Free Printable Unit 3 Review

1.

Find $\frac{dy}{dx}$ for $y=x^3\sqrt{2x+1}$ .

a)

$3x^2\left(2x+1\right)^{\frac{1}{2}}+x^3\cdot\left(2x+1\right)^{-\frac{1}{2}}$

b)

$3x^2\cdot\frac{1}{2}\left(x+1\right)^{-\frac{1}{2}}$

c)

$3x^2\cdot\frac{1}{2}\left(x+1\right)^{-\frac{1}{2}}+x^3\left(x+1\right)^{\frac{1}{2}}$

d)

$3x^2\left(2x+1\right)^{\frac{1}{2}}+x^3\cdot\frac{1}{2}\left(2x+1\right)^{-\frac{1}{2}}$

2.

Find $f'\left(x\right)$ if $f\left(x\right)=\cos^5\left(4x\right)$

a)

$-20\cos^4\left(4x\right)\sin\left(4x\right)$

b)

$-4\sin^5\left(4x\right)\cos\left(4x\right)$

c)

$-20\cos\left(4x\right)\sin x$

d)

$-20\sin^4\left(4x\right)$

3.

Find $f'\left(\theta\right)$ if $f\left(\theta\right)=\sqrt{\sin\left(2\theta\right)}$

a)

$\frac{\cos\left(2\theta\right)}{\sqrt{\sin\left(2\theta\right)}}$

b)

$\frac{1}{2}\tan\left(2\theta\right)$

c)

$-\frac{\cos\left(2\theta\right)}{2\sqrt{\sin\left(2\theta\right)}}$

d)

$\frac{2}{\sqrt{\cos\left(2\theta\right)}}$

4.

Given $f\left(x\right)=x^2\left(2x-5\right)^2$ , find f'(x).

Hint: You're going to have to get creative with your factoring to find the answer.

a)

$2x\left(2x-5\right)\left(4x-5\right)$

b)

$2x\left(4x+5\right)\left(2x-5\right)$

c)

$2x\left(x+1\right)\left(4x+5\right)$

d)

$2x\left(x-1\right)\left(4x-5\right)$

5.

Find $y''$ for $y=\frac{\csc x}{2}$

a)

$\frac{1}{2}\csc x\left(\cot^2x+\csc^2x\right)$

b)

$-\frac{1}{2}\csc x\cot x$

c)

$-\frac{1}{2}\csc^3x\cot x$

d)

$-\frac{1}{2}\csc x\left(\cot^2x+\csc^2x\right)$

6.

Find $y'$ for $y=\ln\left(\sqrt{x^2-4x-7}\right)$

a)

$\frac{x-2}{x^2-4x-7}$

b)

$\frac{2x-4}{\sqrt{x^2-4x-7}}$

c)

$\frac{x-2}{\sqrt{x^2-4x-7}}$

d)

$\frac{2x-4}{x^2-4x-7}$

7.

Find $y'$ for $y=-4e^{\sec x}$

a)

$-4e^{\sec x\tan x}$

b)

$-4e^{\sec x}\sec x\tan x$

c)

$-4e^{\sec^2x\tan x}$

d)

$-4e^{\sec x}\tan^2x$

8.

Find $y'$ for $y=3^{5x}$

a)

$3^{5x\ln3}\cdot5$

b)

$3^{5x}\ln3\cdot5$

c)

$15^{5x}\ln3$

d)

$3^{5x}\ln5$

9.

Suppose that $h(x)=f(g(x))$ . If $g(14)=2$ , $g'(14)=5$ , $f'(14)=15$ , and $f'(2)=12$ . Find $h'(14)$ .

a)

60

b)

12

c)

75

d)

24

10.

Suppose f and g and their derivatives have the values given in the table. Find the derivative of $g\left(f\left(x\right)\right)$ at $x=-1$

a)

8

b)

-12

c)

4

d)

3

11.

Suppose f and g and their derivatives have the values given in the table. Find $h'\left(-1\right)$ given $h\left(x\right)=f\left(g\left(x\right)\right)$

a)

2

b)

-1

c)

-12

d)

8

12.

Suppose f and g and their derivatives have the values given in the table. Find $h'\left(0\right)$ given $h\left(x\right)=g\left(x+f\left(x\right)\right)$ .

a)

-2

b)

-1

c)

0

d)

1

13.

Find $f''(x)$ if $f\left(x\right)=\tan\left(5x\right)+x^2$

a)

$\sec^2\left(5x\right)+2x$

b)

$10\sec^2\left(5x\right)\tan x\left(5x\right)+2$

c)

$50\sec\left(5x\right)\tan\left(5x\right)+2x$

d)

$50\sec^2\left(5x\right)\tan\left(5x\right)+2$

14.

The table gives selected values of f, g, and their derivatives. "a" is a constant. If $h\left(x\right)=\sin\left(f\left(x\right)\right)$ , write an equation of the tangent line to h at $x=1$ .

a)

$y-\frac{\sqrt{2}}{2}=-\left(x-1\right)$

b)

$y-\frac{\sqrt{2}}{2}=-\frac{\sqrt{2}}{2}\left(x-1\right)$

c)

$y+\frac{\sqrt{2}}{2}=\frac{\sqrt{2}}{2}\left(x-1\right)$

d)

$y-\frac{\sqrt{2}}{2}=-\frac{1}{2}\left(x-1\right)$

15.

The table gives selected values of f, g, and their derivatives. "a" is a constant. If $r\left(x\right)=\frac{1}{\sqrt{g\left(2x\right)}}$ , find $r'(2)$ .

a)

$-\frac{1}{9}$

b)

$-\frac{1}{2}$

c)

$-\frac{1}{6}$

d)

-1

16.

Find $\frac{dy}{dx}$ if $ye^x-x=y^2$

a)

$\frac{1-ye^x}{e^x-2y}$

b)

$\frac{2y+1}{e^x}$

c)

$-\frac{1}{e^x-2y}$

d)

$\frac{e^x+2y}{1+ye^x}$

17.

Given the curve $2x+y^2-xy=4$ , find the slope of one of the tangent lines when x=1.

a)

0

b)

$\frac{1}{3}$

c)

$\frac{1}{2}$

d)

-2

18.

Given $x^2+y^2-xy=4$ , find $\frac{dy}{dx}$ .

a)

$\frac{y-2x}{2y-x}$

b)

$-\frac{2x}{2y-1}$

c)

$\frac{4-2x}{2y-1}$

d)

$-2x-2y$

19.

Consider the graph of $x^2+xy+y^2=1$ . Find $\frac{dy}{dx}$ .

a)

$-\frac{2x+y}{x+2y}$

b)

$-\frac{2x}{2y+1}$

c)

$\frac{1-2x}{2y+1}$

d)

$-\frac{2x-y}{-x+2y}$

20.

Given $x^2+xy+y^2=1$ and its derivative $\frac{dy}{dx}=\frac{-2x-y}{x+2y}$ , what is $\frac{d^2y}{dx^2}$ for the point $\left(1,-1\right)$ ?

a)

2

b)

4

c)

0

d)

6

21.

Given $y^2+y+2x^3=0$ , find $\frac{d^2y}{dx^2}$ for the point $\left(-1,-2\right)$

a)

$-\frac{4}{3}$

b)

$\frac{1}{2}$

c)

$-\frac{10}{3}$

d)

$\frac{28}{9}$

22.

Given $f(4)=6$ , $f'(4)=7$ , $f(6)=10$ , and $f'(6)=-5$ . What is $\left(f^{-1}\right)'\left(10\right)$ ?

a)

$-\frac{1}{5}$

b)

$\frac{1}{7}$

c)

$\frac{1}{6}$

d)

$\frac{1}{10}$

23.

Find $\left(f^{-1}\right)'\left(-12\right)$ for the function $f\left(x\right)=\frac{1}{3}x^3+\frac{5}{3}x+2$ given $f\left(-3\right)=-12$

a)

$\frac{3}{32}$

b)

$\frac{5}{16}$

c)

$\frac{8}{3}$

d)

$\frac{437}{3}$

24.

Let $f\left(x\right)=\sqrt{e^x+3x+8}$ and let g be the inverse of f. Given $f(0)=3$ , what is the value of $g'(3)$ ?

a)

$\frac{3}{2}$

b)

$\frac{2}{3}$

c)

$\frac{\sqrt{e}}{8}$

d)

$\frac{8}{\sqrt{e}}$

25.

Find the derivative of the function $f\left(x\right)=\arccos\left(3x^2-1\right)$

a)

$-\frac{6x}{\sqrt{1+\left(3x^2-1\right)^2}}$

b)

$-\frac{3x^2-1}{\sqrt{1-\left(3x^2-1\right)^2}}$

c)

$-\frac{6x}{\sqrt{\left(3x^2-1\right)^2-1}}$

d)

$-\frac{6x}{\sqrt{1-\left(3x^2-1\right)^2}}$

26.

Find the derivative of the function $f\left(x\right)=x^2\arctan\left(5x\right)$

a)

$\frac{10x}{1+25x^2}$

b)

$\frac{10x^2}{1+25x^2}$

c)

$2x\arctan\left(5x\right)+\frac{5x^2}{1+25x^2}$

d)

$2x\arctan\left(5x\right)+\frac{5x^2}{1+5x^2}$

27.

If $f\left(x\right)=\arcsin x-2x$ , find $f'(x)$

a)

$\frac{1}{\sqrt{1-x^2}}-2$

b)

$\frac{x}{\sqrt{1-x^2}}-2$

c)

$\frac{1}{\sqrt{1-x^2}}-2x$

d)

$\frac{x}{\sqrt{1-x^2}}-2x$

28.

If $f\left(x\right)=\arcsin x-2x$ , find the equation of the tangent line to $f(x)$ at $x=0$

a)

$y=-x$

b)

$y=x$

c)

$y=\frac{\sqrt{3}}{2}x+1$

d)

$y=-\frac{\sqrt{3}}{2}x-2$

29.

If $s\left(t\right)=8t^2+24t$ , what is $s''\left(3\right)$ ?

a)

48

b)

16

c)

30

d)

0

30.

If $y=xe^x$ , which of the following would represent $\frac{d^ny}{dx^n}$ ?

Hint: Start taking derivatives and look for a pattern.

a)

$ne^x+xe^x$

b)

$x^ne^x$

c)

$e^x+nxe^x$

d)

$e^x$

31.

Find $h''\left(x\right)$ if $h\left(x\right)=f\left(x^3\right)$

a)

$9x^4f''\left(x^3\right)+6xf'\left(x^3\right)$

b)

$f''\left(6x\right)$

c)

$6xf''\left(x^3\right)$

d)

$3x^2f''\left(x^3\right)+6xf'\left(x^3\right)$

32.

What is the 20th derivative of $y=\sin\left(2x\right)$

a)

$\sin\left(2x\right)$

b)

$2^{20}\sin\left(2x\right)$

c)

$-2^{19}\cos\left(2x\right)$

d)

$-\cos\left(2x\right)$

33.

$\frac{d}{dx}\arccos\left(\sqrt[]{x}\right)$

a)

$-\frac{1}{\sqrt[]{1-x}}$

b)

$-\frac{1}{\sqrt[]{x-x^2}}$

c)

$-\frac{1}{2\sqrt[]{x-x^2}}$

d)

$-\frac{1}{2\sqrt[]{1-x}}$

34.

The derivative for the curve $x^2=-2+y+5\cos y$ is $\frac{dy}{dx}=\frac{2x}{1-5\sin y}$ .

a)

True

b)

False

35.

If $f\left(x\right)=\sin\left(x^2+\pi\right)$ , then $f'\left(\sqrt{2\pi}\right)$ =

a)

$-2\sqrt{2\pi}$

b)

-2

c)

-1

d)

$\cos\left(2\sqrt{2\pi}\right)$

36.

$\frac{\text{d}}{\text{d}x}\left[e^{x^2}\right]$

a)

$2xe^{x^2}$

b)

$x^2e^{x^2}$

c)

$e^{x^2}$

d)

$2xe^{2x}$

37.

$\frac{\text{d}}{\text{d}x}\left[\log_2\left(ex\right)\right]$

a)

$\frac{1}{x\ln2}$

b)

$\frac{e}{x\ln2}$

c)

$\frac{2}{ex}$

d)

$\frac{e}{2\ln x}$

38.

$\frac{\text{d}}{\text{d}x}\left[\ln\left(x^2-1\right)\right]$

a)

$\frac{2x}{x^2-1}$

b)

$\frac{1}{x^2-1}$

c)

$\frac{1}{2x}$

d)

$\frac{2}{x-1}$

39.

$\frac{\text{d}}{\text{d}x}\left[\arctan\left(\pi x\right)\right]$

a)

$\frac{\pi}{\pi^2+x^2}$

b)

$\frac{\pi}{1+\pi^2x^2}$

c)

$\frac{\pi}{\sqrt{1-\pi^2x^2}}$

d)

$\frac{1}{\sqrt{\pi^2-x^2}}$

40.

$\frac{\text{d}}{\text{d}x}\left[\arcsin\left(\pi x\right)\right]$

a)

$\frac{\pi}{\pi^2+x^2}$

b)

$\frac{\pi}{1+\pi^2x^2}$

c)

$\frac{\pi}{\sqrt{1-\pi^2x^2}}$

d)

$\frac{1}{\sqrt{\pi^2-x^2}}$

41.

$\frac{\text{d}}{\text{d}x}\left[2^{ex}\right]$

a)

$2xe^{x^2}\left(x^2+1\right)$

b)

$e\left(2^{ex}\right)\ln2$

c)

$2e^{x^2}\left(1+2x^2\right)$

d)

$\frac{2e^{x^2}\left(x^2-1\right)}{x^3}$

42.

$\frac{\text{d}}{\text{d}x}\left[e^{\pi}\right]$

a)

$\pi e^{\pi}$

b)

$\pi e^{\left(\pi-1\right)}$

c)

$e^{\pi}$

d)

0

43.

If $y=\left(x^3+1\right)^2$ , then $\frac{\text{d}y}{\text{d}x}=$

a)

$\left(3x^2\right)^2$

b)

$2\left(x^3+1\right)$

c)

$2\left(3x^2+1\right)$

d)

$3x^2\left(x^3+1\right)$

e)

$6x^2\left(x^3+1\right)$

44.

If $f\left(x\right)=\ln\left(x+4+e^{-3x}\right)$ , then $f'\left(0\right)$ is

a)

-2/5

b)

1/5

c)

1/4

d)

2/5

e)

nonexistent

45.

What is the slope of the tangent to the curve $3y^2-2x^2=6-2xy$ at the point (3,2)?

a)

0

b)

4/9

c)

7/9

d)

6/7

e)

5/3

46.

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

a)

1/13

b)

1/4

c)

7/4

d)

4

e)

13

47.

If $f\left(x\right)=e^{\frac{2}{x}}$ , then $f'\left(x\right)=$

a)

$2e^{\frac{2}{x}}\ln x$

b)

$e^{\frac{2}{x}}$

c)

$e^{-\frac{2}{x^2}}$

d)

$-\frac{2}{x^2}e^{\frac{2}{x}}$

e)

$-2x^2e^{\frac{2}{x}}$

48.

If $\sin\left(xy\right)=x$ , then $\frac{dy}{dx}=$

a)

$\frac{1}{\cos\left(xy\right)}$

b)

$\frac{1}{x\cos\left(xy\right)}$

c)

$\frac{1-\cos\left(xy\right)}{\cos\left(xy\right)}$

d)

$\frac{1-y\cos\left(xy\right)}{x\cos\left(xy\right)}$

e)

$\frac{y\left(1-\cos\left(xy\right)\right)}{x}$

49.

What is the slope of the line tangent to the curve $y=\arctan\left(4x\right)$ at the point at which x=1/4?

a)

2

b)

1/2

c)

0

d)

-1/2

e)

-2

50.

If $f\left(x\right)=\cos\left(3x\right)$ , then $f'\left(\frac{\pi}{9}\right)=$

a)

$\frac{3\sqrt{3}}{2}$

b)

$\frac{\sqrt{3}}{2}$

c)

$-\frac{\sqrt{3}}{2}$

d)

$-\frac{3}{2}$

e)

$-\frac{3\sqrt{3}}{2}$

51.

If $\sqrt{x}+y^2=xy+2$ , what is $\frac{dy}{dx}$ at the point (4,0)?

a)

-1/16

b)

1/16

c)

-1/4

d)

1/4

52.

For any real number $\lim_{h\rightarrow0}\frac{\cos\left(x+h\right)^2-\cos\left(x^2\right)}{h}$ =

a)

$\cos\left(x^2\right)$

b)

$2x\cos\left(x^2\right)$

c)

$-\sin\left(x^2\right)$

d)

$-2x\sin\left(x^2\right)$

53.

$\frac{d}{dx}\left(\tan\left(\ln x\right)\right)=$

a)

$\frac{\tan\left(\ln x\right)}{x}$

b)

$\sec^2\left(\ln x\right)$

c)

$\frac{\sec^2\left(\ln x\right)}{x}$

d)

$\tan\left(\frac{1}{x}\right)$

54.

If $f\left(x\right)=e^{3x}\left(x^2+2\right)$ then $f'\left(2\right)=$

a)

$4e^6$

b)

$12e^6$

c)

$18e^6$

d)

$22e^6$

55.

The $\lim_{h\rightarrow0}\frac{\tan\left[3\left(x+h\right)\right]-\tan\left(3x\right)}{h}$ is

a)

0

b)

$3\sec^2\left(3x\right)$

c)

$\sec^2\left(3x\right)$

d)

$3\cot\left(3x\right)$

e)

DNE

56.

If $y=2\cos\left(\frac{x}{2}\right)$ , then $\frac{d^2y}{dx^2}=$

a)

$-8\cos\left(\frac{x}{2}\right)$

b)

$-2\cos\left(\frac{x}{2}\right)$

c)

$-\sin\left(\frac{x}{2}\right)$

d)

$-\cos\left(\frac{x}{2}\right)$

e)

$-\frac{1}{2}\cos\left(\frac{x}{2}\right)$

57.

If $f\left(x\right)=\sin\left(e^{-x}\right)$ , then $f'\left(x\right)=$

a)

$-\cos\left(e^{-x}\right)$

b)

$\cos\left(e^{-x}\right)+e^{-x}$

c)

$\cos\left(e^{-x}\right)-e^{-x}$

d)

$e^{-x}\cos\left(e^{-x}\right)$

e)

$-e^{-x}\cos\left(e^{-x}\right)$

58.

An equation of the line tangent to the graph of $f\left(x\right)=x\left(1-2x\right)^3$ at the point $\left(1,-1\right)$ is

a)

$y=-7x+6$

b)

$y=-6x+5$

c)

$y=-2x$

d)

$y=2x-3$

e)

$y=7x-8$

59.

If $y=\left(x^3-\cos x\right)^5$ , then $y'=$

a)

$5\left(x^3-\cos x\right)^4$

b)

$5\left(3x^2+\sin x\right)^4$

c)

$5\left(3x^2+\sin x\right)$

d)

$5\left(x^3-\cos x\right)^4\left(6x+\cos x\right)$

e)

$5\left(x^3-\cos x\right)^4\left(3x^2+\sin x\right)$

60.

If $y=\arcsin\left(5x\right)$ , then $\frac{dy}{dx}=$

a)

$\frac{1}{1+25x^2}$

b)

$\frac{5}{1+25x^2}$

c)

$-\frac{5}{\sqrt[]{1-25x^2}}$

d)

$\frac{1}{\sqrt[]{1-25x^2}}$

e)

$\frac{5}{\sqrt[]{1-25x^2}}$

61.

If $x+2xy-y^2=2$ , then the at the point (1,1) $\frac{dy}{dx}$ is

a)

$\frac{3}{2}$

b)

$\frac{1}{2}$

c)

0

d)

$-\frac{3}{2}$

e)

DNE

62.

If $x^2y-3x=y^3-3$ , then at the point (-1,2), $\frac{dy}{dx}=$

a)

$-\frac{7}{11}$

b)

$-\frac{7}{13}$

c)

$-\frac{1}{2}$

d)

$-\frac{3}{14}$

e)

7

Unit 3 Review

Total questions: 62

Worksheet time: 3hrs 6mins

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