AP Calculus BC Unit 2: Differentiation: Definition and Fundamental Properties Practice Exam

Assessment for Unit 2: Differentiation: Definition and Fundamental Properties

Estimated Time: 60 minutes

Exam Details

Questions: 32 total (29 multiple choice, 3 short answer)
Time: 60 minutes (recommended)
Topics Covered: Unit 2: Differentiation: Definition and Fundamental Properties
Format: Multiple Choice, Short Answer questions matching real exam difficulty
Scoring: Instant results with detailed explanations

A. Multiple-Choice Questions (29 Questions)

Select the one best answer for each question.

1. Let $f$ be the function given by $f(x) = x^2 - 3x$. Which of the following expressions represents the average rate of change of $f$ over the interval $[1, 1+h]$?

(A)$\frac{(1+h)^2 - 3(1+h) + 2}{h}$(B)$\lim_{h \to 0} \frac{(1+h)^2 - 3(1+h) + 2}{h}$(C)$\frac{(1+h)^2 - 3(1+h) - 2}{h}$(D)$\frac{(1+h)^2 - 3(1+h) - (1^2 - 3(1))}{1+h}$

2. Which of the following limits represents the derivative of the function $f(x) = \tan(x)$ at $x = \frac{\pi}{4}$?

(A)$\lim_{h \to 0} \frac{\tan(\frac{\pi}{4} + h) - 1}{h}$(B)$\lim_{x \to \frac{\pi}{4}} \frac{\tan(x) - 1}{x}$(C)$\lim_{h \to 0} \frac{\tan(\frac{\pi}{4} + h) - \tan(h)}{h}$(D)$\lim_{x \to 0} \frac{\tan(x) - 1}{x - \frac{\pi}{4}}$

3. The limit $\lim_{x \to 2} \frac{e^{3x} - e^6}{x - 2}$ represents the value of $f'(c)$ for a function $f$ and a real number $c$. What are $f(x)$ and $c$?

(A)$f(x) = e^{3x}$ and $c = 2$(B)$f(x) = e^x$ and $c = 6$(C)$f(x) = e^{3x}$ and $c = 6$(D)$f(x) = e^x$ and $c = 2$

4. [Skill: 1.E | Topic: 2.10] Let $f$ be the function defined by $f(x) = \frac{\tan(x)}{x}$. Which of the following is an expression for $f'(x)$ ?

(A)$\frac{x \sec^2(x) - \tan(x)}{x^2}$(B)$\frac{\tan(x) - x \sec^2(x)}{x^2}$(C)$\frac{x \sec(x)\tan(x) - \tan(x)}{x^2}$(D)$\frac{\sec^2(x)}{x^2}$

5. [Skill: 1.E | Topic: 2.10] What is the slope of the line tangent to the graph of $y = \cot(x)$ at $x = \frac{\pi}{6}$ ?

(A)$-4$(B)$-2$(C)$-\frac{4}{3}$(D)$4$

6. Which of the following is the value of the limit $\lim_{h \to 0} \frac{2(3+h)^2 - 18}{h}$ ?

(A)0(B)6(C)12(D)18

7. The function $f$ is differentiable for all real numbers and satisfies $\lim_{x \to 2} \frac{f(x) - f(2)}{x - 2} = 3$. If the graph of $f$ passes through the point $(2, -1)$, which of the following is an equation of the line tangent to the graph of $f$ at $x=2$ ?

(A)y = 3x - 7(B)y = 3x - 5(C)y = 3x + 1(D)y = -1x + 1

8. Let $f$ be the function defined by $f(x) = \sqrt{x}$. Which of the following limits represents the derivative of $f$ at $x=4$ ?

(A)$\lim_{h \to 0} \frac{\sqrt{4+h} - 2}{h} $only(B)$\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4} $only(C)Both $\lim_{h \to 0} \frac{\sqrt{4+h} - 2}{h} $and $\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$(D)Neither limit represents the derivative.

9. The table below gives selected values for a differentiable function $f$. | $x$ | 0 | 2 | 5 | 9 | | :--- | :---: | :---: | :---: | :---: | | $f(x)$ | 4 | 10 | 18 | 25 | Based on the values in the table, which of the following is the best estimate for $f'(3.5)$?

(A)2.0(B)2.667(C)3.0(D)8.0

10. The graph of the function $g$ is shown in the figure above. Which of the following is the best estimate for the value of $g'(2)$?

(A)-2(B)-0.5(C)0.5(D)2

11. The function $h$ is differentiable for all real numbers. The table below gives values of $h(x)$ for values of $x$ near 4. | $x$ | 3.9 | 3.99 | 4.01 | 4.1 | | :--- | :---: | :---: | :---: | :---: | | $h(x)$ | 5.75 | 5.96 | 6.04 | 6.25 | Which of the following expressions gives the best estimate for $h'(4)$?

(A)$\frac{6.25 - 5.75}{4.1 - 3.9}$(B)$\frac{6.04 - 5.96}{4.01 - 3.99}$(C)$\frac{6.04 - 5.75}{4.01 - 3.9}$(D)$\frac{6.25 - 5.96}{4.1 - 3.99}$

12. Let $f$ be the function defined by $$f(x) = \begin{cases} ax + b & \text{for } x < 1 \\ x^2 - 3x & \text{for } x \ge 1 \end{cases}$$ where $a$ and $b$ are constants. If $f$ is differentiable at $x=1$, what is the value of $a + b$?

(A)$-2$(B)$-1$(C)$0$(D)$2$

13. The graph of the function $f$ is shown in the figure above. The graph consists of two line segments that meet at the point $(2, 3)$. Which of the following statements about $f$ is true at $x=2$?

(A)$f$ is continuous at $x=2$, but not differentiable at $x=2$.(B)$f$ is differentiable at $x=2$, but not continuous at $x=2$.(C)$f$ is both continuous and differentiable at $x=2$.(D)$f$ is neither continuous nor differentiable at $x=2$.

14. Let $g$ be a function such that $\lim_{h \to 0} \frac{g(5+h) - g(5)}{h} = 4$. Which of the following must be true? I. $g$ is continuous at $x=5$. II. $g'(5) = 4$. III. $\lim_{x \to 5} g(x) = 4$.

(A)I only(B)II only(C)I and II only(D)I, II, and III

15. Let $f$ be the function given by $f(x) = \frac{4}{\sqrt[3]{x}}$. Which of the following is an expression for $f'(x)$?

(A)$-\frac{4}{3x^{2/3}}$(B)$-\frac{4}{3x^{4/3}}$(C)$\frac{4}{3x^{2/3}}$(D)$\frac{2}{x^{3/2}}$

16. If $f(x) = \frac{x^3 + 2}{x}$, which of the following is equal to $f'(x)$?

(A)$3x$(B)$2x + 2$(C)$2x - \frac{2}{x^2}$(D)$\frac{3x^2}{1}$

17. What is the slope of the line tangent to the graph of $y = 2x^3 - 4x + 1$ at $x = -1$?

(A)2(B)3(C)6(D)10

18. Let $f$ be the function given by $f(x) = \frac{4x^3 - 2x^2 + 5x}{\sqrt{x}}$ for $x > 0$. Which of the following defines $f'(x)$?

(A)$10x^{3/2} - 3x^{1/2} + \frac{5}{2}x^{-1/2}$(B)$12x^2 - 4x + 5$(C)$4x^{3/2} - 2x^{1/2} + 5x^{-1/2}$(D)$10x^{5/2} - 3x^{3/2} + \frac{5}{2}x^{1/2}$

19. The table below gives values for the differentiable functions $f$ and $g$ and their derivatives at $x=2$. | $x$ | $f(x)$ | $f'(x)$ | $g(x)$ | $g'(x)$ | |:---:|:---:|:---:|:---:|:---:| | 2 | 5 | -3 | 1 | 4 | If $h(x) = 3f(x) - 2g(x) + 4$, what is the value of $h'(2)$?

(A)-17(B)-13(C)13(D)17

20. If $y = \frac{1}{2}x^4 - \pi^2 x + e^3$, then $\frac{dy}{dx} =$

(A)$2x^3 - \pi^2 + 3e^2$(B)$2x^3 - 2\pi x + 3e^2$(C)$2x^3 - \pi^2$(D)$4x^3 - 2\pi x$

21. Let $f$ be the function given by $f(x) = 3e^x + 2\sin x$. What is the slope of the line tangent to the graph of $f$ at $x=0$?

(A)1(B)2(C)3(D)5

22. $$\lim_{h \to 0} \frac{\ln(e+h) - 1}{h} \text{ is}$$

(A)$\frac{1}{e}$(B)1(C)e(D)nonexistent

23. Let $f$ be the function defined by $f(x) = x + 2\cos x$. For $0 < x < 2\pi$, at what value(s) of $x$ does the graph of $f$ have a horizontal tangent?

(A)$\frac{\pi}{6} \text{ and } \frac{5\pi}{6}$(B)$\frac{7\pi}{6} \text{ and } \frac{11\pi}{6}$(C)$\frac{\pi}{3} \text{ and } \frac{5\pi}{3}$(D)$\frac{2\pi}{3} \text{ and } \frac{4\pi}{3}$

24. Selected values for the differentiable functions $f$ and $g$ and their derivatives are shown in the table below. | $x$ | $f(x)$ | $f'(x)$ | $g(x)$ | $g'(x)$ | |:---:|:---:|:---:|:---:|:---:| | 2 | 3 | $-4$ | 1 | 5 | If $h(x) = f(x)g(x)$, what is the value of $h'(2)$?

(A)$-20$(B)$1$(C)$11$(D)$19$

25. The graphs of the functions $f$ and $g$ are shown in the figure above. $f$ is a line segment passing through $(0,0)$ and $(4,8)$, and $g$ is a line segment passing through $(0,6)$ and $(4,2)$. Let $P(x) = f(x) \cdot g(x)$. What is the value of $P'(2)$?

(A)$-12$(B)$-2$(C)$0$(D)$4$

26. If $y = x^2 \sin x$, then $\frac{dy}{dx} =$

(A)$2x \cos x$(B)$2x \sin x + x^2 \cos x$(C)$2x \sin x - x^2 \cos x$(D)$x^2 \sin x + 2x \cos x$

27. The functions $f$ and $g$ are differentiable for all real numbers. The table below gives values of the functions and their first derivatives at $x=3$. | $x$ | $f(x)$ | $f'(x)$ | $g(x)$ | $g'(x)$ | |:---:|:---:|:---:|:---:|:---:| | 3 | 4 | -2 | 2 | 5 | If $h(x) = \frac{f(x)}{g(x)}$, what is the value of $h'(3)$?

(A)-6(B)-1(C)4(D)6

28. Let $y = \frac{5x}{x^2 + 1}$. Which of the following is an expression for $\frac{dy}{dx}$?

(A)$\frac{5}{2x}$(B)$\frac{5 - 5x^2}{(x^2 + 1)^2}$(C)$\frac{15x^2 + 5}{(x^2 + 1)^2}$(D)$\frac{5x^2 - 5}{(x^2 + 1)^2}$

29. The graph of the function $f$ is a line segment connecting the points $(0, 0)$ and $(2, 4)$. The graph of the function $g$ is a line segment connecting the points $(0, 4)$ and $(4, 0)$. Let $k(x) = \frac{f(x)}{g(x)}$. What is the value of $k'(1)$?

(A)-2(B)-8/9(C)4/9(D)8/9

B. Short Answer Questions (3 Questions)

Answer all parts of each question. Answers must be in essay form. Outlines or lists alone are not acceptable.

Question 30:

Question 31:

Question 32:

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