AP Calculus BC 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

What is the average rate of change of the function f(x) = x² + 1 over the interval [1, 3]?

A)4B)8C)2D)5

All Questions (13)

What is the average rate of change of the function f(x) = x² + 1 over the interval [1, 3]?

A) 4

B) 8

C) 2

D) 5

Correct Answer: A

The average rate of change is calculated using the difference quotient (f(b) - f(a)) / (b - a). Here, a=1 and b=3. So, (f(3) - f(1)) / (3 - 1) = ((3² + 1) - (1² + 1)) / 2 = (10 - 2) / 2 = 8 / 2 = 4.

The expression (f(b) - f(a)) / (b - a) is a difference quotient that represents which of the following?

A) The instantaneous rate of change of f at x=a.

B) The value of the function f at the midpoint of the interval [a, b].

C) The average rate of change of f over the interval [a, b].

D) The derivative of f at x=b.

Correct Answer: C

According to the provided content, the difference quotient (f(x) - f(a)) / (x - a) expresses the average rate of change of a function over an interval. The form (f(b) - f(a)) / (b - a) is an equivalent representation for the interval [a, b].

Which of the following limits represents the instantaneous rate of change of the function f(x) = cos(x) at the point x = π/2?

A) lim (h→0) [ (cos(π/2 + h) - cos(π/2)) / h ]

B) lim (x→π/2) [ (cos(x) - cos(π/2)) / (x + π/2) ]

C) (cos(π/2 + h) - cos(π/2)) / h

D) lim (h→0) [ cos(π/2 + h) / h ]

Correct Answer: A

The instantaneous rate of change, or the derivative f'(a), is defined as lim (h→0) [ (f(a+h) - f(a)) / h ]. For f(x) = cos(x) at a = π/2, this becomes lim (h→0) [ (cos(π/2 + h) - cos(π/2)) / h ].

The limit lim (x→4) [ (x³ - 64) / (x - 4) ] represents the derivative of a function f at a point a. What are f(x) and a?

A) f(x) = x, a = 4

B) f(x) = x³, a = 64

C) f(x) = x³, a = 4

D) f(x) = x - 4, a = 4

Correct Answer: C

This limit is in the form lim (x→a) [ (f(x) - f(a)) / (x - a) ], which defines f'(a). By comparing the expressions, we can identify a = 4, f(x) = x³, and f(a) = f(4) = 4³ = 64.

Which of the following expressions represents the derivative, f'(3), for the function f(x) = 1/x?

A) lim (x→3) [ ( (1/x) - (1/3) ) / (x - 3) ]

B) lim (h→0) [ ( (1/h) - (1/3) ) / (h - 3) ]

C) ( (1/x) - (1/3) ) / (x - 3)

D) lim (x→3) [ (1/x) / (x - 3) ]

Correct Answer: A

The derivative f'(a) can be expressed as lim (x→a) [ (f(x) - f(a)) / (x - a) ]. For f(x) = 1/x and a = 3, this becomes lim (x→3) [ ( (1/x) - (1/3) ) / (x - 3) ].

The instantaneous rate of change of a function f at a point x=a is best described as:

A) The average rate of change over a very large interval containing a.

B) The value of the function at a, f(a).

C) The limit of the average rate of change over an interval [a, x] as x approaches a.

D) The slope of the line connecting (a, f(a)) and (a+1, f(a+1)).

Correct Answer: C

The content defines the instantaneous rate of change at x=a as the limit of a difference quotient, such as lim (x→a) [ (f(x) - f(a)) / (x - a) ]. The difference quotient itself represents the average rate of change, so the instantaneous rate of change is the limit of this average rate as the interval shrinks to zero.

The limit lim (h→0) [ (√(16 + h) - 4) / h ] represents the derivative f'(a) for which function f and value a?

A) f(x) = √x, a = 4

B) f(x) = √(16 + x), a = 0

C) f(x) = √x, a = 16

D) f(x) = x, a = 16

Correct Answer: C

This limit is in the form lim (h→0) [ (f(a+h) - f(a)) / h ]. By comparing the expressions, we can identify f(a+h) as √(16+h) and f(a) as 4. This implies a=16 and the function is f(x) = √x, since f(16) = √16 = 4.

Using a difference quotient, find the average rate of change of g(t) = 2t² - t on the interval [1, 4].

A) 27

B) 9

C) 1

D) 13.5

Correct Answer: B

The average rate of change is (g(4) - g(1)) / (4 - 1). First, calculate g(4) = 2(4)² - 4 = 32 - 4 = 28. Then, calculate g(1) = 2(1)² - 1 = 2 - 1 = 1. The average rate of change is (28 - 1) / (4 - 1) = 27 / 3 = 9.

The definition of the derivative f'(a) as lim (h→0) [ (f(a+h) - f(a)) / h ] is equivalent to which other limit expression?

A) (f(x) - f(a)) / (x - a)

B) lim (x→a) [ (f(x) - f(a)) / (x - a) ]

C) lim (h→a) [ (f(a+h) - f(a)) / h ]

D) lim (x→0) [ (f(x) - f(a)) / (x - a) ]

Correct Answer: B

The provided content explicitly states that lim (h→0) [ (f(a+h) - f(a)) / h ] and lim (x→a) [ (f(x) - f(a)) / (x - a) ] are equivalent forms for the definition of the derivative, f'(a).

Which expression represents the average rate of change of a function g over the interval starting at x=c with a length of h?

A) lim (h→0) [ (g(c+h) - g(c)) / h ]

B) (g(c+h) - g(c)) / c

C) (g(h) - g(c)) / (h-c)

D) (g(c+h) - g(c)) / h

Correct Answer: D

The interval starts at c and has length h, so it ends at c+h. The average rate of change is the change in the function value divided by the change in the input value. This is (g(c+h) - g(c)) / ((c+h) - c), which simplifies to (g(c+h) - g(c)) / h. This is one of the standard forms of the difference quotient.

Provided the limit exists, the expression lim (h→0) [ (f(2+h) - f(2)) / h ] represents:

A) The average rate of change of f on an interval containing x=2.

B) The value of the function at x=2.

C) The derivative of f at x=2.

D) An undefined quantity.

Correct Answer: C

This expression is the direct definition of the derivative of a function f at the point a=2, denoted f'(2). It represents the instantaneous rate of change of the function at that specific point.

What is the average rate of change of the linear function f(x) = 3x + 5 over the interval [a, a+h]?

A) 3h

B) 3

C) 3a + 5

D) 6a + 3h + 10

Correct Answer: B

Using the difference quotient (f(a+h) - f(a)) / h, we get: ( (3(a+h) + 5) - (3a + 5) ) / h = (3a + 3h + 5 - 3a - 5) / h = (3h) / h = 3. The average rate of change of a linear function is always its constant slope.

The difference quotient (f(x) - f(a)) / (x - a) gives the average rate of change of f on the interval [a, x]. What does the limit of this expression as x approaches a represent?

A) The average rate of change over a larger interval.

B) The value of the function at x=a.

C) The value zero.

D) The instantaneous rate of change of f at x=a.

Correct Answer: D

The content explicitly defines the instantaneous rate of change at x=a as the limit of the average rate of change as the interval shrinks. Taking the limit of the difference quotient (f(x) - f(a)) / (x - a) as x approaches a is the definition of the derivative, f'(a), which is the instantaneous rate of change.