AP Calculus AB Unit 3 Practice Test & Exam | 18 Questions

Estimated Time: 60 minutes

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

(A)$-6$(B)$-4$(C)$-3$(D)8

2. Let $f$ be a differentiable function such that $f(1) = 2$ and $f'(1) = -3$. Let $g$ be the function defined by $g(x) = 2x^3 - 3x^2 + 2$. If $k(x) = f(g(x))$, what is the equation of the line tangent to the graph of $k$ at $x=1$?

(A)$y - 2 = -3(x - 1)$(B)$y - 2 = 0$(C)$y - 2 = -9(x - 1)$(D)$y - 2 = -18(x - 1)$

3. Let $h$ be the function defined by $h(x) = f(x^2)$, where $f$ is a twice-differentiable function. Which of the following is an expression for $h''(x)$?

(A)$2f'(x^2) + 4x^2 f''(x^2)$(B)$4x^2 f''(x^2)$(C)$2x f''(x^2)$(D)$f''(x^4)$

4. If $\sin(x+y) = 2x$, then $\frac{dy}{dx} =$

(A)2 $\sec$(x+y) - 1(B)2 $\sec$(x+y)(C)2 $\cos$(x+y) - 1(D)$\frac{2 - \cos(x+y)}{\sin(x+y)}$

5. Consider the curve defined by the equation $x^2 + 3xy - y^2 = 3$. What is the slope of the tangent line to the curve at the point $(1, 2)$?

(A)-8(B)2(C)8(D)10

6. If $y^2 - x^2 = 16$ and $y > 0$, which of the following is an expression for $\frac{d^2y}{dx^2}$ in terms of $y$?

(A)-$\frac{16}{y^3}$(B)$\frac{16}{y^3}$(C)$\frac{y^2 - x^2}{y^3}$(D)-$\frac{x}{y^2}$

7. The table below gives selected values for a differentiable and one-to-one function $f$ and its derivative $f'$. | $x$ | $f(x)$ | $f'(x)$ | | :---: | :---: | :---: | | 1 | -3 | 4 | | 2 | 1 | 5 | | 3 | 2 | -2 | If $g$ is the inverse function of $f$, what is the value of $g'(1)$?

(A)1/5(B)1/4(C)5(D)-1/2

8. Let $f$ be the function defined by $f(x) = x^3 + 4x - 2$. If $g(x) = f^{-1}(x)$, what is the value of $g'(3)$?

(A)1/31(B)1/7(C)1/3(D)7

9. The function $f$ is differentiable and one-to-one. The line $y = 3x - 1$ is tangent to the graph of $f$ at $x=2$. Let $g$ be the inverse function of $f$. What is the value of $g'(5)$?

(A)3(B)1/3(C)-1/3(D)1/5

10. Let $f$ be the function given by $f(x) = \arcsin(5x)$. Which of the following is an expression for $f'(x)$?

(A)$\frac{1}{\sqrt{1-25x^2}}$(B)$\frac{5}{\sqrt{1-25x^2}}$(C)$\frac{5}{\sqrt{1-5x^2}}$(D)$-\frac{5}{\sqrt{1-25x^2}}$

11. If $y = \arctan(\sqrt{x})$, then $\frac{dy}{dx} =$

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

12. Let h(x) = f(x^2) $\cdot $g(x) . It is known that f and g are differentiable functions. Which of the following is the correct expression for h'(x) ?

(A) f'(x^2) $\cdot $g'(x) (B) f'(x^2) $\cdot $2x $\cdot $g(x) + f(x^2) $\cdot $g'(x) (C) f'(x^2) $\cdot $g(x) + f(x^2) $\cdot $g'(x) (D) f'(x^2) $\cdot $2x $\cdot $g'(x)

13. Consider the curve defined by the equation x^2 y - $\sin$(y) = 2x . Which of the following procedures correctly leads to an expression for $\frac{dy}{dx} $?

(A)Differentiate implicitly to get 2xy - $\cos$(y) = 2 , then solve for y .(B)Differentiate implicitly to get 2x $\frac{dy}{dx} $- $\cos$(y) $\frac{dy}{dx} $= 2 , then solve for $\frac{dy}{dx} $.(C)Apply the product rule to x^2 y and the chain rule to $\sin$(y) to get 2xy + x^2 $\frac{dy}{dx} $- $\cos$(y) $\frac{dy}{dx} $= 2 , then solve for $\frac{dy}{dx} $.(D)Solve the equation for y explicitly as a function of x , then apply the quotient rule.

14. The table below gives values for the differentiable functions $f$ and $g$ and their first and second derivatives at selected values of $x$. If $h(x) = f(g(x))$, what is the value of $h''(2)$? | $x$ | $f(x)$ | $f'(x)$ | $f''(x)$ | $g(x)$ | $g'(x)$ | $g''(x)$ | |:---:|:---:|:---:|:---:|:---:|:---:|:---:| | 2 | 3 | $-1$ | 0 | 3 | $-2$ | 1 | | 3 | 5 | 4 | $-2$ | 1 | 0 | 4 |

(A)$-8$(B)$-4$(C)$-2$(D)0

15. Consider the curve defined by the equation $y^2 - x^2 = 16$. Which of the following expressions represents $\frac{d^2y}{dx^2}$ in terms of $y$?

(A)$\frac{16}{y^3}$(B)$-\frac{16}{y^3}$(C)$\frac{y^2 - x^2}{y^2}$(D)$\frac{1}{y}$