Practice Final Exam

Question 1

What is the limit of the function in the graph at x = 4?

Answer 1

2

Question 2

Evaluate the following as true or false. The average rate of change of a function f(x) between x=x1 and x=x2 is the slope of the line connecting the two points (x1,f(x1)) and (x2,f(x2)), and the derivative of the function f(x) at x is the slope of the line connecting the origin (0,0) and the point (x,f(x)).

Answer 2

false

Question 3

Consider the function y = x^ 2 + x + 9. What is the equation of the tangent line at x = 2?

Answer 3

y=5x+5

Question 4

Evaluate the following as true or false.The function f(x)=5 can be written as f(x)=5^1. Therefore, f′(x)=1⋅5^1−1
=1⋅5^0
=1⋅1
=1.

Answer 4

false

Question 5

Find the derivative of:

v(x)=√(x+2)/^3√(x−3)

Answer 5

v′(x)=(x−13)/6(x−3)^4/3(x+2)^1/2

Question 6

What is the derivative of the function

f(x) = cos^3(x^2−x)?

Answer 6

-3(2x−1)cos^2(x^2−x)sin(x^2−x)

Question 7

Find the derivative of A(x)=4e^[(cos4x)(sin3x)]

Answer 7

A’(x)=4e^[(cos4x)(sin3x)]⋅[3cos4xcos3x−4sin4xsin3x]

Question 8

Find f′(x) if f(x)=cos(2x^2)

Answer 8

f′(x)=−4xsin(2x^2)

Question 9

Use implicit differentiation to find an equation of the line tangent to the curve x ^2 + y ^2 = 10 at the point (3, 1).

Answer 9

y = −3x + 10

Question 10

What is the inverse of f(x)=^3√1+x/2x?

Answer 10

f^−1(x)=1/2x^3−1

Question 11

Which of the following is an antiderivative of arccsin x ?

Answer 11

xarcsinx+√1−x^2

Question 12

A particle moves along the x-axis, with its position x given by x (t) = t − cos t. At which of the following times is the velocity of the particle equal to 0?

Answer 12

3π/2

Question 13

Which of the following is the second derivative off(x)=x^2−4/3x−6?

Answer 13

0

Question 14

A manufacturer wants to make open tin boxes from pieces of tin with dimensions 8 in. by 15 in. by cutting equal squares from the four corners and turning up the sides. Find the side of the square cutout that gives the box the largest possible volume.

Answer 14

5/3 in.

Question 15

A television camera at ground level is filming at the lift-off of a space shuttle that is rising vertically according to the position equation
S = 50t ^2, where S is measured in feet and t is measured in seconds. The camera is 2000 ft from the launch pad. What is the rate of change in the angle of elevation of the camera 10 seconds after lift-off?

Answer 15

2/29 rad/sec

Question 16

What are the critical points of the function

f(x)=√x^2+1 ?

Answer 16

x = 0

Question 17

On which of the following intervals is the graph of

f(x)=x^2−1/2x+1 concave up?

Answer 17

(−∞,−1/2)

Question 18

Which of the following equations has no horizontal asymptote?

Answer 18

y=x^2−4/x

Question 19

Evaluate the indefinite integral∫⎛⎜⎝x^2+2x+1/x+1⎞⎟⎠dx.

Answer 19

x^2/2+x+C

Question 20

Evaluate the indefinite integral ∫t^2√t^3+1dt

Answer 20

2/9(t^3+1)^3/2+C

Question 21

Evaluate ∫cosxsin^5xdx .

Answer 21

sin^6x/6+C

Question 22

What is the area bound by the curve h(x)=2πe4x+3x and the x-axis from x=0to x=2?

Answer 22

π/2e^8+6−π/2

Question 23

Suppose you put a baseball machine at ground level, point it straight up, and fire a baseball into the air at 96 ft / s. How far has the baseball traveled after 5 s?

Answer 23

208 ft

Question 24

What is the area of the region enclosed by

y=x^2 and y=|x|?

Answer 24

1/3

Question 25

What is the area between the curves y=−x^2+4 and y=−3 ?

Answer 25

28√7/3

Question 26

What is the volume of the solid of revolution obtained by rotating the region bounded by y=2x^2+1,x=1, and x=0 around the x-axis?

Answer 26

47π/15

Question 27

The half-life of a newly discovered radioactive element is 30 seconds. To the nearest tenth of a second, how long will it take for a sample of 9 grams to decay to 0.72 grams?

Answer 27

109.3 seconds

Question 28

When a particle is located at a distance x ft from the origin, a force of 3x^ 2 − 2x + 10 pound acts on it. How much work is done in moving it from x = 1 to x = 5?

Answer 28

140 ft-lb

Question 29

Evaluate lim x→0√x+1/ln(x+1).

Answer 29

The limit does not exist.

Question 30

Compute dy/dx for y=(3^x2−x)^2/3−x^2

Answer 30

2(3x^2−x)(6x−1)(3−x^2)+2x(3x^2−x)2/(3−x^2)^2