Maths - Chapter 2 Midterm Exam

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Chapter 2

1. The equation of the line tangent to the graph of at x = 7 is

A) y = 16x – 7

B) y = 16x – 49

C) y = 16x – 343

D) y = 16x – 686

2. The equation of the line tangent to the graph of at x = 1 is

A)

B)

C)

D)

3. Let f(x) = x^2. Compute the slope of the secant line joining the points on the graph of f whose x coordinates are x= -2 and x= -1.9. Use calculus to compute the slope of the line that is tangent to the graph when x=-2 and compare with the slope found on the secant line.

A) m(sec) = 2 and m(tan) = -3

B) m(sec) = -3.9 and m(tan) = -4

C) m(sec) = -4 and m(tan) = -3.9

D) m(sec) = 00868 m(tan) = 0

4. Differentiate:

A) 7

B)

C) 0

D)

5. Find the equation of the tangent line to the graph of at the point

(3, 12).

A) y = 12

B) y = 6x – 6

C) x = 3

D) Not defined

6. Find the rate of change of the given function f (x) with respect for x for the

prescribed value x = –2.

f (x) = x3 + 3x + 9

A) 6

B) –3

C) 15

D) 24

7. An appliance store manager estimates that for x television ads run per day, refrigerators will be sold per month. Find and interpret what it tells us about sales.

A) 4.52; sales will be increasing at about 5 refrigerators per month per ad when they're running 4 ads.

B)203.36; the cost of refrigerators will be rising by $203.36 if they're selling 4 per day.

C)4.52; they’ll sell about 5 refrigerators if they run 4 ads per day.

D)203.36; they’ll sell about 203 refrigerators if they run 4 ads per day.

8. An object moves along a line in such a way that its position at time t is . Find the velocity and acceleration of the object at time t. When is the object stationary?

A) ; a(t) = 6t – 18; t = 3

B) ; a(t) = 6t – 6; t = 1

C) ; a(t) = 6t – 18; t = 1

D) ; a(t) = 6t – 18; t = 1 and 5

9. Differentiate:

A)

B)

C) 2x + 1

D) 12x + 1

10. What is the rate of change of with respect to t when t = 2?

A)

B) 4

C)

D)

11. Differentiate:

A) 2x

B) –x

C)

D)

12. Find if .

A)

B)

C)

D)

13. Find if

A)

B)

C)

D)

14. An equation for the tangent line to the curve at the point where

x = 0 is

A) y = 5x + 1

B) y = 5x – 1

C) y = 10x – 1

D) y = 10x + 1

15. Find all points on the graph of the function where the tangent line is horizontal.

A) (0, 0) and (–4, –8)

B) There are none.

C) (0, 0)

D) (2, 1)

16. True or False: If , then .

A) False

B) True

17. When a certain commodity is sold for p dollars per unit, consumers will buy units per month. It is estimated that t months from now, the price of the commodity will be dollars per unit. The approximate rate at which the monthly demand will be changing with respect to time in 27 months is

A) –32 units per month

B) –35 units per month

C) –132 units per month

D) 35 units per month

18. True or False: The function will decrease by approximately 0.6 as x decreases from 3 to 2.7.

A) False

B) True

19. The largest percentage error you can allow in the measurement of the radius of a sphere if you want the error in the calculation of its surface area using the formula to be no greater than 6 percent is about:

A) 2%

B) 6%

C) 3%

D) 1%

20. You measure the side of a cube to be 14 centimeters long and conclude that the volume of the cube is cubic centimeters. If your measurement of the side is accurate to within 2%, approximately how accurate is your calculation of this volume? Round to two decimal places, if necessary. (Bonus Question)

A) Maximum error in volume is about ±164.64 cm3

B) Maximum error in volume is about ±0.84 cm3

C) Maximum error in volume is about ±2,304.96 cm3

D) Maximum error in volume is about ±11.76 cm3

21. An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 7:00 A.M. will have assembled transistor radios x hours later. Approximately how many radios will the worker assemble between 9:00 and 9:30 A.M.?

A) Approximately 7 radios

B) Approximately 13 radios

C) Approximately 14 radios

D) Approximately 390 radios

22. Find , where .

A)

B)

C)

D)

23. Find the equation of the tangent line to the given curve at the specified point. ; (0, 8)

A)

B) y = –46x + 8

C) y = 46x + 8

D)

24. Use implicit differentiation to find for . (Bonus Question)

A)

B)

C)

D)

25. Suppose the output at a certain factory is units, where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor. The current labor force consists of 20 hours of skilled labor and 10 hours of unskilled labor. Use calculus to estimate the change in unskilled labor y that should be made to offset a 1-hour increase in skilled labor x so that output will be maintained at its current level. Round you answer to two decimal places, if necessary.

A) –2.67 hours

B) 2.67 hours

C) –1 hours

D) –0.38 hours

()5

21

x

fx

x

=-

+

2

4

π

Sr

=

3

142,744

=

32

()73

fxxxx

=-+-

dy dx

dy

dx

3

1

2

yx

=-

32

37

xyxy

-=

3

2

6

37

xy

xy

-

-

3

6

yx

-

2

3

6

x

y

3

67

yx

--

55

568

xyxyxy

-=+-

1

8

46

yx

=-+

1

8

46

yx

=+

2

2

dy

dx

5

411100

xy

+=

11

22

yx

=-

60x 2 +11

60x

2

+11

3

80

x

2

60100

x

-

3

80

11

x

-

3343

45 + 3

Qxxyy

=+

33

22

yx

=+

7

()

fxx

x

=+

3

17

2

2

x

x

+

3

17

2

2

x

x

-

2

()3

fxx

=+

32

()0.013200

Rxxxx

=-+-+

(4)

R

¢

(4)

R

¢

=

32

()9152

stttt

=-++

2

()31815

vttt

=-+

2

()3615

vttt

=-+

v(t) = 3t 2 −18t +15

v(t)=3t

2

-18t+15

2

()(1)(6)

fxxx

=++

2

1

x

+

2

3121

xx

++

75

()

2

t

ft

t

-

=

+

2

()2

fxxx

=+

9

4

19

16

17

4

2

()

6

x

fx

x

=

-

(

)

2

2

312

6

xx

x

+

-

(

)

2

2

12

6

xx

x

-

-

(4)

()

fx

5432

()87774

fxxxxxx

=-+-+-

(4)

()120192

fxx

=-

(4)2

()8

fxxx

=-

()3

fxx

=

(4)2

()87

fxxx

=-+

(4)2

()6019242

fxxx

=-+

()

fx

¢¢¢

2

13

()7

3

fx

x

x

=-+

5

3

1572

()

163

fx

x

xx

¢¢¢

=-+

5

3

1572

()

83

fx

x

xx

¢¢¢

=-+

5

3

572

()

723

fx

x

xx

¢¢¢

=-+

3

3

33

()

83

fx

x

xx

¢¢¢

=-+

25

(71)

yxx

=+-

2

()

2

x

fx

x

=

+

11

22

yx

=+

2

()13

fxx

=-

23/2

3

"()

(13)

fx

x

-

=

-

31,500

()

Dp

p

=

2/3

()5.15

ptt

=+

Chapter 2

1.

The equation of the line tangent to the graph of

2

()2

fxxx

=+

at x = 7 is

A) y = 16x

–

7

B) y = 16x

–

49

C) y = 16x

–

343

D) y = 16x

–

686

2.

The equation of the line tangent to the graph of

()3

fxx

=

at x = 1 is

A)

11

22

yx

=+

B)

3

1

2

yx

=-

C)

11

22

yx

=-

D)

33

22

yx

=+

3.

Let f(x) = x^2. Compute the slope of

the secant line joining the points on the

graph of f whose x coordinates are x=

-

2 and x=

-

1.9. Use calculus to compute the

slope of the line that is tangent to the graph when x=

-

2 and compare with the

slope found on the secant line.

A)

m(sec) = 2 and m(

tan) =

-

3

B)

m(sec) =

-

3.9 and m(tan) =

-

4

C)

m(sec) =

-

4 and m(tan) =

-

3.9

D)

m(sec) = 00868 m(tan) = 0

Chapter 2

1. The equation of the line tangent to the graph of

2

()2fxxx at x = 7 is

A) y = 16x – 7

B) y = 16x – 49

C) y = 16x – 343

D) y = 16x – 686

2. The equation of the line tangent to the graph of ()3fxx at x = 1 is

A)

11

22

yx

B)

3

1

2

yx

C)

11

22

yx

D)

33

22

yx

3. Let f(x) = x^2. Compute the slope of the secant line joining the points on the

graph of f whose x coordinates are x= -2 and x= -1.9. Use calculus to compute the

slope of the line that is tangent to the graph when x=-2 and compare with the

slope found on the secant line.

A) m(sec) = 2 and m(tan) = -3

B) m(sec) = -3.9 and m(tan) = -4

C) m(sec) = -4 and m(tan) = -3.9

D) m(sec) = 00868 m(tan) = 0