common ratio in a geometric sequence is between 0 and 1?

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

    What type of graph is formed when plotting the terms of an arithmetic sequence against their positions?

    <p>Straight line</p> Signup and view all the answers

    What is 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 defines a series in the context of sequences?

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

    What will the sum of the first n terms of a geometric sequence be in sigma notation?

    <p>$S_n = \sum_{i=1}^n ar^{i-1}$</p> Signup and view all the answers

    What happens to an infinite geometric series when the common ratio is greater than 1?

    <p>It diverges to infinity.</p> Signup and view all the answers

    What does the geometric mean between two numbers a and b equal?

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

    What is the value of (\log_a 1)?

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

    Which of the following is NOT a law of logarithms?

    <p>Inverse Rule</p> Signup and view all the answers

    What is the domain of the function (f(x) = \log x)?

    <p>All positive real numbers</p> Signup and view all the answers

    Which of the following is the formula for calculating the pH level of a solution?

    <p>(\text{pH} = -\log_{10}[\text{H}^+])</p> Signup and view all the answers

    If a population triples in size, what is the value of (n) in the formula (3P = P(1 + i)^n)?

    <p>Cannot be determined without knowing the value of (i)</p> Signup and view all the answers

    Which of the following is the correct formula for the change of base rule for logarithms?

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

    What is the first step in finding the inverse of the function $y = ax + q$?

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

    Which of the following statements about the properties of the inverse of a function is true?

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

    For the quadratic function $y = ax^2$, what is necessary for its inverse to also be a function?

    <p>The function must be restricted to its vertex.</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

    What occurs to the graph of an exponential function when the base $b$ is between 0 and 1?

    <p>The function decreases rapidly.</p> Signup and view all the answers

    Which of the following statements about logarithms is correct?

    <p>The logarithm is undefined when the input is less than or equal to zero.</p> Signup and view all the answers

    In the function $y = b^x$, what does the horizontal asymptote represent?

    <p>The value it approaches at infinity.</p> Signup and view all the answers

    Which of the following represents the general form for the inverse of the function $y = ax^2$?

    <p>$y = ext{sqrt}{rac{x}{a}}$</p> Signup and view all the answers

    For which condition is the function $y = b^x$ not defined?

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

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