Free Printable Unit 5 Test Review

1.

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?

a)

3 ft/sec

b)

-7.2 ft/sec

c)

7.2 ft/sec

d)

12

2.

A baseball diamond is a square with a side of 90 ft. Tucker hits the ball and runs towards first base at 16 ft/sec. How fast is the distance between second base and Tucker changing when he is 30 ft from first base? Use a calculator for the very end.

a)

8.875 ft/sec

b)

5.060 ft/sec

c)

-5.060 ft/sec

d)

-8.875 ft/sec

3.

A construction worker pulls a five meter plank up the side of a building under construction by means of a rope tied to one end of the plank. Assume the opposite end of the plank follows a path perpendicular to the wall of the building and the worker pulls the rope at a rate of .15 meters per second. How fast is the end of the plank sliding along the ground when when it is 2.5 meters from the wall of the building?

a)

a²+b²=c²

b)

d=√(x²+y²)

c)

h=r⋅sinθ

d)

1=sin²θ+cos²θ

4.

Find dy/dx
xy+y²=2

a)

-y/(x+2y)

b)

y/(x+2y)

c)

-3y/x

d)

-3x/y

5.

Find dy/dx by Implicit Differentiation
x³ +y³ = 36

a)

6 -x

b)

3x² +3y²

c)

−x²/y²

d)

0

6.

Louisa and Karis were each dropped off at the same bus stop. Louisa’s bus drops her off at 3:30 whereas Karis is dropped off ten minutes later. Louisa runs home at a constant rate of 6 mph and Karis runs home at 3 mph. Louisa lives north of the bus stop and Karis lives to the east. How fast is the distance between them changing at 4:00? Use a calculator at the very end.

a)

6.512 mph

b)

7.115 mph

c)

6.708 mph

d)

6.641 mph

7.

Devin set up a toy rocket. For safety, he stands 6 meters from the rocket. He sets off the rocket and it heads straight up at a constant rate of 4 m/s. How fast is the distance between the rocket and Devin changing after 2s? Use a calculator at the very end.

a)

-2.5 m/s

b)

2.5 m/s

c)

3.2 m/s

d)

-3.2 m/s

8.

a)

-2

b)

-2/3

c)

2/3

d)

2

9.

Find dy/dx at the given point

x³ +2xy -y²=11 at (2,3)

a)

-4/7

b)

12

c)

-9

d)

9

10.

Differentiate the following $y=x^2\ln\ x+2e^{3x}$

a)

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

b)

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

c)

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

11.

Using Implicit Differentiation, find $\frac{\text{d}y}{\text{d}x}$ for $\sin\ \left(x^2y^2\right)=x$

a)

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

b)

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

c)

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

12.

a)

-3

b)

8

c)

0

d)

9

13.

a)

b)

c)

d)

14.

Find an equation of the tangent line to the graph of f(x) at the point (1, 100)
f(x) = (5x⁵ + 5)²

a)

y = 500x + 400

b)

y = 100x + 400

c)

y = -500 x - 400

d)

y = 500x - 400

15.

What is the equation of the tangent line at x = 1 $of$ $f\left(x\right)=\ \sqrt{x}$

a)

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

b)

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

c)

$y=2x-1$

d)

$y=2x$

16.

Find y'

y=\sec^{-1}\sqrt{x}

a)

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

b)

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

c)

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

d)

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

e)

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

17.

$f\left(x\right)=\tan^{-1}\left(4x\right)$

Find the derivative of f(x).

a)

$\frac{4}{16x^2+1}$

b)

$\frac{4}{4x^2+1}$

c)

$\frac{1}{16x^2+1}$

d)

$-\frac{1}{4x^2+1}$

e)

$-\frac{4}{4x+1}$

18.

Find the second derivative

3x^2+y^3=9

a)

$\left(\frac{2y^3-8x^2}{y^5}\right)$

b)

$-\left(\frac{2y^3+8x^2}{y^5}\right)$

c)

$-\frac{\left(-8x^2+2y^3\right)}{y^4}$

d)

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

e)

$\frac{2x^2}{3y^3}$

19.

$\frac{d}{dx}\left(\cot^{-1}x\right)=$

a)

$\frac{1}{\sqrt{1-x^2}},\ x\ne\pm1$

b)

$\frac{-1}{\sqrt{1-x^2}},\ x\ne\pm1$

c)

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

d)

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

20.

$\frac{d}{dx}\left(\cos^{-1}x\right)=$

a)

$\frac{1}{\sqrt{1-x^2}},\ x\ne\pm1$

b)

$\frac{-1}{\sqrt{1-x^2}},\ x\ne\pm1$

c)

$\frac{1}{x\sqrt{x^2-1}},\ x\ne\pm1,0$

d)

$\frac{-1}{x\sqrt{x^2-1}},\ x\ne\pm1,0$

21.

Find the second derivative of f(x) = x²+ e^x - cosx

a)

f"(x) = 2 + e^x + cosx

b)

f"(x) = 2x + e^x + cosx

c)

f"(x) = 2x + xe^x - cosx

d)

f"(x) = 2x + e^x + sinx

22.

Find dy/dx at a given point.

a)

5/4

b)

4/5

c)

1

d)

-5

23.

a)

-2

b)

-2/3

c)

2/3

d)

2

24.

$x^2+xy+y^2=9\ @\ \left(1,2\right)$ Use implicit differentiation to write the equation of the tangent line to the curve at the given point.

a)

$y-1=\frac{-4}{5}\left(x-2\right)$

b)

$y-2=\frac{-4}{5}\left(x-1\right)$

c)

$y-2=\frac{4}{5}\left(x-1\right)$

d)

$y+2=\frac{-4}{5}\left(x+1\right)$

25.

Find d/dx if $2x^4y^4=20$

a)

$\frac{dy}{dx}=-\frac{y}{x}$

b)

$\frac{dy}{dx}=\frac{y}{x}$

c)

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

d)

$\frac{dy}{dx}=\frac{y^3}{x^3}$

26.

Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The radius of the spill increases at a rate of 5 m/min. How fast is the area of the spill increasing when the radius is 5 m?

a)

50π m²/min

b)

47π m²/min

c)

52π m²/min

d)

40π m²/min

27.

Water leaking onto a floor forms a circular pool. The radius of the pool increases at a rate of 4 cm/min. How fast is the area of the pool increasing when the radius is 10 cm?

a)

86π cm²/min

b)

89π cm²/min

c)

80π cm²/min

d)

71π cm²/min

28.

A hypothetical square shrinks at a rate of 64 m²/min. At what rate are the sides of the square changing when the sides are 10m each?

a)

-17/11 m/min

b)

-6/7 m/min

c)

-16/5 m/min

d)

-2 m/min

29.

A crowd gathers around a movie star, forming a circle. The radius of the crowd increases at a rate of 8 ft/sec. How fast is the area taken taken up by the crowd increasing when the radius is 13 ft?

a)

$208\pi\ \frac{ft^2}{\sec}$

b)

$211\pi\ \frac{ft^2}{\sec}$

c)

$216\pi\ \frac{ft^2}{\sec}$

d)

$218\pi\ \frac{ft^2}{\sec}$

30.

Given: Water slowly evaporates from a circular shaped puddle. The area of the puddle decreases at a rate of $36\pi\ \frac{in^2}{hr}$ . Assuming the puddle retains its circular shape, at what rate is the radius of the puddle changing when the radius is 5 in?

a)

$\frac{dr}{dt}=-1.8\ \frac{in}{hr}$

b)

$\frac{dr}{dt}=-3.6\ \frac{in}{hr}$

c)

$\frac{dr}{dt}=-\frac{36}{25}\ \frac{in}{hr}$

d)

$\frac{dr}{dt}=6\ \frac{in}{hr}$

Unit 5 Test Review

Total questions: 30

Worksheet time: 4hrs 35mins

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