Free Printable Derivatives Chapter 3

1.

Find $\frac{dy}{dx}$ .

y=\frac{3}{2}x^5+2x^3-\frac{5}{3}x^2

a)

$\frac{dy}{dx}=\frac{15}{2}x^4+6x^2-\frac{10}{3}x^{ }$

b)

$\frac{dy}{dx}=\frac{15}{2}x^2+6x^3-\frac{10}{3}x^{ }$

c)

$\frac{dy}{dx}=\frac{3}{2}x^4+2x^2-\frac{5}{3}x$

d)

$\frac{dy}{dx}=\frac{15}{2}x+6x-\frac{10}{3}x^{ }$

2.

Find $\frac{dy}{dx}$

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

a)

$\frac{dy}{dx}=60x^3+48x$

b)

$\frac{dy}{dx}=30x^3+18x$

c)

$\frac{dy}{dx}=3x^2+6x+3$

d)

$\frac{dy}{dx}=15x^4+84x^2+9$

3.

Find $\frac{dy}{dx}$ .

$y=\frac{5x^2+4}{4x^2+5}$

a)

$\frac{dy}{dx}=\frac{18x}{\left(4x^2+5\right)^2}$

b)

$\frac{dy}{dx}=\frac{18x}{4x^2+5}$

c)

$\frac{dy}{dx}=\frac{40x^3+32x}{\left(4x^2+5\right)^2}$

4.

Find y'.

$y=5x^2\sec x$

a)

$y'=10x\sec x+5x^2\sec x\tan x$

b)

$y'=5x\sec^2x+10x\sec x\tan x$

c)

$y'=10x\sec x\tan x$

d)

$y'=5x^2\sec x+10x\sec x\tan x$

5.

$y=x^2\sin x\ +\ \cos x$

Find y'.

a)

$y'=\left(2x-1\right)\sin x+x^2\cos x$

b)

$y'=x\sin x+\cos x$

c)

$y'=2x\cos x-\sin x$

d)

$y'=2x\sin x+\cos x-x^2\sin x$

6.

Find the derivative.

$y=\frac{1}{x^2}$

a)

$y'=-2x$

b)

$y'=-\frac{2}{x^3}$

c)

$y'=2x$

d)

$y'=\frac{x}{2}$

7.

Find the derivative.

y = x⁴ tan x

a)

y' = x⁴sec²x - 4x³tanx

b)

y' = x⁴sec²x + 4x³tanx

c)

y' = 4x³sec²x

d)

y' = -4x³ tanx

8.

Find the derivative y = tanxcosx

a)

y ' = sec²xcosx - tanxsinx

b)

y ' = sec²xcosx + tanxsinx

c)

y ' = sec²xsinx

d)

y ' = sec²xcosx - tanxcosx

9.

Find the derivative

y = 5sinx + 3x³cosx

a)

y' = -5cosx − 3x³sinx + 9x²cosx

b)

y' = 5sinx − 3x³sinx + 9x²cosx

c)

y' = 5cosx − 3x³sinx + 9x²cosx

d)

y' = 5cosx + 3x³sinx + 9x²cosx

10.

A particle moves along a horizontal line. It's position function, s(t) = t³ -11t² + 24t , measured in feet. Find the displacement over interval of time 1 < t < 5 seconds.

a)

-44 feet

b)

116 feet

c)

236 feet

d)

-176 feet

11.

If you shoot a paper clip straight up in the air, the paper clip will be s(t) = 64t - 16t² feet above your hand at t seconds after firing. How long does it take the paper clip to reach its maximum height?

a)

2 seconds

b)

1.8 seconds

c)

3 seconds

d)

2.4 seconds

12.

If you shoot a paper clip straight up in the air, the paper clip will be s(t) = 64t - 16t² feet above your hand at t seconds after firing.

If the paper clip reaches maximum height 2 seconds after being launched, what is the maximum height?

a)

64 feet

b)

32 feet

c)

128 feet

d)

16 feet

13.

Find the slope of the curve y = sinx cosx when x = π.

a)

1

b)

-1

c)

0

d)

2

14.

Find the equation of the tangent line to the curve y = 8x^-² at x = 2

a)

y - 2 = -2(x - 2)

b)

y - 32 = -2(x - 2)

c)

y - 2 = 2( x - 2)

d)

y + 2 = -2(x - 2)

15.

Find the coordinate points where the tangent line(s) are parallel to the x-axis, for the function y = x³ - 3x² + 6.

a)

(-2, -14) and (0, 6)

b)

(0, 6) and (1, 4)

c)

No tangents parallel to the x-axis exist.

d)

(0, 6) and (2, 2)

16.

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

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

a)

64 m/sec²

b)

60 m/sec⁹

c)

66 m/sec²

d)

70 m/sec²

17.

The average rate of change is the same thing as

a)

the slope between two points.

b)

the derivative.

c)

marginal cost.

d)

the cost of a stamp.

18.

The instantaneous rate of change is the same things as

a)

the derivative.

b)

the slope between two points.

c)

the difference quotient.

d)

the times of our lives.

19.

Match formulas for particle motion.

position of the particle at any time is s = f(t)

where the interval of time is [a, b].

a)

Displacement of particle

1.

f (b) - f (a)

b)

Average velocity of the particle

2.

$\frac{f\left(b\right)-f\left(a\right)}{b-a}$

c)

instantaneous velocity

v (t )

3.

s' (t)

d)

acceleration

a (t )

4.

s'' (t)

e)

Jerk

j (t)

5.

s"' (t)

20.

A particle can change direction when (a) is equal to zero.

Choose from the below words

velocity

acceleration

position

average velocity

jerk

21.

Which types of graph features should you look for to indicate when a function is not differentiable at a point?

Select all that apply.

a)

Cusp

b)

Corner

c)

Vertical Tangent

d)

Discontinuity

22.

Why is the graph NOT differentiable at x = 1?

a)

The graph is discontinuous.

b)

The graph oscillates.

c)

There is a vertical asymptote.

d)

There is a vertical tangent line.

23.

At which x-values is the graph of f(x)

NOT differentiable?

a)

x = -4, x = 1,

and x= 4

b)

x = 1 only

c)

x = -4 and x = 4

d)

x = -4 and x = 1

24.

At what x-values between -4 and 6

is the graph of f(x) NOT differentiable?

a)

x = 1 and x = 4

b)

x = 1 and x = 2

c)

x = 0, x = 1,

and x = 4

d)

x = -2 and x = 2

25.

If f(x) is differentiable at x = c,

then f(x) MUST be continuous at x = c.

a)

True

b)

False

26.

At which value(s) of x is the graph

continuous but NOT differentiable?

a)

x = 4 and x = 6

b)

x = 8

c)

x = 4

d)

x = 6

27.

a)

A

b)

B

c)

C

d)

D

28.

Which one of the following statements is always true?

a)

When a graph is increasing, its derivative is negative.

b)

When a graph is decreasing, so is its derivative.

c)

When a graph is decreasing, its derivative is negative.

d)

When a graph is increasing, so is its derivative.

29.

Select the correct derivative of f(x).

a)

A.

b)

B.

c)

C.

d)

D.

30.

Which graph is the derivative of the top graph?

a)

A

b)

B

c)

C

d)

D

31.

The derivative of a function is its

a)

Slope

b)

Maximum/Minimum

c)

Instantaneous rate of change

d)

Common Denominator

Derivatives Chapter 3

Total questions: 31

Worksheet time: 2hrs 36mins

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