AP Calculus AB 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: 28 total (25 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 (25 Questions)

Select the one best answer for each question.

1. [Skill: 1.C | Topic: 2.1] The function $f$ is defined 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)$\frac{(1+h)^2 - 3(1+h) - 2}{h}$(C)$\lim_{h \to 0} \frac{(1+h)^2 - 3(1+h) + 2}{h}$(D)$\frac{(1+h)^2 - 3(1+h)}{h}$

2. [Skill: 2.B | Topic: 2.1] $\lim_{x \to 3} \frac{2^x - 8}{x - 3}$ The limit above represents the derivative $f'(c)$ for a function $f$ at a number $c$. Which of the following defines $f(x)$ and $c$?

(A)$f(x) = 2^x$ and $c = 3$(B)$f(x) = 2^x$ and $c = 8$(C)$f(x) = x^3$ and $c = 2$(D)$f(x) = x^3$ and $c = 3$

3. [Skill: 3.A | Topic: 2.1] The graph of a continuous function $f$ passes through the points $P(a, f(a))$ and $Q(b, f(b))$, where $b \neq a$. Which of the following limits represents the slope of the line tangent to the graph of $f$ at $x=a$?

(A)$\lim_{b \to a} \frac{f(b) - f(a)}{b - a}$(B)$\lim_{b \to 0} \frac{f(b) - f(a)}{b - a}$(C)$\frac{f(b) - f(a)}{b - a}$(D)$\lim_{b \to a} \frac{f(b) + f(a)}{b + a}$

4. The function $f$ is defined by $f(x) = 2\sec x - \tan x$ for $-\frac{\pi}{2} < x < \frac{\pi}{2}$. At which value of $x$ does the graph of $f$ have a horizontal tangent?

(A)x = $\frac{\pi}{6}$(B)x = $\frac{\pi}{3}$(C)x = -$\frac{\pi}{6}$(D)x = 0

5. If $f(x) = x^2 \cot x$, what is $f'\left(\frac{\pi}{2}\right)$?

(A)-$\frac{\pi^2}{4}$(B)$\frac{\pi^2}{4}$(C)0(D)-$\pi$

6. Let $y = \frac{\tan x - 1}{\sec x}$. Which of the following is an expression for $\frac{dy}{dx}$?

(A)$\cos $x + $\sin $x(B)$\cos $x - $\sin $x(C)$\sec^2 $x - $\sec $x $\tan $x(D)$\sec $x ($\tan $x + 1)

7. Which of the following expressions is equivalent to the limit $\lim_{h \to 0} \frac{(2+h)^4 - 16}{h}$ ?

(A)4(B)12(C)32(D)The limit does not exist.

8. The graph of a function $f$ passes through the point $(3, 5)$. If $\lim_{x \to 3} \frac{f(x) - f(3)}{x - 3} = -2$, which of the following is an equation of the line tangent to the graph of $f$ at $x = 3$ ?

(A)y = -2x + 1(B)y = -2x + 11(C)y = 3x - 4(D)y = 5x - 17

9. Let $f$ be the function defined by $f(x) = \frac{1}{x+2}$. Which of the following limits represents $f'(x)$ ?

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

10. The function $f$ is continuous and differentiable on the interval $[0, 10]$. Selected values of $x$ and $f(x)$ are shown in the table below. | $x$ | 0 | 2 | 5 | 9 | | :--- | :--- | :--- | :--- | :--- | | $f(x)$ | 3 | 8 | 17 | 27 | Based on the values in the table, what is the best estimate for $f'(4)$?

(A)2.4(B)3(C)9(D)12.5

11. A tank contains water. The volume of water in the tank at time $t$ minutes is given by a differentiable function $W(t)$, where $W(t)$ is measured in gallons. Selected values of $W(t)$ are given in the table below. | $t$ (minutes) | 10 | 20 | 30 | 40 | | :--- | :--- | :--- | :--- | :--- | | $W(t)$ (gallons) | 150 | 142 | 130 | 112 | Using the data in the table, what is the best estimate for the rate at which the volume of water is changing at time $t = 25$ minutes?

(A)-12 gallons per minute(B)-1.2 gallons per minute(C)1.2 gallons per minute(D)12 gallons per minute

12. Let $f$ be the function defined by: $$ f(x) = \begin{cases} cx + d & \text{for } x \le 2 \\ x^2 - cx & \text{for } x > 2 \end{cases} $$ where $c$ and $d$ are constants. If $f$ is differentiable at $x = 2$, what is the value of $c + d$?

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

13. Let $f$ be a function such that $\lim_{h \to 0} \frac{f(5+h) - f(5)}{h} = 3$. Which of the following statements must be true? I. $f$ is continuous at $x = 5$. II. $f$ is differentiable at $x = 5$. III. The derivative of $f$ is continuous at $x = 5$.

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

14. [Skill: 1.E | Topic: 2.5] If $f(x) = 4x^3 - \frac{2}{x^2} + 5$, which of the following expressions represents $f'(x)$?

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

15. [Skill: 1.C | Topic: 2.5] Which of the following is the derivative of $y = \frac{3}{x} + 2\sqrt{x^3}$ ?

(A)$y' = -\frac{3}{x^2} + 3\sqrt{x}$(B)$y' = -\frac{3}{x^2} + \frac{1}{\sqrt{x}}$(C)$y' = 3 + 3x$(D)$y' = -\frac{3}{x^2} + \frac{4}{3\sqrt[3]{x}}$

16. [Skill: 1.C | Topic: 2.6] If $y = \frac{3x^4 - 2x + \sqrt{x}}{x}$, then $\frac{dy}{dx} =$

(A)12x^3 - 2 + $\frac{1}{2\sqrt{x}}$(B)9x^2 - $\frac{1}{2}$x^{-3/2}(C)9x^2 + $\frac{1}{2}$x^{-1/2}(D)3x^3 - 2 + x^{-1/2}

17. [Skill: 3.A | Topic: 2.7] Which of the following is equivalent to the limit $\lim_{h \to 0} \frac{\cos(\frac{\pi}{6} + h) - \cos(\frac{\pi}{6})}{h}$ ?

(A)-$\frac{1}{2}$(B)$\frac{1}{2}$(C)-$\frac{\sqrt{3}}{2}$(D)$\frac{\sqrt{3}}{2}$

18. [Skill: 1.E | Topic: 2.7] If $f(x) = 4e^x - 2\sin x$, what is the value of $f'(0)$ ?

(A)2(B)4(C)6(D)0

19. [Skill: 1.E | Topic: 2.7] Let $g$ be the function defined by $g(x) = 3\ln x + x^2$. What is the instantaneous rate of change of $g$ at $x = 2$ ?

(A)$\frac{3}{2} $+ 4(B)3$\ln $2 + 4(C)$\frac{3}{2} $+ 2(D)$\frac{1}{2} $+ 4

20. Selected values for the differentiable functions $f$ and $g$ and their derivatives are shown in the table below.$\n\n$| $x$ | $f(x)$ | $f'(x)$ | $g(x)$ | $g'(x)$ |$\n$|:---:|:---:|:---:|:---:|:---:|$\n$| 2 | 3 | $-4$ | 1 | 5 |$\n$| 3 | $-1$ | 6 | 4 | $-2$ |$\n\nLet $$h$ be the function defined by $h(x) = f(x)g(x)$. What is the value of $h'(2)$?

(A)$-20$(B)$-19$(C)11(D)15

21. The graphs of the functions $f$ and $g$ consist of line segments. $\nAt $$x=4$, the graph of $f$ passes through the point $(4, 2)$ with a slope of $\frac{1}{2}$. $\nAt $$x=4$, the graph of $g$ passes through the point $(4, -3)$ with a slope of $-1$. $\n\nIf $$p(x) = f(x) \cdot g(x)$, what is the value of $p'(4)$?

(A)$-3.5$(B)$-0.5$(C)0.5(D)3.5

22. Let $f$ be a differentiable function. The line tangent to the graph of $f$ at $x = -1$ is given by the equation $y = 3x + 5$. $\n\nLet $$h$ be the function defined by $h(x) = x^2 f(x)$. What is the value of $h'(-1)$?

(A)$-6$(B)$-1$(C)2(D)7

23. Let $f$ be the function defined by $f(x) = \frac{x^2 - 3}{x - 1}$. Which of the following is an expression for $f'(x)$ ?

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

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

(A)$\frac{4}{5}$(B)$\frac{2}{9}$(C)$\frac{22}{9}$(D)-$\frac{22}{9}$

25. If $y = \frac{\cos x}{x^3}$, then $\frac{dy}{dx} =$

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

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 26:

Question 27:

Question 28:

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