MTH 280 Wk 7 – Final Exam

Question 1

A car starts from a point at 2:00 p.m. and travels north at 40 mph. Another car starts from the same point at 3:00 p.m. and travels west at 50 mph. After the second car has traveled 1 h, at what rate is the distance between the two cars changing?

60.42 mph

94.34 mph

30.17 mph

64.03 mph

Question 2

A spherical ball is measured to have a radius of 6 space c m with a possible measurement error of plus-or-minus 0.1 rm c m. Use the differentials to estimate the percentage error in computing the volume of the ball.

1%

3%

5%

10%

Question 3

A company determines a cost function of c equals 6 x squared minus 180 x plus 2000, where c is the cost (in dollars) of producing x number of items. How many items should the company manufacture to minimize the cost?

12

15

24

30

Question 4

The position of an object is given by the equation s left parenthesis t right parenthesis equals 2 x squared plus x minus 6. Find the time t at which the instantaneous velocity of the object equals the average velocity in the interval open square brackets 0 comma 3 close square brackets.

t equals 1.5 semicolon space s

t equals 3 semicolon space s

t equals 2 semicolon space s

t equals 0 semicolon space s

Question 5

Find the locations of local minimum and maximum of x to the power of 9 minus 4 x to the power of 8 using the second derivative test.

Local minimum at x equals 0, local maximum at x equals 32 over 9

Local minimum at x equals 32 over 9, no local maximum

Local minimum at x equals 32 over 9, local maximum at x equals 0

Local minimum at x equals 0, no local maximum

Question 6

In a shop, the revenue and the cost of a product are determined by R left parenthesis x right parenthesis equals 22 x and C left parenthesis x right parenthesis equals 2 x squared plus 2 x plus 1, respectively. If x represents the number of products, how many products should the shop sell to maximize the profit?

11

5

6

10

Question 7

Evaluate limit as x rightwards arrow 0 of fraction numerator x squared over denominator e to the power of x minus x minus 1 end fractionby applying L’Hôpital’s rule.

0

1

2

infinity

Question 8

Let f left parenthesis x right parenthesis equals x cubed minus x squared minus 1 and x subscript 0 equals 1. To the nearest three decimal places, find x subscript 5 using Newton’s method of approximation.

1.466

1.486

1.625

2.000

Question 9

Evaluate integral sin 2 x cos 2 x comma space d x

1 fourth cos 4 x plus C

negative 1 over 8 cos 4 x plus C

negative 1 fourth cos 4 x plus C

1 over 8 cos 4 x plus C

Question 10

Evaluate integral subscript negative 1 end subscript superscript 1 left parenthesis t squared plus t plus 1 right parenthesis d t using the Fundamental Theorem of Calculus, Part 2.

8 over 3

negative 5 over 6

10 over 6

Question 11

Water is flowing into a tank at a rate of r left parenthesis t right parenthesis equals 3 square root of t over 2 end root cubic meters per minute. How much water entered the tank between 2 and 8 minutes?

3 cubic meters

6 cubic meters

14 cubic meters

28 cubic meters

Question 12

To the nearest two decimal places, calculate R subscript 5 for f left parenthesis x right parenthesis equals x cubed plus 1 on open square brackets 0 comma space 4 close square brackets.

44.96

65.00

33.77

96.16

Question 13

Use substitution to evaluate integral subscript 0 superscript straight pi over 2 end superscript sin 2 x square root of 4 plus 9 sin squared x end root space d x.

1 third left parenthesis 26 square root of 13 minus 16 right parenthesis

1 over 27 left parenthesis 26 square root of 13 minus 16 right parenthesis

2 over 27 square root of straight pi cubed end root

2 over 3 square root of straight pi cubed end root

Question 14

Evaluate the integral integral fraction numerator 1 plus tan x over denominator 1 minus tan x end fraction d x.

ln open vertical bar sin x minus cos x close vertical bar plus C

negative ln open vertical bar cos x plus sin x close vertical bar plus C

negative ln open vertical bar cos x minus sin x close vertical bar plus C

ln open vertical bar cos x plus sin x close vertical bar plus C

Question 15

Evaluate integral subscript 1 superscript 2 fraction numerator 6 over denominator open vertical bar 3 x close vertical bar square root of 9 x squared minus 4 end root end fraction d x.

sin to the power of negative 1 end exponent 3 minus sin to the power of negative 1 end exponent 3 over 2

tan to the power of negative 1 end exponent 3 minus tan to the power of negative 1 end exponent 3 over 2

s e c to the power of negative 1 end exponent 3 minus s e c to the power of negative 1 end exponent 3 over 2

Question 16

Find the area between the curves f left parenthesis x right parenthesis equals 1 minus 2 x and g left parenthesis x right parenthesis equals negative x minus 1 over the interval open square brackets negative 4 comma space minus 1 close square brackets.

13.5 space u n i t s squared

21 space u n i t s squared

27 space u n i t s squared

28.5 space u n i t s squared

Question 17

If R denotes a region bounded above by the graph of a continuous function f left parenthesis x right parenthesis, below by the x-axis, and on the left and right by the lines x equals a and x equals b, respectively, then which of the following integrals gives the mass of the lamina with density rho?

m equals rho integral subscript a superscript b open square brackets f left parenthesis x right parenthesis close square brackets squared over 2 space d x

m equals rho integral subscript a superscript b x f left parenthesis x right parenthesis space d x

m equals rho integral subscript a superscript b f left parenthesis x right parenthesis space d x

m equals rho integral subscript a superscript b open square brackets f left parenthesis x right parenthesis close square brackets squared space d x

Question 18

Evaluate fraction numerator d over denominator d x end fraction cos h left parenthesis 2 x squared plus 1 right parenthesis.

4 x sin h left parenthesis 2 x squared plus 1 right parenthesis

2 x sin h left parenthesis 2 x squared plus 1 right parenthesis

x sin h left parenthesis 2 x squared plus 1 right parenthesis

left parenthesis 2 x squared plus 1 right parenthesis sin h left parenthesis 2 x squared plus 1 right parenthesis

Question 19

Evaluate integral fraction numerator negative 1 over denominator open vertical bar x close vertical bar square root of 1 plus begin display style x squared over 25 end style end root end fraction space d x.

c s c h to the power of negative 1 end exponent open vertical bar x close vertical bar plus C

1 fifth c s c h to the power of negative 1 end exponent open vertical bar x over 5 close vertical bar plus C

c s c h to the power of negative 1 end exponent open vertical bar x over 5 close vertical bar plus C

1 fifth c s c h to the power of negative 1 end exponent open vertical bar x close vertical bar plus C

Question 20

Find the vertical and horizontal asymptotes of f left parenthesis x right parenthesis equals x plus sin x.

x equals 0 comma space y equals 0

x equals 1, no vertical asymptote

x equals negative straight pi over 2 comma space y equals straight pi over 2

No vertical and horizontal aymptotes