math 10360 { final review - university of notre damem10360/handout/10360finalreview.pdf · math...

Math 10360 – Final Review

1. Find the equation of the tangent line to the curve y = ln

(5x2 − 1

x2 + 3

)at x = 1.

2. The density of a circular disc of 10 cm is given by the function in polar coordinates ρ(r, θ) =2θ√

400− r2gram/cm2. Find the total mass of the disc.

3. Find the equation of the tangent line to the graph of the function f(x) = ex + xe at x = 1.

4. It was observed that a radioactive substance reduces its mass from 10 grams to 2 grams after 3 hours. (a)Find a formula for its mass y(t) after t hours, (b) What is the half-life of the radioactive substance?

Answer: (a) y(t) = 10et3

ln(0.2)= 10(0.2)t/3; (b) Half-life =

3 ln(0.5)

ln(0.2)

5. Consider the region bounded by y = 1− x, y = ex, x = 1.

(a) Find the area of the region.

(b) Find the volume of the solid obtained by rotating the region about (i) x = 1, (ii) about y = −1.

(c) Consider the solid whose base the given region. If the cross-sections perpendicular to the x-axis are semi-circles.

6. A colony of wasps is building an underground nest. They start by finding an empty hole and beginconstructing cells at a rate of

dy

dt= e−0.5y

in tens of cells per day. Find the number of cells y in terms of t. Assume that y(0) = 0

7. An open tank has the shape of a right circular cone is filled with a fluid of weight-density 900 kg/m3. Thetank is 8 meters across the top and 6 meters high. How much work is done in emptying the tank by pumpingthe water over the top edge? Take the acceleration due to gravity as 10m/s2.Hint: Similar triangles.

8. Parametrize the shaded region R as a singleregion. Evaluate the following integral∫∫

R(x+ y2)dA

9. Consider a solid whose base is the region R. If theslice perpendicular to the y−axis are semicircles, findthe volume of the solid. 2-2 0 x

y

2

R

10. Let y(t) be the solution to the initial value problem y′ = y3 − 2y, y(−1) = 1.(a) Find T3(t), the 3rd-degree Taylor polynomial of y(t) about −1. (Ans: 1 − (t + 1) − 1

2(t + 1)2 + 5

6(t + 1)3)

(b) Use (a) to approximate y(−1.1) (Ans: T3(−1.1) = 1.09)

11a. Solve the following initial value problems:dy

dx= xy2 + x; y(0) = 1.

11b. Find the equation of y = f(x) if its slope is given by 4 sin2(x) and that it passes through the point (π, 5).

1

12. Assuming that the pattern continues, which of the following series are geometric series? Find the sum ofthe geometric series that converges.

(a)0.5

(0.03)2+

0.5

(0.03)3+

0.5

(0.03)4+ · · · (Ans: Divergent geometric series)

(b) 5 + 6(0.03) + 7(0.03)2 + 8(0.03)3 + 9(0.03)4 + · · · (Ans: Not geometric)

(c) 5− 5( eπ

)+ 5

( eπ

)2− 5

( eπ

)3+ 5

( eπ

)4− · · · (Ans:

5

1 + eπ

=5π

π + e)

(d) 2 +4

e+

8

e2+

16

e3+

32

e4+ · · · (Ans:

2

1 − 2/e)

13a. The velocity of a particle moving on a straight line is given by v(t) =1

6− 5t+ t2for t > 3. What is the

total change in the position of the particle over the time interval 4 < t < 5.

13b. Find the Nth partial sum of the series

∞∑n=4

1

6− 5n+ n2. Hence find the sum of the series.

14. Evaluate the following indefinite integrals:

a.

∫e2x − 2e−x + e+ x3/2 dx d.

∫cos2(2x) dx g.

∫xe3x dx

b.

∫cos(4x) cos(5x) dx e.

∫ √30

x√9− x2

dx h.

∫ 3

1x2 lnx dx

c.

∫arcsin(2x)√

1− 4x2dx f .

∫sin5(2x) dx i.

∫x+ 3

2x2 − 3x+ 1dx

15a. The Maclaurin series of ex is given as follows: ex =∞∑k=0

1

k!xk for all real number x.

(i) Find the 3rd Taylor polynomial of ex at 0 and estimate e0.2. (Ans: 1.22133)

(ii) Write down the error of your estimate in Part (b) as an infinite series.(iii) Use the Maclaurin polynomial for e−x

2/2 of degree 4 to estimate∫ 10 e−x2/2dx. (Ans: 0.858333)

(iv) Find the Taylor series for e2x−2 centered at 1. Be sure to simplify the coefficients.

15b. Use (i) Trapezoidal rule, (ii) Simpson’s rule to estimate∫ 10 e−x2/2dx.

16. Explain why the integral

∫ 1

0

xdx3√

1− x2is improper. Evaluate the integral.

17. The 4th-degree polynomial of the function f(x) about −2 is given by

P4(x) = 5− 2(x+ 2) + 3(x+ 2)2 − (x+ 2)4.

(a) What is the slope of f(x) at x = −2? Find the equation of the tangent line to the graph of f(x) at x = −2.Give your answer in the form y = mx+ b. (Ans: y = −2x + 1)

(b) Write down the values of f(−2), f ′′(−2), f ′′′(−2), and f (4)(−2).(Ans: f(−2) = 5, f ′′(−2) = 6, f ′′′(−2) = 0, and f(4)(−2) = −24)

18. Find the following liimit: limn→∞

(1 +

1

n

)3n

.

19. Using Lagrange Multipliers, find the critical points for the optimization of f(x, y) = 5x0.4y0.6 subjected tothe constraint g(x, y) = x+ 2y − 20.

2

20. A chain 10 meter long weighs 4 kg per meter with a 20 kg pounds load attached at the end is hung froma platform 15 meter above the ground. How much work is required to lift the whole chain and the load to thetop of the platform?

21. A tank has 100 liter of solution containing 4 kg of salt. A mixture with 0.01kg of salt per liter of water isadded at a rate of 0.2 liters per minute. At the same time, 0.1 liter of well-stirred mixture are removed eachminute. Find the amount of salt in the tank as a function of time.

22. Consider the following initial value problem

dy

dt= t− y2; y(1) = 1.

Use Euler’s method with three equal steps to estimate y(1.3).

23. Suppose a patient take 5 milligrams of a certain drug at the beginning of each day. The drug is eliminatedfrom the patient’s system in such a way that a single dose will only have 80% remaining in the patient’s systemafter one day. (a) By the end of the third day, the patient has taken a total of three doses. How many milligramsremain in his system just before he takes the next dose? (b) Assuming the treatment is continued indefinitely,use an infinite geometric series to approximate the amount remaining in the patient’s system after a very longperiod if measurement is take before the next dose.

24. A 20 m long tank has uniform cross-section as show below. Suppose the tank is completely filled with aliquid with weight density 800 kg/m3. Find the work required to pump all the liquid over the top of the tank.You may take the acceleration due to gravity as g = 10m/sec2.

4m

12m

25. Find the values of x for which the following power series is (a) convergent and (b) divergent. Write downthe radius of convergence. Ignore end-point behavior.

a.∞∑n=1

n2xn

5nb.

∞∑n=1

(−1)n(x− 3)n

(2n)!c.

∞∑n=1

5(x− 3)n

2n

26. Solve the equation:dy

dt= y + t

27. A population of animal species in a nature reserve has birth rate 0.15 per year and a death rate of 0.05per year. It is also observed that a 3 thousand animal of the same species migrates into the nature reserve peryear. Find the number of animal after 50 years. Discuss how large the population of animals can get as timeprogresses.

28. The population of a kind of fish grows at an intrinsic rate of 0.36 per year in a lake with carrying capacity90 thousand. If the fish is caught at a rate of 6.5 thousand per year, write down the logistic model for thepopulation p(t) (in thousands) of this fish leaving in this lake at time t years. Find the threshold and steadystate solution and draw graphs to illustrate your answers.

3

Name:

Class Time:Math 10360: Calc B, Final ExamDecember 17, 2029

• Be sure that you have all 15 pages of the test.• No calculators are to be used.• The exam lasts for two hours.• You may remove this answer sheet if you prefer. Return the entire exam

including questions and the answer sheet properly attached.• Sign the pledge. “On my honor, I have neither given nor received unauthorized

aid on this Exam”:

Good Luck!

PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!

1. a b c d e

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Name:

Class Time:

Multiple Choice

1.(6 pts.) The population density of a city measured from the city center is given by theradial function

ρ(r) =1

1 + r2,

where r is the distance from the city center measured in kilometers, and ρ is in thousandsper km2. Find the the number of people (in thousand) within a 5km radius from the citycenter.

(a)25π

26thousand.

(b) tan(26) thousand.

(c)1

2ln(26) thousand.

(d) π ln(26) thousand.

(e) 2π tan(26) thousand.

2.(6 pts.) Evaluate the integral

∫ 2

1

10

x(5− x)dx.

(a) 10 ln

(8

3

)

(b)1

5ln

(8

3

)

(c) 2 ln

(8

3

)

(d) 2 ln

(3

2

)

(e) 10 ln

(3

2

)

2

Name:

Class Time:

3.(6 pts.) Let f(x, y) = ln(3x− 2y). Find the value of the limit

limh→0

f(1, 1 + h)− f(1, 1)

h.

(a) −3 (b) 2 (c) Does not exist

(d) 3 (e) −2

4.(6 pts.) Find the Nth partial sum of the infinite series:∞∑n=1

ln

(n+ 1

n+ 2

).

(a) − ln(N + 2)

(b) ln(2)

(c) −∞

(d) ln(2)− ln(N + 2)

(e) ln(N + 1)

3

Name:

Class Time:

5.(6 pts.) The Taylor series for the function ex centered at 0 is:∞∑n=0

xn

n!.

Use the given series to find the Taylor series for the function e(2−x) centered at 2.

(a)∞∑n=0

(−1)n

n!(x− 2)n

(b)∞∑n=0

(−1)n+1

n!(x− 2)n

(c)∞∑n=0

(−1)n

n!(2− x)n

(d)∞∑n=0

(−1)n

n!(x+ 2)n

(e)∞∑n=0

(−1)n

n!xn

6.(6 pts.) Suppose that the 6th Taylor polynomial for f(x) at −1 is

P6(x) = 5− (x+ 1)− 3(x+ 1)3 + 2(x+ 1)5 + (x+ 1)6.

What is the value of f (5)(−1)?

(a) 120

(b) 720

(c) 2

(d) 240

(e) 1

4

Name:

Class Time:

7.(6 pts.) Consider the initial value problem:

y′ = (y − t)2, y(1) = 0.

Using Euler’s method, with TWO, steps of equal size estimate y(1.2)

(a) 1.1

(b) 0.2

(c) 0.1

(d) 0.3

(e) 1.0


∫xe3x dx.

(a)x

3e3x − 1

9e3x + C

(b) xe3x − e3x + C

(c)x2

6e3x + C

(d) 3xe3x − 9e3x + C

(e) None of these.

5

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Class Time:

9.(6 pts.) Optimize the function f(x, y) = x2+y2 subjected to the constraint x4+y4 = 1.

(a) Minimum =1

2, Maximum =

√2.

(b) Minimum = 1, Maximum =√

3.

(c) Minimum = 1, Maximum =√

2.

(d) Minimum =√

2, Maximum =√

3.

(e) Minimum =√

2, Maximum = 2.

10.(6 pts.) Consider the volume of solid formed when the region enclosed by the graphsof the x-axis, y-axis, y = ex and x = 1 is revolved about the line x = 1. If you useSHELL method to find this volume, which of the following integrals will you obtain?

(a)

∫ e

1

2πy(1− ln y) dy.

(b)

∫ 1

0

2π(1− x)ex dx.

(c)

∫ e

0

π(1− ey)2 dy.

(d)

∫ 1

0

2πx(1− ex) dx.

(e)

∫ 1

0

πe2x dx.

6

Name:

Class Time:

11.(6 pts.) The rate of change of temperate in a weather simulation chamber is given by

r(t) = 20 sin(2t) sin(t)

where r is oC/hour. What is the total change in temperature over the time duration

0 ≤ t ≤ π

2?

(a)20

3oC

(b)10

3oC

(c) 0 oC

(d) −20

3oC

(e)40

3oC

12.(6 pts.) Convert the following integral to an integral with variable θ using the sub-stitution x = 3 tan θ. ∫ 3

0

x3√9 + x2

dx

(a)

∫ 3

0

9 tan3 θ

csc θdθ

(b)

∫ π/4

0

9 tan3 θ

sec θdθ

(c)

∫ π/4

0

27 tan3 θ sec θ dθ

(d)

∫ 3

0

27 tan3 θ sec θ dθ

(e)

∫ 3

0

9 tan3 θ

sec θdθ

7

Name:

Class Time:

13.(6 pts.) A tank contains 100 gal of syrup with sugar concentration 0.5 lb/gal. Thetank is kept well stirred as syrup with sugar concentration 0.1 lb/gal is pumped in ata rate of 2 gal/min. If syrup is pumped out of the tank at rate 4 gal/min, and x(t)denotes the amount of sugar in the tank, in pounds, which one of the following initialvalue problems models the the amount of sugar in the tank.

(a)dx

dt= 0.2− 4x

100 + 2t, x(0) = 50

(b)dx

dt= 0.2− 4x

100− 2t, x(0) = 50

(c)dx

dt= 0.4− 2x

100− 4t, x(0) = 50

(d)dx

dt= 0.1− x

100 + 2t, x(0) = 0.5

(e)dx

dt= 0.1− x

100− 2t, x(0) = 0.5

14.(6 pts.) Evaluate the following geometric series

1 +π

3+π2

9+π3

27+ · · ·

(a)22

7

(b)1

3− π

(c) The series is divergent.

(d)3

3− π

(e)3

3 + π

8

Name:

Class Time:

15.(6 pts.) The series∞∑n=1

2n(x− 1)n

n+ 1converges on which of the following interval?.

(a) −1

2< x <

1

2

(b) x = 1

(c) −1 < x < 1

(d) −∞ < x <∞

(e)1

2< x <

3

2

16.(6 pts.) Solve for the general solution of the equation below for 0 < x < π/2:

xdy

dx− y = x2 sec2 x

(a) y = −Cx tanx.

(b) y = x secx tanx+ Cx.

(c) y = Cx tanx.

(d) y = −x tanx+ Cx.

(e) y = x tanx+ Cx.

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Name:

Class Time:

17.(6 pts.) Areas bounded between the curves y = f(x) and y = g(x) over 1 ≤ x ≤ 5

are as given below. Find the value of the definite integral

∫ 5

1

[f(x)− g(x)]dx.

0 1 53 4

y = f(x)

y = g(x)

y

x

6 unit2

3 unit2(a) 9

(b) 2

(c) −2

(d) −3

(e) 3

18.(6 pts.) A patient is injected once a day with 20 units of a certain drug. Suppose only70% of each dose remains at the end of each day. Using a geometric series, approximatethe number of units of the drug remaining in the patient’s system after a very long timeif the measurement is done before the next dose is given.

(a)140

3units.

(b) 70 units.

(c)20

1− e−0.7

(d)200

17units.

(e)140

17units.

10

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Class Time:

19.(6 pts.) Let (x, y) denote the usual rectangular coordinates. Write the followingfunction f(x, y) in polar coordinates (r, θ).

f(x, y) = 2x2 + 2y2 +x

y.

(a) f(r, θ) = 2r2 + cot θ.

(b) f(r, θ) = 2√r + cot θ.

(c) f(r, θ) = 2r + tan θ.

(d) f(r, θ) = r2 cos(2θ) + cot θ.

(e) f(r, θ) = 2r2 + tan θ.

20.(6 pts.) A 10 meter long uniform chain weighing 30 kg (in total) is hung from the topof a 50 meter building. How much work is required to lift the whole chain to the top ofthe platform? Take acceleration due to gravity as 10m/s2.

(a) 15000 J

(b) 37500 J

(c) 5000 J

(d) 3000 J

(e) 1500 J

11

Name:

Class Time:

21.(6 pts.) Perform the integral

∫cos3(4x) dx.

(a) sin(4x)− sin3(4x)

3+ C

(b)sin(4x)

4− sin3(4x)

12+ C

(c) −sin(4x)

4+

sin3(4x)

12+ C

(d)sin(4x)

16− sin3(4x)

48+ C

(e) − sin(4x) +sin3(4x)

3+ C


y′ = sin(y)− x2; y(−1) = 0

Find the 2nd Taylor polynomial for y(x) centered at −1.

(a) T2(x) = −x+1

2x2

(b) T2(x) = −(x+ 1) +1

2(x+ 1)2

(c) T2(x) = −1

2(x+ 1) +

1

6(x+ 1)2

(d) T2(x) = −(x− 1) +1

2(x− 1)2

(e) T2(x) = −1

2(x− 1) +

1

6(x− 1)2

12

Name:

Class Time:

23.(6 pts.) Using Trapezoidal rule with 4 equal subintervals, estimate the area under thecurve y = f(t) over the interval 0 ≤ t ≤ 4.

1 2 3 4

1

2

3

4

t0

y = f(t)(a) 8

(b) 16

(c) 9

(d) 17

(e)17

2

24.(6 pts.) Solve the initial value problem:

dy

dx=

y√1− x2

; y(0) = 3e

(a) y = earctan(x) + 3e

(b) y = earcsin(x) + 3e

(c) y = 3earctan(x)+1

(d) y = 3earcsin(x)+1

(e) y = 3e− arccos(x)

13

Name:

Class Time:

25.(6 pts.) The density of a metal plate as show below is given by

ρ(x, y) = y kg/m2.

Find the total mass of the plate in kg.

(a) 8 kg. (b) 2 kg. (c)8

3kg.

(d)4

3kg. (e)

16

3kg.

2

0 2-2

14

Name:

Class Time:

You may find these identiies helpful in the test:

cos2 θ + sin2 θ = 1

1 + tan2 θ = sec2 θ

cot2 θ + 1 = csc2 θ

cos2 θ =1 + cos(2θ)

2

sin2 θ =1− cos(2θ)

2

sin(2θ) = 2 sin θ cos θ

sinA cosB =1

2[sin(A+B) + sin(A−B)]

cosA cosB =1

2[cos(A+B) + cos(A−B)]

sinA sinB = −1

2[cos(A+B)− cos(A−B)]

15

Name:

Class Time: ANSWERSMath 10360: Calc B, Final ExamDecember 17, 2029




Good Luck!


1. a b c • e

2. a b • d e

3. a b c d •4. a b c • e

5. • b c d e

6. a b c • e

7. a • c d e

8. • b c d e

9. a b • d e

10. a • c d e

11. a b c d •12. a b • d e

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17. a b c • e

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19. • b c d e

20. a b c d •

21. a • c d e

22. a • c d e

23. a b c d •24. a b c • e

25. a b • d e

Name:

Class Time:Math 10360: Calc B, Final ExamDecember 11, 2032




Good Luck!


1. a b c d e

2. a b c d e

3. a b c d e

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6. a b c d e

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Name:

Class Time:

Multiple Choice

1.(6 pts.)

Vel

oci

ty (m

/sec)

Time (sec) 42 60

1

2

3

1 3 5

The graph of the velocity of a particle traveling on a straight line is given above. Esti-mate the total change in its position over the duration 0 ≤ t ≤ 6 using the trapezoidrule with three equal sub-intervals (T3).

(a) 14 meters

(b) 15 meters

(c) 13 meters

(d) 10 meters

(e) 12 meters

2.(6 pts.) Find the value of x for which the two curves intersect

y = ln(2x)− 3 and y = ln(x+ 1).

(a)e3

2− e3

(b)e3

2 + e3

(c)2 + e3

e3

(d)1

e3 − 2

(e)e3 − 1

3

2

Name:

Class Time:

3.(6 pts.) The annual birth rate of a population of animals is 0.1 per year, and annualdeath rate is 0.4 per year. Find the size of the population after two years if the currentpopulation is 100 thousand.

(a) 64 thousand.

(b) 49 thousand.

(c) 9 thousand.

(d) 25 thousand.

(e) 169 thousand.

4.(6 pts.) If, in addition to death and birth rates above, an immigration rate of 2thousand animal is added to the population per year. What is estimated size of thepopulation after a very long time?

(a)20

3thousand.

(b) 10 thousand.

(c)40

3thousand.

(d)20

7thousand.

(e)40

7thousand.

3

Name:

Class Time:

5.(6 pts.) Find the Maclaurin series for the function f(x) =3

4 + xusing the known series

1

1− x= 1 + x+ x2 + · · ·+ xn + . . . .

Give the also the values of x for which the series in your answer is convergent.

(a)∞∑n=0

1

4nxn for −1

4< x <

1

4.

(b)∞∑n=0

3(−1)n

4nxn for −4 < x < 4.

(c)∞∑n=0

(−1)n

4n+1xn for −1

4< x <

1

4.

(d)∞∑n=0

3

4n+1xn for −4 < x < 4.

(e)∞∑n=0

3(−1)n

4n+1xn for −4 < x < 4.


y′ = 2x+ y2, y(1) = 0.

Using Euler’s method with TWO steps of equal size, estimate y(2).

(a) y(2) ≈ 3

(b) y(2) ≈ 11/2

(c) y(2) ≈ 5/2

(d) y(2) ≈ 4

(e) y(2) ≈ 7/2

4

Name:

Class Time:

7.(6 pts.) The mass of a circular disc is distributed according to the polar density function

ρ(r, θ) =θ2√

1− r2,

where r is the distance (in m) from the center of the disc, and ρ is in kg per m2. Findthe total mass (in kg) if the radius of the disc is 1/2 meter.

(a)2π3

3ln

(4

3

)

(b)8π3

3

(π

4− arctan

(3

4

))

(c)4π4

9

(d)4π3

3ln

(4

3

)

(e)8π3

3

(1−

√3

4

)

8.(6 pts.) Find the sum of the first 70 terms of the series∞∑n=3

(2 sin(n)− 2 sin(n− 1)).

Note that the index n starts from n = 3

(a) 2 sin(N + 1)− 2 sin(2)

(b) −2 sin(2)

(c) 2 sin(72)− 2 sin(2)

(d) 2 sin(70)− 2 sin(2)

(e) Divergent

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∫ 1

0

arctan(x) dx.

(a)π

4− 1

(b) x arctan(x)− 1

(c) ln

(1

2

)(d)

π

4− 1

2ln(2)

(e) x arctan(x)− 1

2ln(2)

10.(6 pts.) Find the equation of the tangent line to the curve y = 1+arcsin(2x) at x = 0

(a) y = x+ 1

(b) y = 2x+ 1

(c) y =x

2+ 1

(d) y =

(1

1 + 4x2

)x+ 1

(e) y =

(2√

1− 4x2

)x+ 1

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11.(6 pts.) The quantity y(t) of a radioactive substance decays according to the rateequation

dy

dt= −0.5y

where y is in grams and t is in days. Find the amount of time that must lapse for thesubstance to reduce to 80% of its initial amount.

(a)ln(8)

ln(5)days

(b)ln(0.8)

2days

(c)ln(2)

0.5days

(d) ln(0.8)− 0.5 days

(e) −2 ln(0.8) days

12.(6 pts.) Use Lagrange Multipliers to find the maximum and the minimum value ofH(x, y) = xy subjected to the constraint 4x2 + 9y2 = 32.

(a) Minimum = −8

3and Maximum =

8

3

(b) Minimum = − 2√3

and Maximum =2√3

(c) Minimum = −1

3and Maximum =

1

3

(d) Minimum = − 4√3

and Maximum =2√3

(e) Minimum = −4

3and Maximum =

4

3

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∫ π/3

0

12 sin3(x) dx.

(a)45

16

(b)27

2

(c)27

16

(d)5

2

(e)3

4

14.(6 pts.) Consider the solid whose base is given by the shaded region between thelines y = −x + 2, y = 1, the x−axis and y−axis. Find the volume of the solid if thecross-sections perpendicular to the y-axis are squares.

y

x0

1

2

(a)3

2

(b)7

3

(c) 2

(d)1

3

(e)8

3

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15.(6 pts.) For the same region above, which of the following set of inequalities describethe shaded region?

(a) 0 ≤ y ≤ 1; 2− y ≤ x ≤ 2

(b) 0 ≤ x ≤ 2; 0 ≤ y ≤ 2− x

(c) 0 ≤ y ≤ 1; 0 ≤ x ≤ 2− y

(d) 0 ≤ x ≤ 2; 2− x ≤ y ≤ 1

(e) 0 ≤ x ≤ 1; 2− x ≤ y ≤ 1

16.(6 pts.) Consider the volume of solid formed when the region enclosed by the graphsof the y-axis, y = 2ex, y = 1, and x = 1 is revolved about the x-axis. Find the volume ofthe solid?

(a)4π

3

(b) 2π (2e− 3)

(c) π (2e2 − 4e+ 3)

(d) π (2e2 − 3)

(e)8π

3

!"

#

$

%

!

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17.(6 pts.) Find the interval of convergence for the series∞∑n=1

n3(x− 1)n

5n.

(a) −4 < x < 6

(b) −1 < x < 1

(c) −5 < x < 5

(d)4

5< x <

6

5

(e) −∞ < x <∞

18.(6 pts.) Solve for the general solution of the equation below:

dy

dx= e−x

3 − 3x2y.

(a) y = (x+ C)e−x3

(b) y = (x− 1)ex3

+ C

(c) y = (x+ 1)e−x3

+ C

(d) y = C(x+ 1)e−x3

(e) y = xex3

+ C

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19.(6 pts.) Let R be the region bounded by the lines y = 2, y = x, and y = −x. Evaluatethe following double integral ∫∫

R

(3x2 + y2) dA

(a) 6

(b) 8

(c) 16

(d) 4

(e) 12

20.(6 pts.) The Maclaurin series for the function f(x) is:∞∑n=0

(−1)nx2n

n!.

Find the 6th derivative of f(x) at x = 0 that is f (6)(0).

(a) 60

(b)1

6

(c) −120

(d) −720

(e) −1

6

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y′ =1

x+ y; y(1) = 0.

Find the 2nd Taylor polynomial of y(x) centered at x = 1.

(a) T2(x) = (x− 1)− 2(x− 1)2

(b) T2(x) = x− x2

(c) T2(x) = x− 2x2

(d) T2(x) = (x+ 1)− (x+ 1)2

(e) T2(x) = (x− 1)− (x− 1)2

22.(6 pts.) Find the general solution of the following differential equation:

dy

dx= 2xy2 + 2x.

(a) y = tan(x2 + C)

(b) y = C + sin(x2)

(c) y = C + tan(x2)

(d) y = sin(x2 + C)

(e) y = C tan(x2)

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23.(6 pts.) The Maclaurin series for the function ln(x2 + 1) is:∞∑n=1

(−1)n−1

nx2n.

Use the 4th Taylor Polynomial for the function ln(x2+1) to estimate the value of ln(1.25).

(a) (0.5)2 − (0.5)4

2+

(0.5)6

3− (0.5)8

4

(b) (0.25)2 − (0.25)4

2+

(0.25)6

3− (0.25)8

4

(c) (0.5)2 − (0.5)4

2

(d) (0.25)2 − (0.25)4

2

(e) (1.25)2 − (1.25)4

2+

(1.25)6

3− (1.25)8

4

24.(6 pts.) Evaluate the integral:∫ 3

2

8

(x− 1)(5− x)dx.

(a) 2 ln(3)

(b) 8 ln(4)− 8 ln(3)

(c) ∞

(d) 0

(e) 4 ln(2)− 2 ln(3)

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25.(6 pts.) A cubical tank (Figure 1) has dimensions 4 m× 4 m× 4 m. It is completelyfilled with a water ( 1000 kg/m3). Find the work required to pump all of the fluid out ofthe tank through a pipe with opening 8 m from the bottom of the tank. YOu may takethe acceleration due to gravity as g = 10 m/s2.

Fluid

Out

8 m 4 m

4 m

4 m

Figure 1.

(a) 288000 J.

(b) 384000 J.

(c) 192000 J.

(d) 512000 J.

(e) 576000 J.

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You may find these formulae helpful in the test:

cos2 θ + sin2 θ = 1

1 + tan2 θ = sec2 θ

cot2 θ + 1 = csc2 θ

cos2 θ =1 + cos(2θ)

2

sin2 θ =1− cos(2θ)

2

sin(2θ) = 2 sin θ cos θ

sinA cosB =1

2[sin(A+B) + sin(A−B)]

cosA cosB =1

2[cos(A+B) + cos(A−B)]

sinA sinB = −1

2[cos(A+B)− cos(A−B)]

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Class Time: ANSWERSMath 10360: Calc B, Final ExamDecember 11, 2032




Good Luck!


1. a b • d e

2. • b c d e

3. a • c d e

4. • b c d e

5. a b c d •6. • b c d e

7. a b c d •8. a b • d e

9. a b c • e

10. a • c d e

11. a b c d •12. • b c d e

13. a b c • e

14. a • c d e

15. a b • d e

16. a b c • e

17. • b c d e

18. • b c d e

19. a b • d e

20. a b • d e

21. a b c d •22. • b c d e

23. a b • d e

24. • b c d e

25. a • c d e

math 10360 { final review - university of notre damem10360/handout/10360finalreview.pdf · math...

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