5.3 the fundamental theorem of calculus. (5.1 & 5.2 ...cjashley/spring2020/day23.pdf · 5.2 the...

Spring 2020 Math 115: Calculus I

5.3 The Fundamental Theorem of Calculus.(5.1 & 5.2 Measuring Distance −→ Definite Integral)

1. Water is being poured into the cone-shaped container, see figure 1 below. When the depth of thewater is 2.5 in, it is increasing at 3 in/min. At that time, how fast is the surface area, A, that iscovered by water increasing? [Hint: A = πrs, where r, s are as shown.]

Figure 1:

2. Grit, which is spread on roads in winter, is stored in mounds which are the shape of a cone. Asgrit is added to the top of a mound at 2 cubic meters per minute, the angle between the slant sideof the cone and the vertical remains 45 deg. How fast is the height of the mound increasing whenit is half a meter high? [Hint: Volume V = πr2h

3, where r is radius and h is height.]

5.1

3. Baby growth . The table below gives the expected growth rate, g(t), in ounces per week, of theweight of a baby in its first 54 weeks of life. Assume for this problem that g(t) is a decreasingfunction.

week t 0 9 18 27 36 45 54growth rate g(t) 6 6 4.5 3 3 3 2

(a) Using six subdivisions, find an overestimate and underestimate for the total weight gained bya baby over its first 54 weeks of life.

(b) How frequently over the 54 week period would you need the data for g(t) to be measured tofind overestimates and underestimates for the total weight gain over this time period that differby 8 oz?

4. Freaky fast again! . The record time for the 100 meter dash is 9.58 seconds, set by Usain Boltat a race in 2009. Let v(t) be Bolt’s velocity, in meters per second, t seconds after Bolt starts therace. Several values of v(t) are shown below. Assume the v(t) is an increasing function for the firstthree seconds of the race.

t 0 0.5 1 1.5 2 2.5 3v(t) 0.6 3.5 5.8 7.7 9.1 10.1 10.6

(a) Estimate Bolt’s instantaneous acceleration 1.75 seconds after Bolt starts the race. Rememberto include units.

Problems are from Calculus by Hughes-Hallett, Gleason, et al., 7th Edition & Exam Shop.


(b) Based on the data provided, give the best possible underestimate of the distance run by Boltduring the first 3 seconds of his race.

(c) How often would we need to measure Bolt’s velocity so that the difference between the bestpossible underestimate and the best possible overestimate of the distance he runs in the first 3seconds is 2 meters?

5. Find the difference between the left sum estimate and the right sum estimate for the displacementat velocity v(t) = 3t+ 4 on the interval a ≤ t ≤ b for n subdivisions.

5.2 The Definite Integral

(a) Compute (in exact form) the left-hand sum of this integral using n subdivisions.

(b) Compute (in exact form) the right-hand sum of this integral using n subdivisions.

(c) Evaluate the limits of the quantities you computed in (a) and (b) when ∆t→ 0.

6. Use the table to estimate∫ 40

0f(x) dx. What values of n and ∆x did you use?

286 Chapter Five KEY CONCEPT: THE DEFINITE INTEGRAL

Exercises and Problems for Section 5.2Exercises

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?

1.

6

g(x)

x

2.

6

g(x)

x

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 the

other one be larger or smaller?

(b) What are a, b, n and ∆x?

20x

Figure 5.29

4. Using Figure 5.30, draw rectangles representing each of

the following Riemann sums for the function f on the

interval 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

2 4 6 8

48

121620242832

f(t)

t

Figure 5.30

In Exercises 5–10, use a calculator or a computer to find the

value of the definite integral.

5.

∫ 4

1

(x2 + x) dx 6.

∫ 3

0

2xdx

7.

∫ 1

−1

e−x2

dx 8.

∫ 3

0

ln(y2 + 1) dy

9.

∫ 1

0

sin(t2)dt 10.

∫ 4

3

√ez + z dz


0f(x)dx. What values of n

and ∆x did you use?

x 0 10 20 30 40

f(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

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.

15. Use Figure 5.31 to estimate∫ 20

0f(x) dx.

4 8 12 16 20

1

2

3

4

5

f(x)

x

Figure 5.31

ca

7. Use the figure to estimate∫ 20

0f(x) dx.


Exercises and Problems for Section 5.2Exercises

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?

1.

6

g(x)

x

2.

6

g(x)

x

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 the

other one be larger or smaller?

(b) What are a, b, n and ∆x?

20x

Figure 5.29

4. Using Figure 5.30, draw rectangles representing each of

the following Riemann sums for the function f on the

interval 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

2 4 6 8

48

121620242832

f(t)

t

Figure 5.30

In Exercises 5–10, use a calculator or a computer to find the

value of the definite integral.

5.

∫ 4

1

(x2 + x) dx 6.

∫ 3

0

2xdx

7.

∫ 1

−1

e−x2

dx 8.

∫ 3

0

ln(y2 + 1) dy

9.

∫ 1

0

sin(t2)dt 10.

∫ 4

3

√ez + z dz


0f(x)dx. What values of n

and ∆x did you use?

x 0 10 20 30 40

f(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

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.


0f(x) dx.

4 8 12 16 20

1

2

3

4

5

f(x)

x

Figure 5.318. (a) What is the area between the graph of f(x) and the x−axis, between x = 0 and x = 5?

(b) What is∫ 5

0f(x) dx?

9. (a) Use the figure below to find∫ 0

−3 f(x) dx.?

(b) If the area of the shaded region is A, estimate∫ 4

−3 f(x) dx?

10. Without computation, decide of∫ 2π

0e−x sin(x) dx is positive or negative.

11. Sketch the graph of a function f (you do not need to give a formula for f) on an interval [a, b] with

the property that with n = 2 subdivisions,∫ baf(x) dx < Left-hand sum < Right-hand sum.

12. Explain what is wrong with the statement:



5.2 THE DEFINITE INTEGRAL 287


−10f(x)dx.

−10 0 10

10

20

30

x

f(x)

Figure 5.32

17. Using Figure 5.33, estimate∫ 5

−3f(x)dx.

−3 −1 2 4

5

−20

−10

10

x

f(x)

Figure 5.33

Problems

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

0.5 1.0

20

40

60

80

100 f(t)

t

Figure 5.34

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?

3

5

✠

Area = 7

✒

Area = 6

x

f(x)

Figure 5.35

20. Find the total area between y = 4 − x2 and the x-axis

for 0 ≤ x ≤ 3.

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.

In Problems 22–28, find the area of the regions between the

curve and the horizontal axis

22. Under y = 6x3 − 2 for 5 ≤ x ≤ 10.

23. Under the curve y = cos t for 0 ≤ t ≤ π/2.

24. Under y = ln x for 1 ≤ x ≤ 4.

25. Under y = 2 cos(t/10) for 1 ≤ t ≤ 2.

26. Under the curve y = cos√

x for 0 ≤ x ≤ 2.

27. Under the curve y = 7 − x2 and above the x-axis.

28. Above the curve y = x4 − 8 and below the x-axis.

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

a b c

f(x)

✠

Area = 13

■Area = 2

x

Figure 5.36

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.

−2 2−2

2

f(x)

x

Figure 5.37


31. (a) Using Figure 5.38, find∫ 0

−3f(x) dx.

(b) If the area of the shaded region is A, estimate∫ 4

−3f(x) dx.

−4 −3 −2 −1 1 2 3

4

5

−1

1

x

f(x)

Figure 5.38

32. Use Figure 5.39 to find the values of

(a)∫ 2

0f(x) dx (b)

∫ 7

3f(x) dx

(c)∫ 7

2f(x) dx (d)

∫ 8

5f(x) dx

2 4 6 8 10−2

−1

1

2f(x)

x

Figure 5.39: Graph consists of a semicircle and

line segments

33. (a) Graph f(x) = x(x + 2)(x − 1).

(b) Find the total area between the graph and the x-axis

between x = −2 and x = 1.

(c) Find∫ 1

−2f(x) dx and interpret it in terms of areas.

34. Compute the definite integral∫ 4

0cos

√x dx and interpret

the result in terms of areas.

35. Without computation, decide if∫ 2π

0e−x sin x dx is posi-

tive or negative. [Hint: Sketch e−x sin x.]

36. Estimate∫ 1

0e−x2

dx using n = 5 rectangles to form a

(a) Left-hand sum (b) Right-hand sum

37. (a) On a sketch of y = lnx, represent the left Riemann

sum with n = 2 approximating∫ 2

1ln x dx. Write

out the terms in the sum, but do not evaluate it.

(b) On another sketch, represent the right Riemann sum

with n = 2 approximating∫ 2

1ln x dx. Write out the

terms in the sum, but do not evaluate it.

(c) Which sum is an overestimate? Which sum is an un-

derestimate?

38. (a) Draw the rectangles that give the left-hand sum ap-

proximation to∫ π

0sin x dx with n = 2.

(b) Repeat part (a) for∫ 0

−πsin x dx.

(c) From your answers to parts (a) and (b), what is

the value of the left-hand sum approximation to∫ π

−πsin x dx with n = 4?

39. (a) Use a calculator or computer to find∫ 6

0(x2 + 1) dx.

Represent this value as the area under a curve.

(b) Estimate∫ 6

0(x2 + 1) dx using a left-hand sum with

n = 3. Represent this sum graphically on a sketch

of f(x) = x2 + 1. Is this sum an overestimate or

underestimate of the true value found in part (a)?

(c) Estimate∫ 6

0(x2+1) dx using a right-hand sum with

n = 3. Represent this sum on your sketch. Is this

sum an overestimate or underestimate?

40. (a) Graph f(x) ={

1 − x 0 ≤ x ≤ 1x − 1 1 < x ≤ 2.

(b) Find

∫ 2

0

f(x) dx.

(c) Calculate the 4-term left Riemann sum approxima-

tion to the definite integral. How does the approxi-

mation compare to the exact value?

41. Estimate∫ 2

1x2 dx using left- and right-hand sums with

four subdivisions. How far from the true value of the in-

tegral could your estimate be?

42. Without computing the sums, find the difference between

the right- and left-hand Riemann sums if we use n = 500

subintervals to approximate∫ 1

−1(2x3 + 4) dx.

43. Sketch the graph of a function f (you do not need to give

a formula for f ) on an interval [a, b] with the property

that with n = 2 subdivisions,

∫ b

a

f(x) dx < Left-hand sum < Right-hand sum.

44. Write a few sentences in support of or in opposition to

the following statement:

“If a left-hand sum underestimates a definite integral

by a certain amount, then the corresponding right-hand

sum will overestimate the integral by the same amount.”

45. Consider the integral∫ 2

1(1/t) dt in Example 1. By divid-

ing the interval 1 ≤ t ≤ 2 into 10 equal parts, we can

show that

0.1(

1

1.1+

1

1.2+ . . . +

1

2

)≤

∫ 2

1

1

tdt

and∫ 2

1

1

tdt ≤ 0.1

(1

1+

1

1.1+ . . . +

1

1.9

).

(a) Now divide the interval 1 ≤ t ≤ 2 into n equal parts

to show that

n∑

r=1

1

n + r<

∫ 2

1

1

tdt <

n−1∑

r=0

1

n + r.

(b) Show that the difference between the upper and

lower sums in part (a) is 1/(2n).

(c) The exact value of∫ 2

1(1/t) dt is ln 2. How large

should n be to approximate ln 2 with an error of at

most 5 · 10−6, using one of the sums in part (a)?

(a) For any function,∫ 3

1f(x) dx is the area between the graph of f and the x−axis on 1 ≤ x ≤ 3.

(b) The left hand sum with 10 subdivisions for the integral∫ 2

1sin(x) dx is

0.1(

sin(1.1) + ...+ sin(2)).

13. Give an example of:

(a) A function f and an interval [a, b] such that∫ baf(x) dx is negative.

(b) A function f such that∫ 3

1f(x) dx <

∫ 2

1f(x) dx.

14. Positive or negative? Consider the function

f(x) = e−x sin(x)

(a) Use calculus to sketch the graph of f(x) on the interval [0, 2π].

(b) Based on your plot, is∫ 2π

0e−x sin(x)dx positive or negative?

15. Consider the integral ∫ 1

0

xdx.

(a) Compute (in exact form) the left-hand sum of this integral using n subdivisions.

(b) Compute (in exact form) the right-hand sum of this integral using n subdivisions.

(c) Evaluate the limits of the quantities you computed in (a) and (b) when ∆t→ 0.



16. Square root . The Figure below shows a plot of y =√t and we are interested in the evaluation of

the definite integral∫ 15

0

√t dt.

(a) For this integral, are left sums always overestimates, always underestimates, or could they beeither? What about right sums?

(b) Use a Riemann sum with 5 equal subdivisions to find a lower estimate for the integral. Showyour answer to three decimal places.

(c) Use a Riemann sum with 5 equal subdivisions to find an upper estimate for the integral. Showyour answer to three decimal places.

(d) Repeat (b) and (c) with 10 equal subdivisions. Show your answers to three decimal places.

17. At a recent UM football game, a football scientist was measuring the excitement density E(x), incheers per foot, in a one hundred foot row of the football stadium where x is the distance in feetfrom the beginnning of the row. He took measurements every twenty feet and the data is recordedin this table.

x 0 20 40 60 80 100E(x) 30 24 19 16 13 7

Assume for this problem that E(x) is a decreasing function for 0 ≤ x ≤ 100.

(a) Write a right sum and a left sum which approximate the total cheers in the row. Be sure towrite all of the terms for each sum.

(b) Indicate whether the right and left sums are overestimates or underestimates for the totalnumber of cheers in the row.

(c) How many measurements must the scientist take to guarantee that the left sum approximatesthe total number of cheers in the row within 5 cheers of the actual number?

5.3: The Fundamental Theorem of Calculus

1. If f(t) is measured in dollars per year and t is measured in years, what are the units of∫ baf(t) dt?

2. If f(x) is measured in pounds and x is measured in feet, what are the units of∫ baf(x) dx?



3. Explain in words what the integral represents and give units:∫ 6

0a(t) dt, where a(t) is acceleration

in km/hr2 and t is time in hours.

4. Let f(t) = F ′(t). Write the integral∫ baf(t) dt and evaluate it using the Fundamental Theorem

of Calculus for F (t) = 7 · 4t; a = 2, b = 3.

5. Pollution is removed from a lake on day t at a rate of f(t) kg/day.

i. Explain the meaning of the statement f(12) = 500.

ii. If∫ 15

5f(t) dt = 4000, give units of the 5, the 15, and the 4000.

iii. Give the meaning of∫ 15

5f(t) dt = 4000.

6. Water is leaking out of a tank at a rate of R(t) gallons/hour, where t is measured in hours.

i. Write a definite integral that expresses the total amount of water that leaks out in the firsttwo hours.

ii. In the figure below, shade the region whose area represents the total amount of water thatleaks out in the first two hours.

iii. Give an upper and lower estimate of the total amount of water that leaks out in the firsttwo hours.

5.3 THE FUNDAMENTAL THEOREM AND INTERPRETATIONS 295

17. (a) If F (t) = 12

sin2 t, find F ′(t).

(b) Find

∫ 0.4

0.2

sin t cos t dt two ways:

(i) Numerically.

(ii) Using the Fundamental Theorem of Calculus.

18. (a) If F (x) = ex2

, find F ′(x).

(b) Find

∫ 1

0

2xex2

dx two ways:

(i) Numerically.

(ii) Using the Fundamental Theorem of Calculus.

19. Pollution is removed from a lake on day t at a rate of

f(t) kg/day.

(a) Explain the meaning of the statement f(12) = 500.

(b) If∫ 15

5f(t) dt = 4000, give the units of the 5, the

15, and the 4000.

(c) Give the meaning of∫ 15

5f(t) dt = 4000.

20. Oil leaks out of a tanker at a rate of r = f(t) gallons per

minute, where t is in minutes. Write a definite integral

expressing the total quantity of oil which leaks out of the

tanker in the first hour.

21. Water is leaking out of a tank at a rate of R(t) gal-

lons/hour, where t is measured in hours.

(a) Write a definite integral that expresses the total

amount of water that leaks out in the first two hours.

(b) In Figure 5.42, shade the region whose area repre-

sents the total amount of water that leaks out in the

first two hours.

(c) Give an upper and lower estimate of the total amount

of water that leaks out in the first two hours.

1 2

2

1

t

R(t)

Figure 5.42

22. As coal deposits are depleted, it becomes necessary to

strip-mine larger areas for each ton of coal. Figure 5.43

shows the number of acres of land per million tons of coal

that will be defaced during strip-mining as a function of

the number of million tons removed, starting from the

present day.

(a) Estimate the total number of acres defaced in ex-

tracting the next 4 million tons of coal (measured

from the present day). Draw four rectangles under

the curve, and compute their area.

(b) Re-estimate the number of acres defaced using rect-

angles above the curve.

(c) Use your answers to parts (a) and (b) to get a better

estimate of the actual number of acres defaced.

1 2 3 4 5

1

2

3

4

0.2

million tons ofcoal extracted(measuredfrom presentday)

acres defacedper million tons

Figure 5.43

23. The rate at which the world’s oil is consumed (in billions

of barrels per year) is given by r = f(t), where t is in

years and t = 0 is the start of 2004.

(a) Write a definite integral representing the total quan-

tity of oil consumed between the start of 2004 and

the start of 2009.

(b) Between 2004 and 2009, the rate was modeled by

r = 32e0.05t. Using a left-hand sum with five subdi-

visions, find an approximate value for the total quan-

tity of oil consumed between the start of 2004 and

the start of 2009.

(c) Interpret each of the five terms in the sum from

part (b) in terms of oil consumption.

24. A bungee jumper leaps off the starting platform at time

t = 0 and rebounds once during the first 5 seconds.

With velocity measured downward, for t in seconds and

0 ≤ t ≤ 5, the jumper’s velocity is approximated5 by

v(t) = −4t2 + 16t meters/sec.

(a) How many meters does the jumper travel during the

first five seconds?

(b) Where is the jumper relative to the starting position

at the end of the five seconds?

(c) What does∫ 5

0v(t) dt represent in terms of the

jump?

25. An old rowboat has sprung a leak. Water is flowing into

the boat at a rate, r(t), given in the table.

(a) Compute upper and lower estimates for the volume

of water that has flowed into the boat during the 15minutes.

(b) Draw a graph to illustrate the lower estimate.

t minutes 0 5 10 15

r(t) liters/min 12 20 24 16

5Based on www.itforus.oeiizk.waw.pl/tresc/activ//modules/bj.pdf. Accessed Feb 12, 2012.

7. The graph of a continuous function f is given in the figure below. Rank the following integralsin ascending numerical order. Explain your reasons.

i.∫ 2

0f(x) dx

ii.∫ 1

0f(x) dx

iii.∫ 2

0

(f(x)

)1/2dx

iv.∫ 2

0

(f(x)

)2dx


26. Annual coal production in the US (in billion tons per

year) is given in the table.6 Estimate the total amount

of coal produced in the US between 1997 and 2009. If

r = f(t) is the rate of coal production t years since

1997, write an integral to represent the 1997–2009 coal

production.

Year 1997 1999 2001 2003 2005 2007 2009

Rate 1.090 1.094 1.121 1.072 1.132 1.147 1.073

27. The amount of waste a company produces, W , in tons per

week, is approximated by W = 3.75e−0.008t , where t is

in weeks since January 1, 2005. Waste removal for the

company costs $15/ton. How much does the company

pay for waste removal during the year 2005?

28. A two-day environmental cleanup started at 9 am on the

first day. The number of workers fluctuated as shown in

Figure 5.44. If the workers were paid $10 per hour, how

much was the total personnel cost of the cleanup?

8 16 24 32 40 48

10

20

30

40

50

hours

workers

Figure 5.44

29. Suppose in Problem 28 that the workers were paid $10

per hour for work during the time period 9 am to 5 pm

and were paid $15 per hour for work during the rest of the

day. What would the total personnel costs of the cleanup

have been under these conditions?

30. A warehouse charges its customers $5 per day for ev-

ery 10 cubic feet of space used for storage. Figure 5.45

records the storage used by one company over a month.

How much will the company have to pay?

10 20 30

10,000

20,000

30,000

days

cubic feet

Figure 5.45

31. A cup of coffee at 90◦C is put into a 20◦C room when

t = 0. The coffee’s temperature is changing at a rate of

r(t) = −7e−0.1t ◦C per minute, with t in minutes. Esti-

mate the coffee’s temperature when t = 10.

32. Water is pumped out of a holding tank at a rate of

5 − 5e−0.12t liters/minute, where t is in minutes since

the pump is started. If the holding tank contains 1000liters of water when the pump is started, how much water

does it hold one hour later?

Problems 33–34 concern the graph of f ′ in Figure 5.46.

1 2 3 4x

f ′(x)

Figure 5.46: Graph of f ′, not f

33. Which is greater, f(0) or f(1)?

34. List the following in increasing order:f(4) − f(2)

2, f(3) − f(2), f(4) − f(3).

35. A force F parallel to the x-axis is given by the graph in

Figure 5.47. Estimate the work, W , done by the force,

where W =∫ 16

0F (x) dx.

4 8 10

14 16

−2

−1

1

2

x (meter)

force (newton)

F

Figure 5.47

36. Let f(1) = 7, f ′(t) = e−t2 . Use left- and right-hand

sums of 5 rectangles each to estimate f(2).

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.

0 1 2

100

x

f(x)

Figure 5.48

6http://www.eia.doe.gov/cneaf/coal/page/special/tbl1.html. Accessed May 2011.Problems are from Calculus by Hughes-Hallett, Gleason, et al., 7th Edition & Exam Shop.


8. The graphs in the figure below represent the velocity, v, of a particle moving along the x-axisfor time 0 ≤ t ≤ 5. The vertical scales of all graphs are the same. Identify the graph showingwhich particle:

i. Has a constant acceleration.

ii. Ends up farthest to the left of where it started.

iii. Ends up the farthest from its starting point.

iv. Experiences the greatest initial acceleration.

v. Has the greatest average velocity.

vi. Has the greatest average acceleration.

5.3 THE FUNDAMENTAL THEOREM AND INTERPRETATIONS 297

38. The graphs in Figure 5.49 represent the velocity, v, of a

particle moving along the x-axis for time 0 ≤ t ≤ 5.

The vertical 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.

5t

v(I)

t

v

5

(II)

t

v

5

(III)

t

v

5

(IV)

t

v

5

(V)

Figure 5.49

39. A car speeds up at a constant rate from 10 to 70 mph

over a period of half an hour. Its fuel efficiency (in miles

per gallon) increases with speed; values are in the table.

Make lower and upper estimates of the quantity of fuel

used during the half hour.

Speed (mph) 10 20 30 40 50 60 70

Fuel efficiency (mpg) 15 18 21 23 24 25 26

In Problems 40–41, oil is pumped from a well at a rate of

r(t) barrels per day. Assume that t is in days, r′(t) < 0 and

t0 > 0.

40. What does the value of∫ t0

0r(t) dt tells us about the oil

well?

41. Rank in order from least to greatest:

∫ 2t0

0

r(t) dt,

∫ 2t0

t0

r(t) dt,

∫ 3t0

2t0

r(t) dt.

42. Height velocity graphs are used by endocrinologists to

follow 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 and

the onset of puberty occurs between ages 12 and 15

for the average boy and between ages 10 and 12.5

for the average girl. Estimate the height gained by

each average child during this growth spurt.

(d) When fully grown, about how much taller is the av-

erage man than the average woman? (The average

boy and girl are about the same height at age 3.)

2 4 6 8 10 12 14 16 18

2

4

6

8

10

x (years)

y (cm/yr)

Figure 5.50

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

t 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

43.

∫ 0.5

0

f(t) dt 44.

∫ 0.5

0.2

g′(t) dt

45.

∫ 0.3

0

g (f(t)) dt



Math 115 / Final (December 17, 2015) page 8

8. [8 points] Elur Niahc keeps a bucket in his backyard. It contains water, and the water is twoinches deep when a rainstorm starts. The storm lasts 20 minutes.

• Let h be the depth, in inches, of the water in the bucket.

• Let V (h) be the volume, in gallons, of water in the bucket when the water is h inchesdeep. Assume that V (h) is invertible and differentiable.

• Let r(t) be the rate at which the volume of water in the bucket is increasing, measuredin gallons per minute, t minutes after the storm starts. Assume that r(t) > 0 for theentire duration of the rainstorm.

For each of the questions below, circle the one best answer. No points will be given forambiguous or multiple answers.

a. [2 points] Which of the following expressions represents the depth, in inches, of water inthe bucket when the bucket contains 3 gallons of water?

i. V (3) ii. V −1(3) iii. 2 + V (3) iv. 2 + V ′(3)

b. [2 points] Which of the following is the best interpretation of the equation(V −1)′(3) = 0.4?

i. The rate at which the depth of the water in the bucket is changing is increasing by0.4 inches per minute when the bucket contains 3 gallons of water.

ii. During the third minute of the rainstorm, the volume of the water in the bucketincreases by about 0.4 gallons.

iii. When the depth of the water in the bucket increases from 2.8 to 3 inches, thevolume of the water increases by about 0.08 gallons.

iv. When the volume of the water in the bucket is 3 gallons, the depth of the water isabout 0.2 inches less than the depth will be when the volume is 3.5 gallons.

c. [2 points] Which expression represents the volume, in gallons, of water in the bucketafter the rainstorm ends?

i. V

(2 +

∫ 20

0r(t) dt

)

ii. 2 +

∫ 20

0r(t) dt

iii.

∫ 20

0V (2 + r′(t)) dt

iv. V (2) +

∫ 20

0r(t) dt

v. 2 + V (20)

vi.

∫ 20

0V ′(t) dt

d. [2 points] Which of the following represents the average rate of change of the volume, ingallons per minute, of the water in the bucket during the rainstorm?

i.V (20) − V (0)

20

ii.r(20) − r(0)

20

iii.1

20

∫ 20

0r(t) dt

iv.1

20

∫ 20

0r′(t) dt

University of Michigan Department of Mathematics Fall, 2015 Math 115 Exam 3 Problem 8 (rain bucket)



Math 115 / Final (December 17, 2013) page 4

3. [12 points] The function g(t) is the volume of water in the town water tank, in thousands ofgallons, t hours after 8 A.M. A graph of g′(t), the derivative of g(t), is shown below. Notethat g′(t) is a piecewise-linear function.

t

g′(t)

1 2 3 4 5 6 7

−2

−1

1

2

a. [4 points] Write an integral which represents the average rate of change, in thousands ofgallons per hour, of the volume of water in the tank between 9 A.M. and 1 P.M. Computethe exact value of this integral.

b. [2 points] At what time does the tank have the most water in it? At what time does ithave the least water?

Answer: The tank has the most water in it at .

The tank has the least water in it at .

c. [6 points] Suppose that g(3) = 1. Sketch a detailed graph of g(t) and give both coordinatesof the point on the graph at t = 7.

t1 2 3 4 5 6 7

−1

1

2

3

4

5

6

7

University of Michigan Department of Mathematics Fall, 2013 Math 115 Exam 3 Problem 3 (water tank)



Math 115 / Final (Dec 12, 2014) page 8

7. [10 points] A history professor gives a 60 minute lecture, while one eager undergraduatestudent takes notes by typing what the professor says, word for word. Unfortunately, thestudent cannot always type as quickly as the professor is speaking.Functions p and u are defined as follows. When t minutes have passed since the start ofthe lecture, the professor is speaking at a rate of p(t) words per minute (wpm) while theundergraduate student is typing at a rate of u(t) words per minute (wpm). Shown below aregraphs of y = p(t) (dashed) and y = u(t) (solid).

10

20

30

40

50

60

5 10 15 20 25 30 35 40 45 50 55 60t (min)

y (wpm)

y = u(t)

y = p(t)

a. [2 points] How many minutes after the start of the lecture is the student typing mostquickly?

Answer:

b. [3 points] Write a definite integral equal to the number of words the student types betweenthe start of the lecture and the time the professor reaches the 600th word of the lecture.You do not need to evaluate the integral.

Answer:

c. [3 points] How many minutes after the start of the lecture is the student furthest behind intyping up the lecture? (In other words, after how many minutes is the difference betweenthe total number of words the professor has spoken and the total number of words thestudent has typed the greatest?)

Answer:

d. [2 points] What is the average rate, in words per minute, at which the professor is speakingbetween t = 40 and t = 60?

Answer:University of Michigan Department of Mathematics Fall, 2014 Math 115 Exam 3 Problem 7 (typing)



Math 115 / Final (December 19, 2016) do not write your name on this exam page 5

This problem continues the investigation of Xanthippe’s donuts.

4. [10 points] For your convenience, the graphs of p(t) and q(t) are reprinted below. Recall:

• The rate, in donuts per hour, at which Xanthippe makes donuts t hours after 7 am ismodeled by the function p(t).

• The rate, in donuts per hour, at which customers purchase donuts t hours after 7 am ismodeled by the function q(t).

• Assume that at 7 am, Xanthippe begins with no donuts in stock.

0.5 1 1.5 2 2.5 3 3.5 4

20

40

60

80

100

y = p(t)

y = q(t)

t

y

a. [4 points] Estimate the total number of donuts produced by 10 am using a right-handRiemann sum with two equal subintervals. Be sure to write down all the terms in yoursum. Is your answer an underestimate or overestimate?

Answer: donuts produced by 10 am ≈

This is an (circle one) Overestimate Underestimate

b. [4 points] The number of donuts in stock t hours after 7 am is modeled by the functions(t). Estimate the t-values for all critical points of s(t) in the interval 0 < t < 4, andestimate all values of t in the interval 0 < t < 4 at which s(t) has a local extremum. Foreach answer blank write none if appropriate. You do not need to justify your answers.

Answer: Critical point(s) at t =

Local max(es) at t = Local min(s) at t =

c. [2 points] At what time is the number of donuts that Xanthippe has in stock thegreatest? Round your answer to the nearest half hour. You do not need to justify youranswer.

Answer:

University of Michigan Department of Mathematics Fall, 2016 Math 115 Exam 3 Problem 4 (more donuts)


5.3 the fundamental theorem of calculus. (5.1 & 5.2 ...cjashley/spring2020/day23.pdf · 5.2 the...

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