Free Printable Review Derivatives unit 3 test

1.

f'(x)=

a)

$\lim_{h\rightarrow0}\left(\frac{f\left(x+h\right)-f\left(x\right)}{h}\right)$

b)

$\lim_{x\rightarrow c}\left(\frac{f\left(x\right)-f\left(c\right)}{x-c}\right)$

c)

$\frac{\text{d}y}{\text{d}x}$

d)

an equation for the slope of the tangent line

2.

Use the formal definition of a derivative to find f '(x) given:

f\left(x\right)=\sqrt{x-5}

a)

$f'\left(x\right)\ =\ \left(x-5\right)^{\frac{1}{2}}$

b)

$f'\left(x\right)=\frac{1}{2}\left(x-5\right)^{-\frac{1}{2}}$

c)

$f'\left(x\right)=\frac{1}{2\sqrt{x-5}}$

d)

$\lim_{h\rightarrow0}\ \frac{\sqrt{x+h-5}-\sqrt{x-5}}{h}$

3.

How do you rewrite this in power form:

f\left(t\right)=\frac{1}{t}-\frac{3}{t^2}+\frac{5}{t^3}

a)

$f\left(t\right)=t^{-1}-3t^{-2}+5t^{-3}$

b)

$f\left(t\right)=-t^{-2}+6t^{-3}-15t^{-4}$

c)

$f\left(t\right)=-t^{-2}+t^{-3}-t^{-4}$

d)

$f\left(t\right)=\ t^{-2}-6t^{-3}+15t^{-4}$

4.

Find $\frac{dy}{dt}$
$f\left(t\right)\ =\ \frac{1}{t}-\frac{3}{t^2}+\frac{5}{t^3}$

a)

$\frac{dy}{dt}=-t^2-6t^3-15t^4$

b)

$\frac{dy}{dt}=t^{-1}-3t^{-2}+5t^{-3}$

c)

$\frac{dy}{dt}=-\frac{1}{t^2}+\frac{6}{t^3}-\frac{15}{t^4}$

d)

$\frac{dy}{dt}=-t^{-2}-6t^{-3}-15t^{-4}$

5.

.

a)

Only the Chain Rule

b)

The quotient & the Chain Rules

c)

Only the power rule

d)

Product & Quotient Rules

6.

Find the derivative f(x) = xe^x

a)

f'(x) = e^x

b)

f'(x) = xe^x + xe^x

c)

f'(x) = e^x - xe^x

d)

f'(x) = e^x + xe^x

7.

Find the equation of the normal line when f(x) = xe^x and x = 1

a)

$y\ =\ 2e\left(x-1\right)+e$

b)

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

c)

$y\ =\ e\left(x-1\right)+e$

d)

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

8.

Find the derivative of

g\left(x\right)=\frac{3x-2}{x^2+2}

a)

$9x^2+2$

b)

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

c)

$\frac{-3x^2+4x+6}{\left(x^2+2\right)^2}$

d)

$\frac{-3x^2+10}{\left(x^2+2\right)^2}$

9.

$y\ =\ e^{\pi}\ -\ xe\ +\ x^e-e^x-\pi$

Find y ' given

a)

$y\ '\ =\ \pi e^{\pi-1}-x+ex^{e-1}-xe^{x-1}$

b)

$y\ '\ =\ \pi e^{\pi-1}-x+ex^{e-1}-e^x$

c)

$y\ '\ =-e+ex^{e-1}-xe^{x-1}$

d)

$y\ '\ =\ -e+ex^{e-1}-e^x$

10.

$f\left(x\right)=\frac{\cos x}{\sec x}$

Find the derivative of

a)

$f'\left(x\right)=-2\sin x\cos x$

b)

$f'\left(x\right)=\frac{-\sin x}{\sec x\tan x}$

c)

$f'\left(x\right)=\cos^2x$

d)

$f'\left(x\right)=2\left(-\sin x\right)$

11.

Find the 3rd derivative if

y=3x^4-12x^3+11x^2-8x+10

a)

$\frac{d^3y}{dx^3}=36x^2-72x+22$

b)

$\frac{d^3y}{dx^3}=12x^3-36x^2+22x-8$

c)

$\frac{d^3y}{dx^3}=72x\ -\ 72$

d)

$\frac{d^3y}{dx^3}=72$

e)

$\frac{d^3y}{dx^3}=0$

12.

If $y=3\sin x+4\cos x$ , what is $\frac{\text{d}^{120}y}{\text{d}x^{120}}$ ?

a)

$\frac{\text{d}^{120}y}{\text{d}x^{120}}=3\cos x-4\sin x$

b)

$\frac{\text{d}^{120}y}{\text{d}x^{120}}=-3\sin x-4\cos x$

c)

$\frac{\text{d}^{120}y}{\text{d}x^{120}}=-3\cos x+4\sin x$

d)

$\frac{\text{d}^{120}y}{\text{d}x^{120}}=3\sin x+4\cos x$

e)

$\frac{\text{d}^{120}y}{\text{d}x^{120}}=4\sin x-3\cos x$

13.

What type of discontinuity does f '(-2) have?

a)

Jump discontinuity

b)

removable discontinuity

c)

infinite discontinuity

d)

vertical asymptote

14.

Given the graph f(x), determine which statement is true below.

a)

$f'(-3.5)>f'\left(-\frac{1}{2}\right)$

b)

$f'(-3.5)<f'\left(-\frac{1}{2}\right)$

c)

$f'\left(3.5\right)\ \&\ f'\left(-\frac{1}{2}\right)\ >\ 0$

d)

$f'(-3.5)\ \&\ f'\left(-\frac{1}{2}\right)<0$

e)

$f'(-3.5)\ \ \&\ \ f'\left(-\frac{1}{2}\right)=0$

15.

Match the correct derivative graph to this function.

a)

b)

c)

d)

16.

The position of an object is given as a function of time by S(t) = 3t² + 5t³- 2t

What is the acceleration of the object at time t = 2 s?

a)

64 m/s/s

b)

60 m/s/s

c)

66 m/s/s

d)

70 m/s/s

17.

The velocity of an object is given as v(t) = 2t + 3t³. what is the acceleration of the object at t = 2 s?

a)

38 m/s/s

b)

27 m/s/s

c)

16 m/s/s

d)

49 m/s/s

18.

Which of the following is a true statement?

a)

$\lim_{x\rightarrow1^+}g\left(x\right)\ =\ 5$
$\lim_{x\rightarrow1^+}g'\left(x\right)\ =\ 4$
$g\left(1\right)=7$

b)

$\lim_{x\rightarrow1^+}g\left(x\right)\ =\ 7$
$\lim_{x\rightarrow1^+}g'\left(x\right)\ =\ 2$
$g\left(1\right)=7$

c)

$\lim_{x\rightarrow1^+}g\left(x\right)\ =\ 2$
$\lim_{x\rightarrow1^+}g'\left(x\right)\ =\ -1$
$g\left(1\right)=7$

19.

Where is the graph continuous but not differentiable?

a)

@ x = 1

b)

@ x = 2

c)

@ x = 3

d)

@ x = 4

20.

$f\left(x\right)=x^{\frac{2}{3}}$

Tell whether the function has a corner, cusp, vertical tangent, or discontinuity @ x = 0. Show algebraically.

(a)

Review Derivatives unit 3 test

Total questions: 20

Worksheet time: 20mins

Similar Resources on Wayground