Final Exam

Question 1

What is the slope of the tangent line of the function f (x) = −2x ^2 + 3x − 1 at x = −2?

Answer 1

11

Question 2

Consider the function y = x^ 2 − 3x + 2. At what value of x is the slope of the tangent line equal to 5?

Answer 2

4

Question 3

What is the derivative of the function

f (x) = −10x^5?

Answer 3

-50x^4

Question 4

Find the derivative of:

f(x)=(x+3)^3/2 / (x−2)^3/2

Answer 4

f′(x)=−15(x−2)1^/2(x+3)^1/2/2(x−2)^3

Question 5

Compute dy/dx for y=√x2+1+1/√x^2+1

Answer 5

x^3(1+x^2)^−3/2

Question 6

What is the derivative of the function f(x)=xsinx+1cosx?

Answer 6

sinxcosx+x+sinx/cos^2x

Question 7

What is the equation of the line tangent to the curve y + ex = 2e^xy at the point (0, 1)?

Answer 7

y = x + 1

Question 8

A point moves along the x-axis with its position, x, at time t given by x (t) = t ^3 − 3t ^2 + 2t − 1.

Which of the following is the acceleration when
t = 2?

Answer 8

6

Question 9

Determine d^2/dx^2 [e^x−3]

Answer 9

e^x−3

Question 10

A ladder 25 feet long is leaning against a house. The base of the ladder is pulled away from the wall at a rate of 2 ft / sec. How fast is the top moving down the wall when the base of the ladder is 15 feet from the wall?

Answer 10

−3/2 ft/sec

Question 11

A box manufacturer is to make a closed box of volume 288 in3, where the base is a rectangle whose length is three times its width. What is the length of the box that uses the least amount of material?

Answer 11

12 in

Question 12

What are the critical points of the function f (x) = x ^4 − 6x^ 2 + 8x ?

Answer 12

x = -2, 1

Question 13

For which of the following intervals is the graph of f(x)=24/x^2+12 concave down?

Answer 13

(−2,2)

Question 14

Which of the following equations could be an equation of a horizontal asymptote of y=2x^2−7x+4/x−5?

Answer 14

There is no horizontal asymptote

Question 15

Evaluate the indefinite integral∫⎛⎝3√t2+13√t5⎞⎠dt

Answer 15

3/5t^5/3−3/2t^−2/3+C

Question 16

Evaluate the indefinite integral ∫(3t2√t3+t2+1+2t√t3+t2+1)dt.

Answer 16

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

Question 17

Evaluate ∫16/3 x^1/3cot(x^4/3)dx

Answer 17

4ln∣sin(x^4/3)∣+C

Question 18

What is the area beneath the curve f(x)=x2√3−x^3 from x = −2 to x = −1?

Answer 18

2⋅11^3/2−16/9

Question 19

Determine the height from which an object must be dropped so that it will have the same speed in ft / s when it lands as the height in feet from which it was dropped.

Answer 19

64 ft

Question 20

What is the area of the shaded region shown in the figure below bounded by the curves y = 2x^ 2 − 1, y = x^ 2 − x + 1, and y = x^ 2 + x + 1?

Answer 20

7/3

Question 21

Find dG/dx if

G(x)=ln(√5x+1/5x−1)

Answer 21

dG/dx=−5/25x^2−1

Question 22

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

Answer 22

The limit does not exist

Question 23

Compute the derivative of the function f(x)=2x^2+2e^x.

Answer 23

f′(x)=4x+2e^x

Question 24

What is the inverse of f(x)=x/1+2x

Answer 24

f−1(x)=x/1−2x

Question 25

Evaluate ∫dx/|x|√4x^2−1.

Answer 25

arcsec (2x) + C

Question 26

What is the area between the curves x=y^2−9 and x=−y^2+9 ?

Answer 26

72

Question 27

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

Answer 27

π/3

Question 28

The half-life of iodine-126 (I 126 ) is 13 days. Of an original sample of 1,000.0 grams, how may grams of I 126 will remain after 98 days? Give the answer to the nearest tenth of a gram.

Answer 28

5.4 g

Question 29

A force of 2x + 5 pounds at a point x feet from the origin moves an object from
x = 2 to x = 5. What is the work done?

Answer 29

36 ft-lb

Question 30

Evaluate lim x→1 lnx/x^2−1

Answer 30

1/2