Gr12 Mathematics: June Exam Medium P(1)

What is the derivative of the function f(x) = x^2?

2x^2

x^2

x

2x (correct)

What is the derivative of the function f(x) = 3x^2 + 2x - 5?

3x^2 - 2x + 5

6x + 2 (correct)

3x^2 + 2x - 5

6x - 2

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

f(x)'

f^2(x)

f'(x) (correct)

f''(x)

What is the purpose of the rules for differentiation?

To make differentiation simpler and faster Signup and view all the answers

What is the derivative of a constant function f(x) = k?

0 Signup and view all the answers

What is the derivative of the function f(x) = x^3 - 2x^2 + x - 1?

3x^2 - 4x + 1 Signup and view all the answers

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

The gradient of the curve at that point Signup and view all the answers

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

The differential operator Signup and view all the answers

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

To find the equation of a tangent to a curve Signup and view all the answers

What is the general rule for differentiation?

d/dx [x^n] = nx^(n-1) Signup and view all the answers

What is the main purpose of limits in calculus?

To explore how functions behave as they approach specific points Signup and view all the answers

What happens to the function defined as $y = \frac{x^2 + 4x - 12}{x + 6}$ when $x$ approaches -6?

The function approaches a value of -8 Signup and view all the answers

Which of the following statements about the function $y = \frac{(x + 6)(x - 2)}{x + 6}$ is true?

It has a discontinuity at $x = -6$ Signup and view all the answers

In the context of Zeno's Achilles and Tortoise paradox, what does the paradox illustrate about limits?

Infinite quantities can be resolved into finite segments Signup and view all the answers

What can be concluded about the graph of the function $y = \frac{x^2 + 4x - 12}{x + 6}$?

It has a hole at $x = -6$ Signup and view all the answers

What expression does the function simplify to when $x \neq -6$?

$x - 2$ Signup and view all the answers

What is the value of R in the division of a polynomial p(x) by a divisor cx - d?

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

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

One degree less than p(x) Signup and view all the answers

What does the Factor Theorem state?

Both a and b Signup and view all the answers

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

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

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

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

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

To factorize the polynomial 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?

Use the Factor Theorem to find a factor Signup and view all the answers

What is the common difference in an arithmetic sequence?

The difference between two consecutive terms Signup and view all the answers

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

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

What is the relationship between the degree of the quotient polynomial Q(x) and the degree of the polynomial p(x) when dividing by a linear polynomial cx - d?

The degree of Q(x) is one less than p(x) Signup and view all the answers

What is the first step in determining the equation of a tangent to a curve?

Find the derivative using the rules of differentiation. Signup and view all the answers

How is the relationship between the gradients of the tangent and the normal defined?

The product of the gradients equals negative one. Signup and view all the answers

What does the second derivative indicate about a function?

It indicates the change in the direction of the function's slope. Signup and view all the answers

Which of the following notations is NOT commonly used for the second derivative?

$f^{(2)}(x)$ Signup and view all the answers

To find the y-intercept of the cubic function $f(x) = ax^3 + bx^2 + cx + d$, what value should be substituted for $x$?

0 Signup and view all the answers

What is the condition for a stationary point of a cubic function?

The first derivative equals zero. Signup and view all the answers

What happens to the graph of a cubic function when the leading coefficient $a$ is positive?

The graph rises to the right and falls to the left. Signup and view all the answers

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

To identify stationary points and calculate slopes. Signup and view all the answers

In the context of cubic functions, how can local maxima and minima be characterized?

Local maxima are points where the function changes from increasing to decreasing. Signup and view all the answers

What method can be used to find x-intercepts of a cubic function?

Solve $f(x) = 0$ through factoring or using the Rational Root Theorem. Signup and view all the answers

What does it mean for a graph to be concave up?

The second derivative is greater than zero Signup and view all the answers

Which of the following statements is true about points of inflection?

They indicate a change in the concavity of the graph Signup and view all the answers

What is the first step in sketching a cubic graph?

Determining the shape of the graph based on the sign of $a$ Signup and view all the answers

What is the outcome of applying the Remainder Theorem?

It gives the remainder after polynomial division Signup and view all the answers

Which method is NOT used for factorizing cubic polynomials?

Graphing Signup and view all the answers

To find the turning points of a cubic function, what must be solved?

$f'(x) = 0$ Signup and view all the answers

What does the average rate of change measure over an interval?

The overall change in the function divided by the interval length Signup and view all the answers

In synthetic division, which of the following is used?

The coefficients of the dividend polynomial Signup and view all the answers

Given a polynomial of the form $f(x) = ax^3 + bx^2 + cx + d$, which point provides the y-intercept?

Evaluate $f(0)$ Signup and view all the answers

What is true about the end behavior of a cubic polynomial as $x$ approaches positive or negative infinity?

It can always be determined by the sign of the leading coefficient Signup and view all the answers

Which of the following is equivalent to the expression (\log_a(x/y)) using the laws of logarithms?

(\log_a x - \log_a y) Signup and view all the answers

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

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

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

-1 < r < 1 Signup and view all the answers

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

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

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

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

What is the value of the common ratio (r) in a geometric sequence if the second term is 6 and the first term is 2?

3 Signup and view all the answers

In sigma notation, what does the index of summation represent?

The position of a term in the sequence Signup and view all the answers

What is the sum of the first 5 terms of the geometric sequence with a = 2 and r = 3?

242 Signup and view all the answers

What is the sum of the infinite geometric series with a = 1 and r = 1/2?

2 Signup and view all the answers

Which of the following is an example of a finite geometric series?

1 + 3 + 9 + 27 + 81 Signup and view all the answers

What is the sum of the first 10 terms of the arithmetic series 1 + 4 + 7 + 10 + ...?

145 Signup and view all the answers

What is the sum of the first 100 positive integers using Gauss's method?

5050 Signup and view all the answers

What is the formula for the sum of the first $n$ terms in an arithmetic series?

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

Which statement correctly defines a function?

A function is a relation where each input maps to exactly one output. Signup and view all the answers

Which graphical characteristic identifies a one-to-one function?

Every horizontal line intersects the graph at most once. Signup and view all the answers

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

The function must be one-to-one. Signup and view all the answers

What do the graphs of a function and its inverse exhibit?

They are symmetrical about the line $y = x$. Signup and view all the answers

How is the inverse function found algebraically?

By interchanging $x$ and $y$ and then solving for $y$. Signup and view all the answers

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

$f(x) = x^2$ for $x eq 0$. Signup and view all the answers

If a function fails the horizontal line test, what does this indicate?

The function does not have an inverse that is a function. Signup and view all the answers

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

The inverse function of $f$, not the reciprocal. Signup and view all the answers

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

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

Which of the following describes the domain of the inverse function for $y = ax^2$?

Restricted to either $x ightarrow ext{positive}$ or $x ightarrow ext{negative}$ based on the sign of a Signup and view all the answers

What is the range of the inverse function $f^{-1}(x) = rac{1}{a}x - rac{q}{a}$?

All real numbers Signup and view all the answers

Which characteristic is true about the inverse of a linear function?

It remains a linear function. Signup and view all the answers

How is the inverse of $y = b^x$ expressed?

$y = ext{log}_b x$ Signup and view all the answers

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

It has a vertical asymptote at $x = 0$. Signup and view all the answers

What transformation do the graphs of a function and its inverse exhibit?

They are reflections about the line $y = x$. Signup and view all the answers

Which statement about an exponential function where $b > 1$ is correct?

The function rises rapidly. Signup and view all the answers

What describes the behavior of the exponential function $f(x) = b^x$ when $b < 1$?

The function decreases and approaches zero. Signup and view all the answers

What property does the logarithm have when converting from exponential form, such as $5^2 = 25$?

It retains the same base as the original expression. Signup and view all the answers

If the first term of a geometric sequence is 3 and the common ratio is 2, what is the 5th term?

192 Signup and view all the answers

Which of the following sequences is an arithmetic sequence?

1, 3, 5, 7, 9 Signup and view all the answers

What is the common difference in the arithmetic sequence 5, 11, 17, 23, ...?

6 Signup and view all the answers

If the geometric mean between two numbers is 6, and one of the numbers is 4, what is the other number?

9 Signup and view all the answers

What is the sum of the first 10 terms of the arithmetic sequence 2, 5, 8, 11, ...?

155 Signup and view all the answers

What is the 7th term of the geometric sequence 1, 3, 9, 27, ...?

729 Signup and view all the answers

The first term of an arithmetic sequence is 10 and the common difference is -3. What is the 15th term?

-35 Signup and view all the answers

Which of the following describes the graphical representation of a geometric sequence?

An exponential curve Signup and view all the answers

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

3 Signup and view all the answers

The sum of the first 5 terms of an arithmetic series is 45. If the first term is 3, what is the common difference?

3 Signup and view all the answers

What is the primary concept that the Achilles and the Tortoise Paradox illustrates?

Limits Signup and view all the answers

What happens to the function y = (x^2 + 4x - 12)/(x + 6) when x approaches -6?

The function approaches -8 Signup and view all the answers

What is the graphical representation of the function y = (x^2 + 4x - 12)/(x + 6)?

A straight line with a hole Signup and view all the answers

What is the purpose of limits in calculus?

To build on algebra and geometry Signup and view all the answers

Why can't we cancel the (x + 6) terms in the function y = (x^2 + 4x - 12)/(x + 6) when x = -6?

Because the function is not defined at x = -6 Signup and view all the answers

What is the significance of the fact that y approaches -8 as x approaches -6 in the function y = (x^2 + 4x - 12)/(x + 6)?

It indicates a limit Signup and view all the answers

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

To analyze the concavity of the function. Signup and view all the answers

What is the condition for a point of inflection?

f'(x) = 0 and f''(x) changes sign. Signup and view all the answers

What is the outcome of applying the Remainder Theorem?

The remainder is obtained by substituting d/c into the polynomial. Signup and view all the answers

What is the relationship between the degree of the quotient polynomial and the degree of the polynomial p(x) when dividing by a linear polynomial cx - d?

The degree of the quotient polynomial is one less than the degree of p(x). Signup and view all the answers

What is the method used to find the x-intercepts of a cubic function?

Solving the equation f(x) = 0. Signup and view all the answers

What does it mean for a graph to be concave up?

The graph opens upwards. 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?

Factor and Remainder Theorem. Signup and view all the answers

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

To determine the maximum and minimum values of the function. Signup and view all the answers

What is the general method for sketching a cubic graph?

Find the y-intercept, x-intercepts, stationary points, and points of inflection. Signup and view all the answers

What does the average rate of change measure over an interval?

The rate of change over a specific interval. Signup and view all the answers

What is the derivative of the function (f(x) = x^3 - 2x^2 + x - 1)?

(f'(x) = 3x^2 - 4x + 1) Signup and view all the answers

What is the equation of the tangent to the curve (y = x^2 + 2x - 3) at the point ((1, 0))?

(y = 4x - 4) Signup and view all the answers

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

(D_xf(x)) Signup and view all the answers

What is the derivative of the function (f(x) = 5x^2 + 3x - 2)?

(f'(x) = 10x + 3) Signup and view all the answers

What is the derivative of the function (f(x) = rac{1}{x})?

(f'(x) = -rac{1}{x^2}) Signup and view all the answers

What is the derivative of the function (f(x) = 2\sqrt{x})?

(f'(x) = rac{1}{\sqrt{x}}) Signup and view all the answers

What is the derivative of the function (f(x) = rac{x^2 + 1}{x})?

(f'(x) = 1 - rac{1}{x^2}) Signup and view all the answers

What is the derivative of the function (f(x) = (x + 1)^2)?

(f'(x) = 2x + 2) Signup and view all the answers

What is the derivative of the function (f(x) = \sin(x))?

(f'(x) = \cos(x)) Signup and view all the answers

What is the derivative of the function (f(x) = e^x)?

(f'(x) = e^x) 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?

The gradients are multiplicative inverses. Signup and view all the answers

Which of the following correctly describes the effect of the coefficient 'a' on the shape of the cubic function ( y = ax^3 + bx^2 + cx + d ) when ( a > 0 )?

The graph rises to the right and falls to the left. Signup and view all the answers

What is the first step in finding the stationary points of a cubic function ( f(x) = ax^3 + bx^2 + cx + d )?

Find the first derivative ( f'(x) ). Signup and view all the answers

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

( f'(x) ) Signup and view all the answers

To find the equation of a tangent line to the graph of ( f(x) ) at ( x = a ), what should be calculated after finding the derivative ( f'(x) )?

The value of ( f'(a) ) Signup and view all the answers

How can you classify a stationary point of a cubic function as a local maximum or a local minimum?

By looking at the behavior of the function on either side of the stationary point. Signup and view all the answers

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

The rate of change of the gradient. Signup and view all the answers

Which of the following is NOT a common method for finding the x-intercepts of a cubic function ( f(x) = ax^3 + bx^2 + cx + d )?

Finding the stationary points. Signup and view all the answers

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

To determine the concavity of the function. Signup and view all the answers

What are the key steps involved in sketching the graph of a cubic function ( f(x) = ax^3 + bx^2 + cx + d )?

Finding the y-intercept, the x-intercepts, and the stationary points. Signup and view all the answers

What is the value of log_a(a) in logarithms?

1 Signup and view all the answers

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

all real numbers Signup and view all the answers

What is the formula to change the base of a logarithm?

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

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

a reflection about the line y = x Signup and view all the answers

What is the application of logarithms in measuring pH levels?

pH = -log[H+] Signup and view all the answers

What is the formula for population growth?

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

What is the common difference in an arithmetic sequence?

The difference between consecutive terms Signup and view all the answers

In a geometric sequence, how is the common ratio found?

By dividing each term by the previous term Signup and view all the answers

What does the slope of the graph representing an arithmetic sequence indicate?

The common difference of the sequence Signup and view all the answers

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

7 Signup and view all the answers

Which statement is true about geometric sequences?

The ratio between consecutive terms is constant Signup and view all the answers

What happens to a geometric sequence with a common ratio less than 1?

The terms decay exponentially Signup and view all the answers

How is the sum of a finite series of an arithmetic sequence denoted?

S_n Signup and view all the answers

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

The common ratio must be less than 1 in absolute value Signup and view all the answers

Which formula is used to find the n-th term of a geometric sequence?

T_n = ar^{n-1} Signup and view all the answers

What is true about the graphical representation of a geometric sequence?

It yields an exponential graph Signup and view all the answers

In the general form of the sigma notation, what does the term (T_i) represent?

The term of the sequence at index (i) Signup and view all the answers

Which of the following statements accurately describes the difference between a finite and an infinite series?

A finite series sums only a specific number of terms, while an infinite series sums all terms of the sequence. Signup and view all the answers

Given a geometric sequence with a first term (a) and a common ratio (r), what is the formula for the (n)-th term of the sequence?

(T_n = a \cdot r^{n-1}) Signup and view all the answers

What condition must be met for an infinite geometric series to converge?

The common ratio (r) must be between -1 and 1 (exclusive). Signup and view all the answers

If the common difference of an arithmetic sequence is (d) and the first term is (a), what is the formula for the (n)-th term of the sequence?

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

Which of the following is NOT a characteristic of a finite geometric series?

The series always converges to a finite value. Signup and view all the answers

What is the sum of the first 5 terms of the geometric series with the first term (a = 2) and common ratio (r = 3)?

242 Signup and view all the answers

Given the infinite geometric series with the first term (a = 4) and common ratio (r = 1/2), what is the sum of the series?

8 Signup and view all the answers

Which of the following series is NOT an infinite geometric series?

1 + 3 + 5 + 7 + ... Signup and view all the answers

Which of the following scenarios is an example of an infinite geometric series?

The total distance traveled by a ball bouncing repeatedly, where the height of each bounce is half the height of the previous bounce. Signup and view all the answers

What is the inverse of the linear function defined as $f(x) = 2x + 3$?

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

Which restriction is typically placed on the domain of the quadratic function $y = ax^2$ to ensure its inverse is a function?

$x \geq 0$ if $a > 0$ Signup and view all the answers

Which statement is true regarding the inverse of the function $y = b^x$?

The inverse is expressed as $y = \log_b x$. Signup and view all the answers

When converting from exponential form to logarithmic form, which is the correct transformation of the equation $3^4 = 81$?

$\log_3(81) = 4$ Signup and view all the answers

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

$x > 0$ Signup and view all the answers

What can be concluded about the graph of the function $f(x) = b^x$ when $b < 1$?

The graph is decreasing and approaches zero. Signup and view all the answers

Which of the following reflects the correct relationship between the range of $y = ax^2$ and the domain of its inverse?

The range becomes the domain of the inverse. Signup and view all the answers

Which graph characteristic describes the function $y = b^x$ where $b > 1$?

It is increasing and has a horizontal asymptote at $y = 0$. Signup and view all the answers

When finding the inverse of the quadratic function $y = ax^2$, which condition must be satisfied for it to be a valid function?

The function must be one-to-one. Signup and view all the answers

When dividing a polynomial (p(x)) by (cx - d), what is the degree of the quotient polynomial (Q(x)) relative to the degree of the original polynomial (p(x))?

The degree of (Q(x)) is one degree less than the degree of (p(x)). Signup and view all the answers

Which of the following statements about the Factor Theorem is NOT true?

The Factor Theorem states that if (cx - d) is a factor of (p(x)), then (p(x)) can be expressed as ( (cx - d) \cdot Q(x) ), where (Q(x)) is the quotient polynomial. Signup and view all the answers

When using the Factor Theorem to find a factor of a cubic polynomial, what must be true about the potential roots that are substituted into the polynomial?

The potential roots must be factors of the constant term of the polynomial. Signup and view all the answers

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

9 Signup and view all the answers

If (x + 3) is a factor of the polynomial (p(x) = x^3 + 5x^2 + 7x + 3), what is the other factor?

x^2 + 2x + 1 Signup and view all the answers

In an arithmetic sequence, the first term is 5 and the common difference is -2. What is the value of the 10th term?

-9 Signup and view all the answers

Which of the following is NOT a key step in solving a cubic equation using factorization methods?

Apply the Fundamental Theorem of Algebra to find all roots. Signup and view all the answers

The sum of the first five terms of an arithmetic sequence is 35. If the common difference is 2, what is the first term?

3 Signup and view all the answers

Given that (x - 2) is a factor of the polynomial (p(x) = x^3 - 6x^2 + 11x - 6), which of the following is another factor?

x - 1 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, (l) is the last term, and (n) is the number of terms?

[S_n = \frac{n}{2}(2a + (n - 1)d)] Signup and view all the answers

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

Each element in the domain maps to exactly one element in the range. Signup and view all the answers

What is the relationship between the graph of a function (f(x)) and its inverse (f^{-1}(x)) when plotted on the same coordinate plane?

They are symmetrical about the line (y = x). Signup and view all the answers

If a function (f(x)) is not one-to-one, what can be concluded about its inverse?

Its inverse does not exist as a function. Signup and view all the answers

What is the primary difference between a relation and a function?

A function maps each element of one set to exactly one element in another set, while a relation can map to multiple elements. Signup and view all the answers

What is the value of the sum of the first 100 positive integers?

5050 Signup and view all the answers

If the first term of an arithmetic sequence is 3 and the common difference is 5, what is the sum of the first 10 terms?

280 Signup and view all the answers

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

A function must be one-to-one for its inverse to also be a function. Signup and view all the answers

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

[f^{-1}(x) = \frac{x - 1}{2}] Signup and view all the answers

What is the horizontal line test used for?

To determine if a function has an inverse that is also a function. Signup and view all the answers

What does the cancellation of the term $x + 6$ in the function $y = \frac{(x + 6)(x - 2)}{x + 6}$ imply about the behavior of the function at $x = -6$?

The function does not have a defined value at $x = -6$. Signup and view all the answers

In the context of Zeno's paradox with Achilles and the tortoise, what concept does this illustrate related to limits?

That some limits can never be reached. Signup and view all the answers

What value does $y$ approach as $x$ gets closer to -6 in the function $y = \frac{x^2 + 4x - 12}{x + 6}$?

-8 Signup and view all the answers

Which of the following statements best describes the graph of the function $y = \frac{x^2 + 4x - 12}{x + 6}$?

It is a straight line with a discontinuity at $x = -6$. Signup and view all the answers

What are limits primarily used for in calculus?

To analyze behaviors of functions as they approach specific points. Signup and view all the answers

What is the main conclusion about the function $y = x - 2$ derived from the original function when $x eq -6$?

It shows that the function behaves like a linear function everywhere else. 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?

The gradients are negative reciprocals of each other. Signup and view all the answers

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

Find the derivative of the function. Signup and view all the answers

If the second derivative of a function is negative at a given point, what does it indicate about the original function at that point?

The function is decreasing at that point. 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 ) when ( a > 0 )?

The graph rises to the right and falls to the left. Signup and view all the answers

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

Substitute ( x = 0 ) into the function. Signup and view all the answers

What is the condition for a stationary point of a cubic function ( f(x) = ax^3 + bx^2 + cx + d )?

( f'(x) = 0 ) Signup and view all the answers

Which of the following is NOT a common notation for the second derivative of a function?

( rac{dy}{dx} ) Signup and view all the answers

How can you classify a stationary point of a cubic function as a local maximum or a local minimum?

By examining the sign change of the first derivative around the point. Signup and view all the answers

What is the purpose of finding the derivative of a function in the context of sketching a cubic graph?

To find the stationary points of the graph. Signup and view all the answers

What is the best method to find the x-intercepts of a cubic function?

All of the above. Signup and view all the answers

What expression represents the derivative of the function $f(x) = x^5$?

$5x^4$ Signup and view all the answers

For the function defined as $y = k$, where $k$ is a constant, what is the derivative?

$0$ Signup and view all the answers

Which of the following notations indicates differentiation with respect to $x$?

$rac{d}{dx}[f(x)]$ Signup and view all the answers

When applying the limit definition, what is the value of $h$ approaching?

$0$ Signup and view all the answers

What is the derivative of the sum of two functions $f(x) + g(x)$?

$rac{d}{dx}[f(x)] + rac{d}{dx}[g(x)]$ Signup and view all the answers

What does the notation $D_xy$ represent?

Derivative of $y$ with respect to $x$ Signup and view all the answers

What condition must be true for using the rules of differentiation rather than first principles?

When the problem does not specify how to differentiate Signup and view all the answers

Which of the following represents the derivative of the difference $f(x) - g(x)$?

$rac{d}{dx}[f(x)] - rac{d}{dx}[g(x)]$ Signup and view all the answers

What is indicated by the limit process in determining a derivative?

An instantaneous rate of change Signup and view all the answers

What is the primary purpose of using first principles for differentiation?

To derive the rules for differentiation Signup and view all the answers

What does the summation symbol Σ represent?

The sum of a sequence Signup and view all the answers

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

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

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

-1 < r < 1 Signup and view all the answers

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

S_∞ = a / (1 - r) Signup and view all the answers

What is a finite geometric series?

A series with a fixed number of terms Signup and view all the answers

What is the general formula for a finite geometric series?

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

What is an arithmetic sequence?

A sequence where each term is added by a constant Signup and view all the answers

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

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

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

The number of terms Signup and view all the answers

What is the purpose of sigma notation?

To represent the sum of a sequence Signup and view all the answers

What is the value of loga(a)?

1 Signup and view all the answers

What is the product rule of logarithms?

loga(xy) = loga(x) + loga(y) Signup and view all the answers

What is the range of the logarithmic function f(x) = log(x)?

y ∈ ℝ Signup and view all the answers

What is the formula to calculate the population growth?

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

What is the application of logarithms in pH levels?

pH = -log[H+] Signup and view all the answers

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

A reflection of the original function about the line y = x Signup and view all the answers

What does it mean if the remainder R of a polynomial p(x) divided by cx - d equals zero?

The divisor cx - d is a factor of p(x). Signup and view all the answers

How is the quotient polynomial Q(x) related to the original polynomial p(x) when dividing by a linear polynomial cx - d?

Q(x) has a degree one less than p(x). Signup and view all the answers

What is the first step in solving a cubic equation using the Factor Theorem?

Substitute possible rational roots into the polynomial. Signup and view all the answers

In the context of arithmetic sequences, if the first term a is 5 and the common difference d is 3, what is the 10th term T_10?

30 Signup and view all the answers

What is the relationship between the value obtained by substituting x = d/c into p(x) and the factor cx - d?

It will equal zero if cx - d is a factor. Signup and view all the answers

Which of the following correctly states the general form of a cubic polynomial divided by a linear divisor?

p(x) = (cx - d) * Q(x) + R. Signup and view all the answers

When dividing a polynomial by a divisor of the form cx - d, what type of value does the remainder R represent?

A constant value. Signup and view all the answers

Which of the following processes is NOT a standard step in solving cubic equations?

Graphing the cubic equation to find intersections. Signup and view all the answers

If the common difference d of an arithmetic sequence is negative, what does that indicate about the sequence?

The sequence is decreasing. Signup and view all the answers

When using the Quadratic Formula to solve a quadratic polynomial, what indicates that the polynomial has no real solutions?

The discriminant is negative. Signup and view all the answers

Given the linear function ( f(x) = 2x - 3 ), what is the equation of its inverse, ( f^{-1}(x) )?

( f^{-1}(x) = rac{1}{2}x + rac{3}{2} ) Signup and view all the answers

What is the appropriate restriction on the domain of the quadratic function ( y = -2x^2 ) to ensure its inverse is a function?

( x \leq 0 ) Signup and view all the answers

If the graph of a function and its inverse are reflected about the line ( y = x ), what is true about their x-intercepts and y-intercepts?

The y-intercepts of the function and the x-intercepts of the inverse are the same. Signup and view all the answers

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

The range of the function is the same as the domain of the inverse function. Signup and view all the answers

Which of the following statements about the exponential function ( y = b^x ) is TRUE?

The function is not defined if ( b \leq 0 ). Signup and view all the answers

What is the inverse of the exponential function ( y = 3^x )?

( y = \log_3 x ) Signup and view all the answers

Which of the following equations is equivalent to the logarithmic expression ( \log_7 49 = 2 ) in exponential form?

( 7^2 = 49 ) Signup and view all the answers

What is the horizontal asymptote of the logarithmic function ( y = \log_2 x )?

There is no horizontal asymptote. Signup and view all the answers

Given ( \log_3 9 = 2 ), what is the value of ( \log_3 81 )?

( 4 ) Signup and view all the answers

What is the value of ( \log_2 16 )?

( 4 ) Signup and view all the answers

What is the formula used to find the n-th term of an arithmetic sequence?

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

How do you determine if a sequence is arithmetic?

Calculate the differences between consecutive terms. Signup and view all the answers

What indicates that a sequence is geometric?

The ratios between consecutive terms are equal. Signup and view all the answers

What does the geometric mean of two numbers a and b represent?

$ ext{Geometric Mean} = ext{sqrt}(ab)$ Signup and view all the answers

What happens when $r > 1$ in a geometric sequence?

The sequence grows exponentially. Signup and view all the answers

How is the common difference in an arithmetic sequence related to its graph?

It is represented by the slope of a straight line. Signup and view all the answers

What is true about an infinite series?

It sums an infinite number of terms. Signup and view all the answers

Which statement about a series is accurate?

A series results from adding the terms of a sequence. Signup and view all the answers

What characterizes a finite series?

It sums a specific number of terms. Signup and view all the answers

What is the result of plotting a geometric sequence?

An exponential curve. Signup and view all the answers

A cubic polynomial has a y-intercept at (0, -2). What is the value of the constant term, d, in the equation (f(x) = ax^3 + bx^2 + cx + d)?

-2 Signup and view all the answers

What is the relationship between the leading coefficient a of a cubic polynomial and its end behavior?

If a is positive, the graph goes down on the left and up on the right. If a is negative, the graph goes up on the left and down on the right. Signup and view all the answers

A cubic polynomial has a turning point at (x = 2). What can you conclude about the derivative of the function at (x = 2)?

The derivative is zero. Signup and view all the answers

What is the general formula for synthetic division when dividing a cubic polynomial (a_3x^3 + a_2x^2 + a_1x + a_0) by a linear polynomial (cx - d)?

[ q_2 = a_3 ] [ q_1 = a_2 + q_2 \cdot \frac{d}{c} ] [ q_0 = a_1 + q_1 \cdot \frac{d}{c} ] [ R = a_0 + q_0 \cdot \frac{d}{c} ] Signup and view all the answers

If a cubic polynomial has a point of inflection at (x = 3), what can you conclude about its second derivative at (x = 3)?

The second derivative is zero. Signup and view all the answers

Which of the following statements accurately describes the concavity of a cubic function when its second derivative is positive?

The function is concave up. Signup and view all the answers

What is the purpose of the Remainder Theorem in relation to cubic polynomials?

It determines if a linear polynomial is a factor of a cubic polynomial. Signup and view all the answers

A cubic polynomial is divided by (x - 2), resulting in a remainder of 5. What is the value of the polynomial at (x = 2)?

5 Signup and view all the answers

What is the relationship between the average rate of change and the derivative of a function?

The derivative gives the instantaneous rate of change, while the average rate of change is calculated over an interval. Signup and view all the answers

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

Completing the square Signup and view all the answers

What is the sum of the first 100 positive integers?

5050 Signup and view all the answers

Which of the following best describes a relation where each element of the domain is associated with exactly one element of the range?

One-to-One Function Signup and view all the answers

What is the key property that guarantees a function has an inverse function?

The function must be one-to-one. Signup and view all the answers

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

S_n = n/2 (2a + (n - 1)d) Signup and view all the answers

Which of the following is true about the inverse function of f(x) = 2x + 1?

The inverse function is f^{-1}(x) = (x - 1)/2 Signup and view all the answers

What is the graphical representation of a function that has an inverse that is also a function?

Every horizontal line intersects the graph at most once. Signup and view all the answers

Given a function f(x) and its inverse f^{-1}(x), what is the relationship between their graphs?

They are reflections of each other across the line y = x. Signup and view all the answers

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

Its inverse does not exist. Signup and view all the answers

What is the first step in finding the inverse function of f(x) = 3x - 2?

Interchange x and y. Signup and view all the answers

Which of the following functions has an inverse that is also a function?

f(x) = x^3 Signup and view all the answers

What is the sign of f''(x) for a concave up graph?

f''(x) > 0 Signup and view all the answers

What is the application of differential calculus in optimisation problems?

To determine the stationary points of a function Signup and view all the answers

What is the formula for synthetic division?

q_2 = a_3, q_1 = a_2 + q_2 * d/c, q_0 = a_1 + q_1 * d/c, R = a_0 + q_0 * d/c Signup and view all the answers

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

To determine the graph's shape and position Signup and view all the answers

In Zeno's Achilles and the Tortoise paradox, what mathematical concept is demonstrated by the tortoise seemingly always staying ahead, despite Achilles' speed?

The concept of a limit. Signup and view all the answers

What is the condition for a point of inflection?

f''(x) = 0 and changes sign Signup and view all the answers

What is the application of differential calculus in rates of change?

To find the rate of change of a quantity Signup and view all the answers

Why is the function (y = \frac{x^2 + 4x - 12}{x + 6}) undefined when (x = -6)?

Because the denominator becomes zero. Signup and view all the answers

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

The function approaches -8. Signup and view all the answers

What is the general form of a cubic polynomial?

f(x) = ax^3 + bx^2 + cx + d Signup and view all the answers

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

A straight line with a hole at (x = -6). Signup and view all the answers

What is the method used to find the x-intercepts of a cubic polynomial?

Solving the equation f(x) = 0 Signup and view all the answers

What is the purpose of the Remainder Theorem?

To find the remainder of a polynomial division 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?

The limit is -8. Signup and view all the answers

Which of these statements about the concept of a limit is TRUE?

Limits describe how a function behaves as its input approaches a specific value. Signup and view all the answers

What is the derivative of the function (f(x) = x^3 + 2x^2 - 5x + 1) using the rules of differentiation?

(f'(x) = 3x^2 + 4x - 5) Signup and view all the answers

Which of the following notations represents the derivative of (y) with respect to (x)?

(rac{dy}{dx}) Signup and view all the answers

What is the derivative of the function (f(x) = 5x^2 - 3x + 2) using the rules of differentiation?

(f'(x) = 10x - 3) Signup and view all the answers

What is the derivative of the function (f(x) = rac{1}{x}) using the rules of differentiation?

(f'(x) = -rac{1}{x^2}) Signup and view all the answers

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

(f(x')) Signup and view all the answers

What is the gradient of the tangent to the curve (y = x^2 + 3x - 2) at the point where (x = 1)?

5 Signup and view all the answers

What is the equation of the tangent to the curve (y = x^3 - 2x + 1) at the point ((1, 0))?

(y = x - 1) Signup and view all the answers

What is the derivative of the function (f(x) = \sqrt{x}) using the rules of differentiation?

(f'(x) = rac{1}{2\sqrt{x}}) Signup and view all the answers

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

(f'(x) = 2x + 4) Signup and view all the answers

Which of the following statements is TRUE about the derivative of a constant function?

The derivative is always zero. Signup and view all the answers

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

Sn = (n/2)(a + l) Signup and view all the answers

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

The function must be one-to-one Signup and view all the answers

What is the graphical representation of an inverse function?

The graph of the inverse function is the reflection of the original function's graph across the line y = x Signup and view all the answers

What is the formula for finding the sum of the first n integers?

Sn = (n/2)(2n + 1) Signup and view all the answers

What is the definition of a function?

A function is a relation where each element in the domain maps to exactly one element in the range Signup and view all the answers

What is the purpose of the horizontal line test?

To determine if a function is one-to-one Signup and view all the answers

What is the formula for the inverse of a function f(x)?

f^(-1)(x) is the reflection of f(x) across the line y = x Signup and view all the answers

What is the difference between a one-to-one function and a many-to-one function?

A one-to-one function maps each element in the domain to exactly one element in the range, while a many-to-one function maps each element in the domain to at least one element in the range Signup and view all the answers

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

f^(-1)(x) Signup and view all the answers

What is the key property of an inverse function?

An inverse function undoes the operation of the original function Signup and view all the answers

What is the range of the logarithmic function f(x) = log x?

y ∈ ℝ Signup and view all the answers

What is the formula for the population growth of a city at a constant rate?

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

What is the purpose of logarithms in financial calculations?

To calculate loan repayments and interest rates Signup and view all the answers

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

A reflection of the original function about the line y = x Signup and view all the answers

What is the logarithmic identity log_a(a) equal to?

1 Signup and view all the answers

What is the logarithmic identity log_a(1) equal to?

0 Signup and view all the answers

What is the primary method to verify if a sequence is arithmetic?

Check if the difference between consecutive terms is constant. Signup and view all the answers

How is the arithmetic mean of two numbers defined?

The average of the two numbers. Signup and view all the answers

What characteristic defines a geometric sequence?

Each term is a fixed multiple of the preceding term. Signup and view all the answers

In the context of geometric sequences, what does the common ratio represent?

The factor by which each term is multiplied to get the next term. Signup and view all the answers

Which formula is used to find the n-th term of a geometric sequence?

T_n = ar^{n-1} Signup and view all the answers

Which type of graph represents an arithmetic sequence?

A straight line. Signup and view all the answers

What happens to a geometric sequence if the common ratio is less than one but greater than zero?

The terms decay exponentially. Signup and view all the answers

What is the definition of a series in mathematical terms?

The sum of the terms in a sequence. Signup and view all the answers

What notation is commonly used to represent the sum of terms in a sequence?

Sigma notation. Signup and view all the answers

What is an important characteristic of a finite series?

It only includes specific terms of a sequence. Signup and view all the answers

Given a polynomial ( p(x) ) and a divisor ( cx - d ), what is the formula for the remainder ( R ) when ( p(x) ) is divided by ( cx - d )?

( R = p \left( rac{d}{c} ight) ) Signup and view all the answers

What is the relationship between the degree of the original polynomial ( p(x) ) and the degree of the quotient polynomial ( Q(x) ) when dividing ( p(x) ) by a linear polynomial ( cx - d )?

The degree of ( Q(x) ) is one less than the degree of ( p(x) ). Signup and view all the answers

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

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

What is the relationship between the roots of a polynomial and its factors?

If ( x = rac{d}{c} ) is a root of ( p(x) ), then ( cx - d ) is a factor of ( p(x) ). Signup and view all the answers

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

The remainder ( R ) must be equal to 0. Signup and view all the answers

Given a polynomial ( p(x) ) and a factor ( cx - d ), how can ( p(x) ) be expressed in terms of the factor and the quotient polynomial ( Q(x) )?

( p(x) = (cx - d) \cdot Q(x) ) 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 ) using factorization methods?

Identify a factor using the Factor Theorem. Signup and view all the answers

If a polynomial ( p(x) ) has a root ( rac{d}{c} ), what can be concluded about the polynomial?

The polynomial ( p(x) ) can be factorized as ( (cx - d) \cdot Q(x) ), where ( Q(x) ) is the quotient polynomial. Signup and view all the answers

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

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

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

It helps to find a linear factor of the cubic equation, which simplifies the factorization process. Signup and view all the answers

What is the general form of the inverse of a linear function defined as $y = ax + q$?

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

Which of the following restrictions is typically applied to the domain of the quadratic function $y = ax^2$ to ensure the inverse is a function?

$x ext{ must be non-negative or non-positive based on } a$ Signup and view all the answers

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

$y = ext{log}_b(x)$ Signup and view all the answers

Which of the following correctly describes the range of the exponential function $y = b^x$ when $b > 1$?

$y ext{ is greater than or equal to 0}$ Signup and view all the answers

What is the shape of the graph of the logarithmic function $y = ext{log}_b(x)$?

Increasing Signup and view all the answers

What intercept characterizes the logarithmic function $y = ext{log}_b(x)$?

(1, 0) Signup and view all the answers

Which statement is true regarding the summation symbol $\Sigma$?

It denotes the sum of terms in a sequence from a lower bound to an upper bound. Signup and view all the answers

If $x = b^y$, how would you express $y$ in terms of $x$?

$y = ext{log}_b(x)$ Signup and view all the answers

Which of these is a property of the graph of an exponential function when $0 < b < 1$?

It decreases to zero but never touches the x-axis. Signup and view all the answers

What condition must be satisfied for an infinite geometric series to converge?

The common ratio must satisfy $-1 < r < 1$. Signup and view all the answers

Which statement about the domain of the logarithmic function $y = ext{log}_b(x)$ is true?

$x > 0$ Signup and view all the answers

In the formula for a finite geometric series $S_n = \frac{a(1 - r^n)}{1 - r}$, what does the variable $,a,$ represent?

The first term of the geometric sequence. Signup and view all the answers

Which of the following represents a prerequisite for a series to be classified as convergent?

The sum must approach a fixed number as more terms are added. Signup and view all the answers

Which feature is characteristic of the logarithmic function?

Its graph approaches the y-axis. Signup and view all the answers

In an arithmetic sequence characterized by the common difference $d$, how is the $n$-th term $T_n$ determined?

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

What does the notation $S_n$ represent in the context of finite series?

The sum of the first $n$ terms of a sequence. Signup and view all the answers

Which formula is used to calculate the sum of an infinite geometric series, given that it converges?

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

When evaluating the formula for a finite geometric series, what does $r$ represent?

The common ratio between consecutive terms. Signup and view all the answers

What is a characteristic of a finite series compared to an infinite series?

A finite series consists of finite terms summed up to a limit. Signup and view all the answers

Considering the series defined by $T_n = a imes r^{(n-1)}$, what type of sequence does this represent?

Geometric sequence with constant ratio. 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?

The product of the gradients of the tangent and the normal is equal to -1. Signup and view all the answers

Which of the following statements is TRUE about the second derivative of a function?

The second derivative represents the rate of change of the first derivative. Signup and view all the answers

To find the equation of a tangent line to a curve at a given point, what is the first step?

Find the derivative of the function representing the curve. Signup and view all the answers

What does the coefficient 'a' in the cubic function ( f(x) = ax^3 + bx^2 + cx + d ) determine?

The steepness of the graph. Signup and view all the answers

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

( y' ) Signup and view all the answers

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

To find the stationary points on the graph. Signup and view all the answers

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

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

Which of the following statements is TRUE about the stationary points of a cubic function?

Stationary points can be either local maxima or local minima. Signup and view all the answers

What is the relationship between the second derivative and the concavity of a function?

If the second derivative is positive, the function is concave up. Signup and view all the answers

Gr12 Mathematics: June Exam Medium P(1)

Questions and Answers

What is the derivative of the function f(x) = x^2?

What is the derivative of the function f(x) = 3x^2 + 2x - 5?

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

What is the purpose of the rules for differentiation?

What is the derivative of a constant function f(x) = k?

What is the derivative of the function f(x) = x^3 - 2x^2 + x - 1?

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

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

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

What is the general rule for differentiation?

What is the main purpose of limits in calculus?

What happens to the function defined as $y = \frac{x^2 + 4x - 12}{x + 6}$ when $x$ approaches -6?

Which of the following statements about the function $y = \frac{(x + 6)(x - 2)}{x + 6}$ is true?

In the context of Zeno's Achilles and Tortoise paradox, what does the paradox illustrate about limits?

What can be concluded about the graph of the function $y = \frac{x^2 + 4x - 12}{x + 6}$?

What expression does the function simplify to when $x \neq -6$?

What is the value of R in the division of a polynomial p(x) by a divisor cx - d?

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

What does the Factor Theorem state?

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

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

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

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

What is the common difference in an arithmetic sequence?

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

What is the relationship between the degree of the quotient polynomial Q(x) and the degree of the polynomial p(x) when dividing by a linear polynomial cx - d?

What is the first step in determining the equation of a tangent to a curve?

How is the relationship between the gradients of the tangent and the normal defined?

What does the second derivative indicate about a function?

Which of the following notations is NOT commonly used for the second derivative?

To find the y-intercept of the cubic function $f(x) = ax^3 + bx^2 + cx + d$, what value should be substituted for $x$?

What is the condition for a stationary point of a cubic function?

What happens to the graph of a cubic function when the leading coefficient $a$ is positive?

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

In the context of cubic functions, how can local maxima and minima be characterized?

What method can be used to find x-intercepts of a cubic function?

What does it mean for a graph to be concave up?

Which of the following statements is true about points of inflection?

What is the first step in sketching a cubic graph?

What is the outcome of applying the Remainder Theorem?

Which method is NOT used for factorizing cubic polynomials?

To find the turning points of a cubic function, what must be solved?

What does the average rate of change measure over an interval?

In synthetic division, which of the following is used?

Given a polynomial of the form $f(x) = ax^3 + bx^2 + cx + d$, which point provides the y-intercept?

What is true about the end behavior of a cubic polynomial as $x$ approaches positive or negative infinity?

Which of the following is equivalent to the expression (\log_a(x/y)) using the laws of logarithms?

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

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

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

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

What is the value of the common ratio (r) in a geometric sequence if the second term is 6 and the first term is 2?

In sigma notation, what does the index of summation represent?

What is the sum of the first 5 terms of the geometric sequence with a = 2 and r = 3?

What is the sum of the infinite geometric series with a = 1 and r = 1/2?

Which of the following is an example of a finite geometric series?

What is the sum of the first 10 terms of the arithmetic series 1 + 4 + 7 + 10 + ...?

What is the sum of the first 100 positive integers using Gauss's method?

What is the formula for the sum of the first $n$ terms in an arithmetic series?

Which statement correctly defines a function?

Which graphical characteristic identifies a one-to-one function?

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

What do the graphs of a function and its inverse exhibit?

How is the inverse function found algebraically?

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

If a function fails the horizontal line test, what does this indicate?

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

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

Which of the following describes the domain of the inverse function for $y = ax^2$?

What is the range of the inverse function $f^{-1}(x) = rac{1}{a}x - rac{q}{a}$?

Which characteristic is true about the inverse of a linear function?

How is the inverse of $y = b^x$ expressed?

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

What transformation do the graphs of a function and its inverse exhibit?

Which statement about an exponential function where $b > 1$ is correct?

What describes the behavior of the exponential function $f(x) = b^x$ when $b < 1$?

What property does the logarithm have when converting from exponential form, such as $5^2 = 25$?

If the first term of a geometric sequence is 3 and the common ratio is 2, what is the 5th term?