AP Calculus AB Practice Quiz: Defining Average and Instantaneous Rates of Change at a Point

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 13 questions to check your progress.

Question 1 of 13

Which of the following expressions represents the average rate of change of a function f over the interval [a, a+h]?

A)\\frac{f(a+h)-f(a)}{h}B)\lim_{h \to 0}\\frac{f(a+h)-f(a)}{h}C)f'(a)D)\lim_{x \to a}\\frac{f(x)-f(a)}{x-a}

All Questions (13)

Which of the following expressions represents the average rate of change of a function f over the interval [a, a+h]?

A) \\frac{f(a+h)-f(a)}{h}

B) \lim_{h \to 0}\\frac{f(a+h)-f(a)}{h}

C) f'(a)

D) \lim_{x \to a}\\frac{f(x)-f(a)}{x-a}

Correct Answer: A

According to the provided content, the difference quotient \\frac{f(a+h)-f(a)}{h} expresses the average rate of change of a function over an interval. The other options represent the instantaneous rate of change.

The instantaneous rate of change of a function f at a point x=a is also known as the derivative, denoted f'(a). How is it defined?

A) As the value of the difference quotient for a very small h.

B) As the limit of a difference quotient, provided the limit exists.

C) As the average of the function's values around the point a.

D) As the expression \\frac{f(x)-f(a)}{x-a}.

Correct Answer: B

The content states that the instantaneous rate of change, f'(a), can be expressed by \lim_{h \to 0}\\frac{f(a+h)-f(a)}{h} or \lim_{x \to a}\\frac{f(x)-f(a)}{x-a}, provided the limit exists. These are limits of difference quotients.

Which of the following is an equivalent form for the definition of the derivative, f'(a), to \lim_{h \to 0}\\frac{f(a+h)-f(a)}{h}?

A) \\frac{f(x)-f(a)}{x-a}

B) \lim_{x \to a}\\frac{f(x)-f(a)}{x-a}

C) \lim_{h \to a}\\frac{f(a+h)-f(a)}{h}

D) \\frac{f(a+h)+f(a)}{h}

Correct Answer: B

The provided content explicitly states that \lim_{h \to 0}\\frac{f(a+h)-f(a)}{h} and \lim_{x \to a}\\frac{f(x)-f(a)}{x-a} are equivalent forms of the definition of the derivative, f'(a).

The expression \\frac{g(t)-g(c)}{t-c} is used to determine what quantity for a function g?

A) The instantaneous rate of change of g at t=c.

B) The instantaneous rate of change of g at some point between t and c.

C) The average rate of change of g over the interval between c and t.

D) The limit of the function g as t approaches c.

Correct Answer: C

This expression is a difference quotient of the form \\frac{f(x)-f(a)}{x-a}, which the content identifies as expressing the average rate of change of a function over an interval.

The limit \lim_{x \to 5}\\frac{f(x)-f(5)}{x-5} represents which of the following?

A) The average rate of change of f on an interval containing 5.

B) The value of the function f at x=5.

C) The instantaneous rate of change of f at x=5.

D) The change in the function f as x approaches 5.

Correct Answer: C

This limit matches one of the provided forms for the definition of the derivative, f'(a), where a=5. The derivative represents the instantaneous rate of change at that point.

Which limit expression would be used to find the derivative of a function k(x) at the point x=2?

A) \lim_{h \to 0}\\frac{k(2+h)-k(2)}{h}

B) \\frac{k(2+h)-k(2)}{h}

C) \lim_{x \to h}\\frac{k(x)-k(2)}{x-2}

D) \lim_{h \to 2}\\frac{k(2+h)-k(2)}{h}

Correct Answer: A

The content defines the derivative at x=a as \lim_{h \to 0}\\frac{f(a+h)-f(a)}{h}. Substituting k for f and 2 for a gives the expression in option A.

The limit \lim_{h \to 0}\\frac{(c+h)^3 - c^3}{h} represents the derivative of what function f(x) and at what point a?

A) f(x) = x^3 at a=h

B) f(x) = (x+h)^3 at a=c

C) f(x) = c^3 at a=x

D) f(x) = x^3 at a=c

Correct Answer: D

This expression matches the form \lim_{h \to 0}\\frac{f(a+h)-f(a)}{h}. By comparison, f(a+h) is (c+h)^3 and f(a) is c^3. This implies the function is f(x) = x^3 and the point is a=c.

To find the instantaneous rate of change of a function, one must find the limit of the average rate of change as what occurs?

A) The function's value approaches zero.

B) The interval for the average rate of change shrinks to zero.

C) The value of x approaches infinity.

D) The function becomes continuous.

Correct Answer: B

The instantaneous rate of change is defined as the limit of the difference quotient (average rate of change) as h approaches 0 or x approaches a. In both cases, the length of the interval over which the average is calculated is shrinking to zero.

Which of the following correctly sets up the calculation for the average rate of change of the function f(x) between x=3 and x=7?

A) \lim_{x \to 3}\\frac{f(x)-f(3)}{x-3}

B) \\frac{f(7)-f(3)}{7-3}

C) \lim_{h \to 0}\\frac{f(3+h)-f(3)}{h}

D) \\frac{f(3)-f(7)}{7-3}

Correct Answer: B

The average rate of change is calculated using the difference quotient \\frac{f(x)-f(a)}{x-a}. For the interval between 3 and 7, this becomes \\frac{f(7)-f(3)}{7-3}.

The expression \lim_{x \to -1}\\frac{\\cos(x) - \\cos(-1)}{x - (-1)} defines f'(a) for some function f and point a. What are f(x) and a?

A) f(x) = \\cos(x), a = 1

B) f(x) = \\cos(-1), a = x

C) f(x) = \\cos(x), a = -1

D) f(x) = \\frac{\\cos(x)}{x}, a = -1

Correct Answer: C

This expression matches the form \lim_{x \to a}\\frac{f(x)-f(a)}{x-a}. By direct comparison, the function is f(x) = \\cos(x) and the point is a = -1.

The difference quotient is a formula used to express what concept?

A) The instantaneous rate of change at a single point.

B) The value of a function at a single point.

C) The average rate of change of a function over an interval.

D) The limit of a function as x approaches a point.

Correct Answer: C

The provided content explicitly states that the difference quotients \\frac{f(a+h)-f(a)}{h} and \\frac{f(x)-f(a)}{x-a} express the average rate of change of a function over an interval.

A student correctly calculates the average rate of change of a function f on the interval [a, x] using \\frac{f(x)-f(a)}{x-a}. To find the instantaneous rate of change at a, what is the next logical step?

A) Set x equal to a.

B) Take the limit of the expression as a approaches 0.

C) Take the limit of the expression as x approaches a.

D) Multiply the expression by (x-a).

Correct Answer: C

The definition of the instantaneous rate of change, f'(a), is the limit of the average rate of change as the interval shrinks. For the difference quotient \\frac{f(x)-f(a)}{x-a}, this means taking the limit as x approaches a.

Both the average and instantaneous rates of change are derived from the same fundamental expression. What is this expression called?

A) The derivative

B) The limit quotient

C) The function evaluator

D) The difference quotient

Correct Answer: D

The average rate of change is expressed by the difference quotient. The instantaneous rate of change is expressed as the limit of the difference quotient. Therefore, the difference quotient is the fundamental expression for both concepts.