calculus i-review problems new york university final exam is...

Calculus I-Review ProblemsNew York University

Final Exam is cumulative!!!This Review Sheet covers material from 2.3-2.6, 2.8, 3.1-3.5, 3.7, 4.1, 4.2, 4.3, 4.4, 4.5, 4.7, 5.1-5.5. For the first partof the semester refer to the Exam 1 Review Sheet.

1. Compute the following limits:

(a) limx→3− e2/(x−3)

(b) limx→10− ln(100− x2)

(c) limx→∞ e−x sinx

(d) limx→∞ 1+2x

1−2x

(e) limx→∞x3e−x

(f) limx→1+xx−1 − 1

ln x

(g) limx→π/2− tanxcos x

(h) lim→∞ x[ln(x+ 5)− lnx]

2. Calculate y′.

(a) y = 2x√x2+1

(b) y = 1sin(x−sin x)

(c) y = sec 2θ1+tan 2θ

(d) xy4 + x2y = x+ 3y

(e) sin(xy) = x2 − y(f) y = ln(x2ex)

(g) y = (cosx)x

(h) y = ln(1/x) + 1ln x

(i) y = arctan(arcsin√x)

(j) xey = y − 1

(k) y = (tanx)1/x

(l) ex2y = x+ y

3. The graph of y = x3 − 9x2 − 16x+ 1 has a slope of 5 at two points. Find the coordinates of the points.

4. Find the equation of the line tangent to f(x) at x = 2, if

f(x) =x3

2− 4

3x.

5. (a) Find the slope of the graph of f(x) = 1− ex at the point where it crosses the x-axis.

(b) Find the equation of the tangent line to the curve at this point.

(c) Find the equation of the line perpendicular to the tangent line at this point. (This is the normal line.)

6. Suppose f and g are differentiable functions with the values shown in the following table. For each of the followingfunctions h, find h′(2).

(a) h(x) = f(x) + g(x)

(b) h(x) = f(x)g(x)

(c) h(x) =f(x)

g(x)

x f(x) g(x) f ′(x) g′(x)2 3 4 5 −2

7. If you invest P dollars in a bank account at an annual interest rate of r%, then after t years you will have Bdollars, where

B = P(

1 +r

100

)t.

(a) Find dB/dt, assuming P and r are constant. In terms of money, what does dB/dt represent?

(b) Find dB/dr, assuming P and t are constant. In terms of money, what does dB/dr represent?

8. Use the following graph to calculate the derivative.

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3.6 THE CHAIN RULE AND INVERSE FUNCTIONS 153

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Job: chap3-temp Sheet: 39 Page: 153 (March 8, 2012 10 : 10) [ex-6]

ins3-6w51-53 In Problems 60–62, use Figure 3.32 to calculate the derivative.

ins3-6w51-53fig

(2, 5)

(2.1, 5.3)

f(x)

Figure 3.32

3-6w51 60. h!(2) if h(x) = (f(x))3

3-6w52 61. k!(2) if k(x) = (f(x))"1

3-6w53 62. g!(5) if g(x) = f"1(x)3-6w54 63. (a) Given that f(x) = x3, find f !(2).

(b) Find f"1(x).(c) Use your answer from part (b) to find (f"1)!(8).(d) How could you have used your answer from part (a)

to find (f"1)!(8)?3-6w55 64. (a) For f(x) = 2x5 + 3x3 + x, find f !(x).

(b) How can you use your answer to part (a) to deter-mine if f(x) is invertible?

(c) Find f(1).(d) Find f !(1).(e) Find (f"1)!(6).

3-6w57 65. Use the table and the fact that f(x) is invertible and dif-ferentiable everywhere to find (f"1)!(3).

x f(x) f !(x)

3 1 76 2 109 3 5

3-miscw117 66. At a particular location, f(p) is the number of gallons ofgas sold when the price is p dollars per gallon.

(a) What does the statement f(2) = 4023 tell you aboutgas sales?

(b) Find and interpret f"1(4023).(c) What does the statement f !(2) = !1250 tell you

about gas sales?(d) Find and interpret (f"1)!(4023)

3-6w58 67. Let P = f(t) give the US population6 in millions inyear t.

(a) What does the statement f(2005) = 296 tell youabout the US population?

(b) Find and interpret f"1(296). Give units.(c) What does the statement f !(2005) = 2.65 tell you

about the population? Give units.(d) Evaluate and interpret (f"1)!(296). Give units.

3-6w59 68. Figure 3.33 shows the number of motor vehicles,7 f(t),in millions, registered in the world t years after 1965.With units, estimate and interpret

(a) f(20) (b) f !(20)

(c) f"1(500) (d) (f"1)!(500)

3-6w59fig ’65 ’70 ’75 ’80 ’85 ’90 ’95 2000

200

400

600

800

(year)

(millions)

Figure 3.33

3-6w60 69. Using Figure 3.34, where f !(2) = 2.1, f !(4) = 3.0,f !(6) = 3.7, f !(8) = 4.2, find (f"1)!(8).

3-6w60fig2 4 6 8

8

16

24

x

f(x)

Figure 3.34

3-miscw118 70. If f is increasing and f(20) = 10, which of the two op-tions, (a) or (b), must be wrong?

(a) f !(10)(f"1)!(20) = 1.(b) f !(20)(f"1)!(10) = 2.

3-6w62 71. An invertible function f(x) has values in the table. Eval-uate

(a) f !(a) · (f"1)!(A) (b) f !(b) · (f"1)!(B)

(c) f !(c) · (f"1)!(C)

x a b c d

f(x) A B C D

3-6w63 72. If f is continuous, invertible, and defined for all x, whymust at least one of the statements (f"1)!(10) = 8,(f"1)!(20) = !6 be wrong?

3-6w64 73. (a) Calculate limh#0(ln(1 + h)/h) by identifying thelimit as the derivative of ln(1 + x) at x = 0.

(b) Use the result of part (a) to show thatlimh#0(1 + h)1/h = e.

(c) Use the result of part (b) to calculate the relatedlimit, limn#$(1 + 1/n)n.

6Data from www.census.gov/Press-Release/www/releases/archives/population/006142.html, accessed May 27, 2007.7www.earth-policy.org, accessed May 18, 2007.

(a) h′(2) if h(x) = (f(x))3

(b) k′(2) if k(x) = (f(x))−1

(c) g′(5) if g(x) = f−1(x)

9. (a) Find dy/dx given that x2 + y2 − 4x+ 7y = 15.

(b) Under what conditions on x and/or y is the tangent line to this curve horizontal? Vertical?

10. In the following problems find the local linearization of f(x) near x = 0 and use this to approximate the value ofa.

(a) f(x) = (1 + x)r, a = (1.2)3/5

(b) f(x) = ekx, a = e0.3

(c) f(x) =√b2 + x, a =

√26

11. Let y2 + 4x = 4xy2

(a) Compute dydx .

(b) Find the equation for the tangent line to this curve at (1/3, 2)

(c) Find the x and y-coordinates of all points at which the tangent line to this curve is vertical.

12. Find the local linearization L(x) of the function f(x) = (1 + x)k near x = 0, where k is a positive constant.Suppose you wnat to use L(x) to find an approximation of the number

√1.1. What number should k be, and

what number should x be? Approximate√

1.1.

13. Use a linear approximation to estimate the given number

(a) (8.06)2/3

(b)√

99.8

14. The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Estimate the maximumpossible error in computing

(a) the volume of the cube

(b) the surface area of the cube.

15. For what values of r does the function y = erx satisfies the differential equation y′′ + 5y′ − 6y = 0?

16. If f(x) = 3 + x+ ex, find (f−1)′(4)

17. A bacteria culture contains 200 cells initially and grows at a rate proportional to its sze. After half an hour thepopulation has increased to 360 cells.

(a) Find the number of bacteria after t hours.

(b) Find the bacteria after 4 hours.

(c) Find the rate of growth after 4 hours.

(d) When will the population reach 10,000.

18. At what point on the curve y = [ln(x+ 4)]2 is the tangent horizontal?

19. (a) Find an equation of the tangent to the curve y = ex that is parallel to the line x− 4y = 1.

(b) Find an equation of the tangent to the curve y = ex that passes through the origin.

20. Find the equation of the tangent line to the curve y = 3 arccos(x/2) at (1, π).

21. Bismuth-210 has a half-life of 5 days. A sample population after 1 day is decaying at a rate of 2 mg/day. Findthe initial mass of the sample. Write a model describing the population of the Bismuth-210 sample.

22. Find the critical points of the function.

(a) f(x) = x3 + 3x2 − 24x

(b) h(p) = p−1p2+4

(c) f(θ) = 2 cos θ + sin2 θ

(d) f(x) = x lnx

(e) f(x) = xe2x

23. Find the local max/min and the absolute max/min of f on the given interval.

(a) f(x) = x4 − 2x2 + 3, [−2, 3]

(b) f(x) = xx2+4 , [0, 3]

(c) f(x) = x− 2 cosx, [−π, π]

(d) f(x) = x− lnx, [1/2, 2]

24. Show that 5 is a critical number of the function g(x) = 2 + (x− 5)3 but g does not have a local extreme value at5.

25. Consider the piecewise linear function f(x) graphed below.!!

!!!

Math 115 / Exam 2 (March 20, 2012) page 7

6. [16 points] Consider the piecewise linear function f(x) graphed below:

8

10

f(x)

x

(10,!6)

(4, 6)

For each function g(x), find the value of g!(3):

a. [4 points] g(x) = sin![f(x)]3

"

Solution:g!(x) = cos(f(x)3) · 3f(x)2 · f !(x)

g!(3) = cos(73) · 3 · 72 · (!1) = 124.0442.

b. [4 points] g(x) =f(x2)

x

Solution:

g!(x) =x · f !(x2) · 2x ! f(x2)

x2

g!(3) =3(!3)6 ! (!3)

9= !5.667.

c. [4 points] g(x) = ln(f(x)) + f(2)

Solution:

g!(x) =1

f(x)f !(x) + 0

g!(3) =1

7· (!1) =

!1

7.

d. [4 points] g(x) = f"1(x)

Solution:

g!(x) =1

f !(f"1(x))

g!(3) =1

f !(6)= !2

3.

For each function g(x), find the value of g′(x):

(a) g(x) = sin(f(x)3)

(b) g(x) = f(x2)x

(c) g(x) = ln(f(x)) + f(2)

(d) g(x) = f−1(x)

26. Verify that f(x) = x3 + x − 1 satisfies the hypothesis of the Mean Value Theorem on the interval [0, 2]. Thenfind all numbers c that satisfy the conclusion of the Mean Value Theorem.

27. Does there exist a function f such that f(0) = −1, f(2) = 4 and f ′(x) ≤ 2 for all x?

28. The graph of f is given below.

!

1

Job: 4-1w17-main Sheet: 1 Page: 1 (May 3, 2012 14 : 23) [4-1w17-main]

1.

!1 1x

f !(x)

(a) Over what intervals is f increasing? decreasing?

(b) Does f have a local maxima or minima, if so which and where?

29. (a) Find the intervals of increase or decrease.

(b) Find the local maximum and minimum values.

(c) Find the intervals of concavity and the inection points.

(d) Use the information from parts (a)(c) to sketch the graph.

I. h(x) = (x2 − 1)3

II. ln(x4 + 27)

III.ex

1 + ex

IV. x√

5− x

30. Find values of a and b so that the function f(x) = x2 + ax+ b has a local minimum at the point (6,−5).

31. Find the global maximum and minimum of the function f(x) =x+ 1

x2 + 3on the interval −1 ≤ x ≤ 2

32. If you have 100 feet of fencing and want to enclose a rectangular area up against a long, straight wall, what isthe largest area you can enclose?

33. A line goes through the origin and a point on the curve y = x2e−3x, for x ≥ 0. Find the maximum slope of sucha line. At what x-value does it occur?

34. If 1200 cm of material is available to make a box with a square base and an open top, nd the largest possiblevolume of the box.

35. Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two verticesabove the x-axis and lying on the parabola y = 8− x2.

36. A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent intoan equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum? (b) Aminimum?

37. Find f if f ′′(θ) = sin θ + cos θ with f(0) = 3 and f ′(0) = 4.

38. A particle is moving with the accelaration a(t) = 10 + 3t− 3t2. If s(0) = 0 and s(2) = 10 find s(t).

39. Figure below gives your velocity during a trip starting from home. Positive velocities take you away from homeand negative velocities take you toward home. Where are you at the end of the 5 hours? When are you farthestfrom home? How far away are you at that time?

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264 Chapter Five KEY CONCEPT: THE DEFINITE INTEGRAL

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Job: chap5-temp Sheet: 10 Page: 264 (May 3, 2012 14 : 19) [ex-1]

5-1w21 16. The velocity of a particle moving along the x-axis isgiven by f(t) = 6 ! 2t cm/sec. Use a graph of f(t)to find the exact change in position of the particle fromtime t = 0 to t = 4 seconds.

ins5-1w17-20 In Problems 17–20, find the difference between the upper andlower estimates of the distance traveled at velocity f(t) on theinterval a " t " b for n subdivisions.

5-1w17 17. f(t) = 5t + 8, a = 1, b = 3, n = 100

5-1w18 18. f(t) = 25 ! t2, a = 1, b = 4, n = 500

5-1w19 19. f(t) = sin t, a = 0, b = !/2, n = 100

5-1w20 20. f(t) = e!t2/2, a = 0, b = 2, n = 20

5-1w22 21. A baseball thrown directly upward at 96 ft/sec has veloc-ity v(t) = 96 ! 32t ft/sec at time t seconds.

(a) Graph the velocity from t = 0 to t = 6.(b) When does the baseball reach the peak of its flight?

How high does it go?(c) How high is the baseball at time t = 5?

5-miscw32 22. Figure 5.14 gives your velocity during a trip starting fromhome. Positive velocities take you away from home andnegative velocities take you toward home. Where are youat the end of the 5 hours? When are you farthest fromhome? How far away are you at that time?

5-miscw32fig1 2 3 4 5

!20

!10

10

20

30

40

t (hours)

v (km/hr)

Figure 5.14

5-3w22 23. When an aircraft attempts to climb as rapidly as possi-ble, its climb rate decreases with altitude. (This occursbecause the air is less dense at higher altitudes.) The ta-ble shows performance data for a single-engine aircraft.

Altitude (1000 ft) 0 1 2 3 4 5

Climb rate (ft/min) 925 875 830 780 730 685

Altitude (1000 ft) 6 7 8 9 10

Climb rate (ft/min) 635 585 535 490 440

(a) Calculate upper and lower estimates for the time re-quired for this aircraft to climb from sea level to10,000 ft.

(b) If climb rate data were available in increments of500 ft, what would be the difference between a lowerand upper estimate of climb time based on 20 subdi-visions?

5-miscw33 24. A bicyclist is pedaling along a straight road for one hourwith a velocity v shown in Figure 5.15. She starts out fivekilometers from the lake and positive velocities take hertoward the lake. [Note: The vertical lines on the graph areat 10 minute (1/6 hour) intervals.](a) Does the cyclist ever turn around? If so, at what

time(s)?(b) When is she going the fastest? How fast is she going

then? Toward the lake or away?(c) When is she closest to the lake? Approximately how

close to the lake does she get?(d) When is she farthest from the lake? Approximately

how far from the lake is she then?

5-miscw33fig

10 20 30 40 50 60

!25

!20

!15

!10

!5

0

5

10

t(minutes)

v (km/hr)

Figure 5.155-1w27 25. Two cars travel in the same direction along a straight

road. Figure 5.16 shows the velocity, v, of each car attime t. CarB starts 2 hours after carA and carB reachesa maximum velocity of 50 km/hr.

(a) For approximately how long does each car travel?(b) Estimate car A’s maximum velocity.(c) Approximately how far does each car travel?

5-1w27fig

Car A

Car B

t (hr)

v (km/hr)

Figure 5.165-1w26 26. Two cars start at the same time and travel in the same

direction along a straight road. Figure 5.17 gives the ve-locity, v, of each car as a function of time, t. Which car:

(a) Attains the larger maximum velocity?(b) Stops first?(c) Travels farther?

5-1w26fig

Car A

Car B

t (hr)

v (km/hr)

Figure 5.17

40. Figure below shows a Riemann sum approximation with n subdivisions to∫ baf(x) dx.

(a) Is it a left- or right-hand approximation? Would the other one be larger or smaller?

(b) What are a, b, n and ∆x?

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5.2 THE DEFINITE INTEGRAL 271

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Exercises and Problems for Section 5.2Exercises

ins5-2dhhwnp001-002 In Exercises 1–2, rectangles have been drawn to approximate! 6

0g(x) dx.

(a) Do the rectangles represent a left or a right sum?(b) Do the rectangles lead to an upper or a lower estimate?(c) What is the value of n?(d) What is the value of!x?

5-2dhhwnp001 1.

6

g(x)

x

5-2dhhwnp005 2.

6

g(x)

x

5-2w1 3. Figure 5.29 shows a Riemann sum approximation with n

subdivisions to! b

af(x) dx.

(a) Is it a left- or right-hand approximation? Would theother one be larger or smaller?

(b) What are a, b, n and!x?

5-2w1fig20

x

Figure 5.29

5-2w2 4. Using Figure 5.30, draw rectangles representing each ofthe following Riemann sums for the function f on theinterval 0 ! t ! 8. Calculate the value of each sum.

(a) Left-hand sum with!t = 4(b) Right-hand sum with!t = 4(c) Left-hand sum with!t = 2(d) Right-hand sum with!t = 2

5-2w2fig2 4 6 8

48

121620242832

f(t)

t

Figure 5.30

ins5-2w11-13 In Exercises 5–10, use a calculator or a computer to find thevalue of the definite integral.

5-2dhhwnp002 5." 4

1

(x2 + x) dx 5-2w11 6." 3

0

2xdx

5-2w13 7." 1

!1

e!x2

dx 5-2dhhwnp003 8." 3

0

ln(y2 + 1) dy

5-2w12 9." 1

0

sin(t2)dt 5-2dhhwnp004 10." 4

3

"ez + z dz

5-2w7 11. Use the table to estimate! 40

0f(x)dx. What values of n

and!x did you use?

x 0 10 20 30 40f(x) 350 410 435 450 460


0f(x) dx.

x 0 3 6 9 12

f(x) 32 22 15 11 9


0f(x) dx.

x 0 3 6 9 12 15

f(x) 50 48 44 36 24 8

5-2w10 14. Write out the terms of the right-hand sum with n = 5

that could be used to approximate" 7

3

1

1 + xdx. Do not

evaluate the terms or the sum.5-miscw4 15. Use Figure 5.31 to estimate

! 20

0f(x) dx.

5-miscw4fig4 8 12 16 20

1

2

3

4

5

f(x)

x

Figure 5.31

41. Write out the terms of the right-hand sum with n = 5 that could be used to approximate∫ 7

31

1+x dx. Do notevaluate the terms or the sum.Is this an over or underestimate?

42. Given∫ 0

−2 f(x)dx = 4 and Figure below, estimate:

(a)∫ 2

0f(x)dx

(b)∫ 2

−2 f(x)dx

(c) The total shaded area.

!


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5-2w4 16. Use Figure 5.32 to estimate! 15

!10f(x)dx.

5-2w4fig !10 0 10

10

20

30

x

f(x)

Figure 5.32

5-miscw5 17. Using Figure 5.33, estimate! 5

!3f(x)dx.

5-miscw5fig

!3 !1 2 4

5

!20

!10

10

x

f(x)

Figure 5.33

Problems5-2w6 18. The graph of f(t) is in Figure 5.34. Which of the fol-

lowing four numbers could be an estimate of! 1

0f(t)dt

accurate to two decimal places? Explain your choice.I. !98.35 II. 71.84 III. 100.12 IV. 93.47

5-2w6fig0.5 1.0

20

40

60

80

100 f(t)

t

Figure 5.345-2w14 19. (a) What is the area between the graph of f(x) in Fig-

ure 5.35 and the x-axis, between x = 0 and x = 5?(b) What is

! 5

0f(x) dx?

5-2w14fig

3

5

!

Area = 7

"

Area = 6

x

f(x)

Figure 5.355-2plwnp1 20. Find the total area between y = 4 ! x2 and the x-axis

for 0 " x " 3.5-2plwnp2 21. (a) Find the total area between f(x) = x3 ! x and the

x-axis for 0 " x " 3.

(b) Find" 3

0

f(x)dx.

(c) Are the answers to parts (a) and (b) the same? Ex-plain.

ins5-2w15-21 In Problems 22–28, find the area of the regions between thecurve and the horizontal axis

5-2w16 22. Under y = 6x3 ! 2 for 5 " x " 10.5-2w15 23. Under the curve y = cos t for 0 " t " !/2.5-2w17 24. Under y = ln x for 1 " x " 4.5-2w19 25. Under y = 2 cos(t/10) for 1 " t " 2.5-2w18 26. Under the curve y = cos

#x for 0 " x " 2.

5-2w20 27. Under the curve y = 7 ! x2 and above the x-axis.5-2w21 28. Above the curve y = x4 ! 8 and below the x-axis.5-2w31 29. Use Figure 5.36 to find the values of

(a)! b

af(x) dx (b)

! c

bf(x) dx

(c)! c

af(x) dx (d)

! c

a|f(x)| dx

5-2w31fig

a b c

f(x)

!

Area = 13

#Area = 2

x

Figure 5.36

5-2w32 30. Given! 0

!2f(x)dx = 4 and Figure 5.37, estimate:

(a)! 2

0f(x)dx (b)

! 2

!2f(x)dx

(c) The total shaded area.

5-2w32fig

!2 2!2

2

f(x)

x

Figure 5.3743. Use the Fundamental Theorem of Calculus to compute the following definite integrals.

(a)∫ 3

12tdt

(b)∫ 5

26t+ 4dt

(c)∫ 5

11t dt

(d)∫ π/20

cos tdt

(e)∫ 3

27 ln(4) · 4tt

(f)∫ 2

0(3x2 + 1) dx.

44. The graphs in Figure ?? represent the velocity, v, of a particle moving along the x-axis for time 0 ≤ t ≤ 5. Thevertical scales of all graphs are the same. Identify the graph showing which particle:

(a) Has a constant acceleration.

(b) Ends up farthest to the left of where it started.

(c) Ends up the farthest from its starting point.

(d) Experiences the greatest initial acceleration.

(e) Has the greatest average velocity.

(f) Has the greatest average acceleration.

!


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5-miscw41 38. The graphs in Figure 5.49 represent the velocity, v, of aparticle moving along the x-axis for time 0 ! t ! 5.The vertical scales of all graphs are the same. Identifythe graph showing which particle:

(a) Has a constant acceleration.(b) Ends up farthest to the left of where it started.(c) Ends up the farthest from its starting point.(d) Experiences the greatest initial acceleration.(e) Has the greatest average velocity.(f) Has the greatest average acceleration.

5-miscw41fig

5t

v(I)

t

v

5

(II)

t

v

5

(III)

t

v

5

(IV)

t

v

5

(V)

Figure 5.49

5-3w40 39. A car speeds up at a constant rate from 10 to 70 mphover a period of half an hour. Its fuel efficiency (in milesper gallon) increases with speed; values are in the table.Make lower and upper estimates of the quantity of fuelused during the half hour.

Speed (mph) 10 20 30 40 50 60 70Fuel efficiency (mpg) 15 18 21 23 24 25 26

ins-s09ex-5-misc-ecnp006-007 In Problems 40–41, oil is pumped from a well at a rate ofr(t) barrels per day. Assume that t is in days, r!(t) < 0 andt0 > 0.

s09ex-5-misc-ecnp005 40. What does the value of! t0

0r(t) dt tells us about the oil

well?s09ex-5-misc-ecnp006 41. Rank in order from least to greatest:

" 2t0

0

r(t) dt,

" 2t0

t0

r(t) dt,

" 3t0

2t0

r(t) dt.

5-3w42 42. Height velocity graphs are used by endocrinologists tofollow the progress of children with growth deficiencies.Figure 5.50 shows the height velocity curves of an aver-age boy and an average girl between ages 3 and 18.

(a) Which curve is for girls and which is for boys? Ex-plain how you can tell.

(b) About how much does the average boy grow be-tween ages 3 and 10?

(c) The growth spurt associated with adolescence andthe onset of puberty occurs between ages 12 and 15for the average boy and between ages 10 and 12.5for the average girl. Estimate the height gained byeach average child during this growth spurt.

(d) When fully grown, about how much taller is the av-erage man than the average woman? (The averageboy and girl are about the same height at age 3.)

5-3w42fig2 4 6 8 10 12 14 16 18

2

4

6

8

10

x (years)

y (cm/yr)

Figure 5.50

ins-s11qz-5-3ecwnp004-006 In Problems 43–45, evaluate the expressions using Table 5.8.Give exact values if possible; otherwise, make the best possi-ble estimates using left-hand Riemann sums.

Table 5.8

tab:2011-02-08-082118t 0.0 0.1 0.2 0.3 0.4 0.5

f(t) 0.3 0.2 0.2 0.3 0.4 0.5

g(t) 2.0 2.9 5.1 5.1 3.9 0.8

s11qz-5-3ecwnp004 43." 0.5

0

f(t) dts11qz-5-3ecwnp005 44." 0.5

0.2

g!(t) dt

s11qz-5-3ecwnp006 45." 0.3

0

g (f(t)) dt

45. The graph of a continuous function f is given in Figure ??. Rank the following integrals in ascending numericalorder. Explain your reasons.

(a)∫ 2

0f(x) dx

(b)∫ 1

0f(x) dx

(c)∫ 2

0(f(x))1/2 dx

(d)∫ 2

0(f(x))2 dx.

!

5.3 THE FUNDAMENTAL THEOREM AND INTERPRETATIONS 281

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

38

40

42

44

46

48

50

52


5-miscw30 26. Annual coal production in the US (in billion tons peryear) is given in the table.6 Estimate the total amountof coal produced in the US between 1997 and 2009. Ifr = f(t) is the rate of coal production t years since1997, write an integral to represent the 1997–2009 coalproduction.

Year 1997 1999 2001 2003 2005 2007 2009Rate 1.090 1.094 1.121 1.072 1.132 1.147 1.073

5-3w27 27. The amount of waste a company produces,W , in tons perweek, is approximated byW = 3.75e!0.008t, where t isin weeks since January 1, 2005. Waste removal for thecompany costs $15/ton. How much does the companypay for waste removal during the year 2005?

5-3w23 28. A two-day environmental cleanup started at 9 am on thefirst day. The number of workers fluctuated as shown inFigure 5.44. If the workers were paid $10 per hour, howmuch was the total personnel cost of the cleanup?

5-3w23fig8 16 24 32 40 48

10

20

30

40

50

hours

workers

Figure 5.44

5-3w24 29. Suppose in Problem 28 that the workers were paid $10per hour for work during the time period 9 am to 5 pmand were paid $15 per hour for work during the rest of theday. What would the total personnel costs of the clean uphave been under these conditions?

5-3w25 30. A warehouse charges its customers $5 per day for ev-ery 10 cubic feet of space used for storage. Figure 5.45records the storage used by one company over a month.How much will the company have to pay?

5-3w25fig10 20 30

10,000

20,000

30,000

days

cubic feet

Figure 5.45

5-3w26 31. A cup of coffee at 90"C is put into a 20"C room whent = 0. The coffee’s temperature is changing at a rate ofr(t) = !7e!0.1t "C per minute, with t in minutes. Esti-mate the coffee’s temperature when t = 10.

5-miscw36 32. Water is pumped out of a holding tank at a rate of5 ! 5e!0.12t liters/minute, where t is in minutes sincethe pump is started. If the holding tank contains 1000liters of water when the pump is started, how much waterdoes it hold one hour later?

ins5-4w42-43 Problems 34–35 concern the graph of f # in Figure 5.46.

ins5-4w42-43fig

1 2 3 4x

f #(x)

Figure 5.46: Graph of f #, not f

5-4w42 33. Which is greater, f(0) or f(1)?5-4w43 34. List the following in increasing order:

f(4) ! f(2)

2, f(3) ! f(2), f(4) ! f(3).

5-3w29 35. A force F parallel to the x-axis is given by the graph inFigure 5.47. Estimate the work, W , done by the force,whereW =

! 16

0F (x) dx.

5-3w29fig4 8 10

14 16

!2

!1

1

2

x (meter)

force (newton)F

Figure 5.47

s09ex-5-misc-ecnp009 36. Let f(1) = 7, f #(t) = e!t2 . Use left- and right-handsums of 5 rectangles each to estimate f(2).

5-miscw37 37. The graph of a continuous function f is given in Fig-ure 5.48. Rank the following integrals in ascending nu-merical order. Explain your reasons.

(i)! 2

0f(x) dx (ii)

! 1

0f(x) dx

(iii)! 2

0(f(x))1/2 dx (iv)

! 2

0(f(x))2 dx.

5-miscw37fig0 1 2

100

x

f(x)

Figure 5.48

6http://www.eia.doe.gov/cneaf/coal/page/special/tbl1.html. Accessed May 2011.

46. The graph of g consists of two straight lines and a semicircle. Use it to evaluate each integral.

!

47. Evaluate the integral by interpreting in terms of areas

∫ 10

0

|x− 5|dx

48. Evaluate the following integrals:

!

49. Use part I of Fundamental Theorem of Calculus to find the derivative of the following functions:

(a) g(x) =∫ x0

√1 + 2tdt

(b) g(x) =∫ x1

ln tdt

(c) g(x) =∫ x1

ln tdt

(d) g(y) =∫ y2t2 sin tdt

50. Find the interval on which the curve

y =

∫ x

0

1

1 + t+ t2dt

is concave up.

51. If f(1) = 12, f ′ is continuous and∫ 4

1f ′(x)dx = 17 what is f(4)?

52. Evaluate the following indefinite integrals using substitution method.

!!

53. Evaluate the following defininte integral using substitution method.

!!

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