AP Calculus Practice Questions

Test your readiness for the AP Calculus exam with this quiz!

AP Calculus Free Practice Question #1

A function f is defined by f(x)=|x+ 4|. For what values of x is the graph of f not differentiable?
A: x = -4
B: x = 0
C: x = 4
D: The function is differentiable over its entire domain.

AIf the graph of a function has a sharp point, the function is not differentiable at that point. This is because the slopes directly to the left and right of the point do not approach the same value.
An absolute value function has a sharp point at its vertex. The graph off(x)=|x + 4| is a horizontal translation (to the left 4 units) of the standard absolute value function, y =|x|, which has vertex (0, 0). Thus, the vertex of fis (-4, 0), which means the function is not differentiable at x =-4. Choice (A) is correct.

AP Calculus Free Practice Question #2

The function f has a removable discontinuity at:
A: x = -2 only
B: x = 0 only
C: x = -2 and x = 0
D: f(x) has no removable discontinuities

D: The only discontinuities that are removable are holes and holes with a point above or below—this function has neither. The function has 2 jump discontinuities (gaps), at x=-2 and x= 0, but neither of these is removable.

AP Calculus Free Practice Question #3

Using only the table of values shown in the quiz above, which of the following best describes f′(x) and f′′(x) over the open interval (-1, 1)?

A: f′(x) < 0; f′′(x) < 0
B: f′(x) < 0; f′′(x) = 0
C: f′(x) > 0; f′′(x) < 0
D: f′(x) > 0; f′′(x) = 0

D: Examine the values in the table: f(x) increases as x gets larger, which indicates that f′(x), the slope of the function, is positive. This means f′(x)> 0, so eliminate (A) and (B).
To choose between (C) and (D), take a closer look at the slopes. The slope between the first pair of points is 3, and the slope between the second pair of points is also 3, so f′(x) is constant. This means f′′(x), which is the derivative of f′(x), must be 0. Choice (D) is correct

AP Calculus Free Practice Question #4
Let f(x) be a differentiable function with f(-1)= 5 and f'(-1)= 2. Use the given information to find the local linear approximation of f(-0.9).
A: 4.9
B: 5.1
C: 5.2
D: 5.3

C: To use a local linear approximation, you need to find the equation of the tangent line. You’ve been given all the information you need in the question stem; you just need to piece it all together. The point on the function is given by f(-1)= 5, which translates to the point (-1, 5). The slope of the tangent line at x=-1 is given by f'(-1)= 2, so the slope is 2. Now the point-slope form of the tangent line is:
y-5 = 2(x-(-1))
y = 5+2(x+1)
Substituting x=-0.9 into the equation of the tangent line yields:
5+2(-0.9+1) = 5+2(0.1) = 5.2
That’s (C).

AP Calculus Free Practice Question #5

The graph of f(x) is shown above. Which of the following is the graph of f'(x) ? (Letters correspond with the image directly to their right)

D: A function has horizontal tangents where its first derivative is equal to 0. The graph of f(x) has horizontal tangents at the points where x=± 1. Of the choices for the graph f′(x), only choice (D) has zeros at x=± 1.

For more practice questions, check out our AP Calculus Prep Plus book.