Question 1

Suppose the function $$f(x) is $$defined as follows:

$$
  f(x) = \begin{cases}
    x^2$ - 1, & x \u003c 2 \
    3x - 5, & x \geq 2
  \end{cases}
$$

Which of the following statements is TRUE about the continuity of $$f(x) at$$ $$x = 2$$?

Accepted Answer

$$f(x) is $$not continuous at $$x = 2$$ because the left and right limits are not equal.

Question 2

Evaluate the following limit:

$$
\lim_{x 	o 0} \frac{\sin(5x)}{x}
$$

Which of the following is correct?

Accepted Answer

$$5$$

Question 3

A car's position at time $$t (in $$seconds) is given by $$s(t) = 2t^3$ - 5t^2$ + 4t. $$What is the car's instantaneous velocity at $$t = 2$$ seconds?

Accepted Answer

$$s'(2) = 8$$

Question 4

Which of the following statements about differentiability and continuity is correct?

Accepted Answer

If a function is differentiable at a point, it must also be continuous at that point.

Question 5

Let $$f(x) = x^2$ \sin(x). $$What is $$f'(x)$$?

Accepted Answer

$$2x \sin(x) + x^2$ \cos(x)$$

Question 6

A tank is being filled with water such that the volume $$V (in $$liters) at time $$t (in $$minutes) is given by $$V = 3t^2$ + 2t. At $$what rate is the volume increasing when $$t = 4$$ minutes?

Accepted Answer

$$V'(4) = 26$$ liters/min

Question 7

Which of the following is the correct application of the Mean Value Theorem for the function $$f(x) = x^3$ on the interval $$[1, 3]$$?

Accepted Answer

There exists $$c \in (1, 3)$$ such that $$f'(c) = \frac{f(3) - f(1)}{3 - 1}$$

Question 8

Given $$y^2$$ + $$x^2 = 25$, find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

$$\frac{dy}{dx} = -\frac{x}{y}$$

Question 9

A medication is administered such that its concentration in the bloodstream at time $$t (in $$hours) is modeled by $$C(t) = 10e$$^{-0.5t}$$. What is the rate of change of the concentration at t = 2$ hours?

Accepted Answer

$$C'(2) = -2.5e$$^{-1}$$

Question 10

Evaluate the definite integral:

$$
\int_{0}^{2} (3x^2$ - 4x + 1)\,dx
$$

Accepted Answer

$$2$$

Question 11

Which of the following best describes the Intermediate Value Theorem?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and $$N is $$between $$f(a)$$ and $$f(b)$$, then there exists $$c \in (a, b)$$ such that $$f(c) = N.$$

Question 12

A particle moves along a line so that its velocity at time $$t is$$ $$v(t) = 6t - 4. $$What is the total displacement of the particle from $$t = 0 to$$ $$t = 3$$?

Accepted Answer

$$21$$

Question 13

Which of the following is the correct linear approximation (linearization) of $$f(x) = \sqrt{x} at$$ $$x = 4$$?

Accepted Answer

$$L(x) = 2 + \frac{1}{4}(x - 4)$$

Question 14

Find the area between the curves $$y = x^2$ and y = 2x$ from x = 0$ to x = 2$.

Accepted Answer

$$\int_{0}^{2} (2x - x^2$)\,dx = 4/3$$

Question 15

Which of the following is the correct setup for the volume of the solid generated by revolving the region bounded by $$y = x^2$ and y = 4$ about the x$-axis?

Accepted Answer

$$V = \pi \int_{-2}^{2} (4^2$ - (x^2$)^2$$)\,dx$

Question 16

Suppose $$f'(x) \u003e 0$$ for all $$x in an $$interval $$I. $$Which of the following must be true about $$f(x) on$$ $$I$$?

Accepted Answer

$$f(x) is $$increasing on $$I.$$

Question 17

A population of bacteria grows at a rate proportional to its size. If $$P(0) = 100$$ and $$P'(t) = 0.03P(t)$$, what is the population after $$10$$ hours?

Accepted Answer

$$P(10) = 100e$$^{0.3}$$

Question 18

Which of the following is the correct expression for the average value of $$f(x) = x^2$ on the interval $$[0, 3]$$?

Accepted Answer

$$\frac{1}{3} \int_{0}^{3} x^2$\,dx$$

Question 19

Given the slope field for $$\frac{dy}{dx} = y - x$$, which of the following statements is true about the solution passing through $$(1, 2)$$?

Accepted Answer

The solution curve is increasing at $$(1, 2)$$ because $$\frac{dy}{dx} = 1$$ there.

AP Calculus AB: College Board-Aligned Study Syllabus (2026-27)

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