MAC 2312 Final Exam Review Spring 2019

This review, produced by the Broward Teaching Center, contains a collection of questions which are representative of the type you

may encounter on the exam. Other resources made

available by the Teaching Center include:

• Walk-In tutoring at Broward Hall

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• Video resources for Math and Science classes at UF

• Written exam reviews and copies of previous exams

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1. Investigate the convergence of the series∞∑n=1

(2n+ 3

3n+ 2

)n.

2. Suppose that∞∑n=0

cn(x+ 1)n converges for x = 1/2 and diverges for x = −4. Consider:

A =∞∑n=0

cn(−5)n B =∞∑n=0

cn C =∞∑n=0

cn2n

What can be said about the convergence of series A, B, and C?

3. Investigate the convergence of∞∑n=1

cos(nπ)

4n.

4. Use the fact that 4 arctan(1) = π, and1

1 + x2=

∞∑n=0

(−1)nx2n to find a series representation

for the number π. [Hint: integrate]

5. Express

∫ x

0

et − 1− tt2

dt as a power series. Find the IOC.

6. Evaluate

∫1√

3− 2x− x2dx

7. Find the sum of the series S =√2−√2π2

42 · 2!+

√2π4

44 · 4!−√2π6

46 · 6!+ · · ·

8. Evaluate the definite integral

∫ 1

0

√4− x2 dx.

9. Set up the partial fraction decomposition for1

x4 − 9x2.

2

10. Complete the partial fraction decomposition from the previous problem and use it to evaluate∫1

x4 − 9x2dx.

11. Evaluate

∫x2√x2 − 4 dx

12. Which of the following integrals converge?

I.

∫ ∞

2

1

xπ/3dx II.

∫ ∞

1

et

1 + e4tdt III.

∫ ∞

2

1

ln(s)ds

13. Find a closed form for the N th partial sum of∞∑n=2

ln

(n

n+ 1

)14. Sketch the polar curve r = 1 + cos(θ), and set up an integral to find the area above the

horizontal axis.

15. Given r = 1− 2 cos(θ) set up an integral for the area of the inner loop.

16. Set up an integral for the area inside r2 and outside r1 where r1 =√3 sin(θ) and r2 = cos(θ),

17. Given x(t) = et + 5t and y(t) = 100, find the arclength from t = 0 to t = 5.

18. At what points (if any) does the parametric curve have horizontal or vertical tangent lines?

x(t) = cos2(t) + cos(t) y(t) = sin(t) cos(t) + sin(t)

19. Consider the solid obtained by rotating the region bounded by y = ln(x), x = e, and y = 0about the x-axis. Set up two integrals for the volume of this solid. One using the disk/washermethod, the other using cylindrical shells.

3

Fall 2018 Final Exam is roughly broken down as follows:

Integration unit (3) (+1)

Convergence tests (5)

Power Series (3)

Parametric (3)

Polar (3)

Volume (5)

Final contains: All multiple choices. 20 ques-

tions worth 3 points/ea and 2 extra credit ques-

tions worth 1.5 points/ea.

’Maclaurin Series’ sheet is provided on the final.

Power Series

1. Evaluate

∫t

1− t7dt as an infinite series.

2. Suppose∑

cnxn converges at x = 6.

Which below MUST be convergent?

P:∑

cn(−4)n, Q:∑

cn(−6)n

(a) P only

(b) Q only

(c) Both

(d) Neither

3. Suppose∑

cnxn converges at x = 6.

Which below is NOT possible?

P:∑

cn(−4)n, diverges

Q:∑

cn(−6)n converges

(a) P only

(b) Q only

(c) Both

(d) Neither

4. Find a power series for e2x, centered at 6.

(a)

∞∑n=0

(x− 6)n

n!

(b)

∞∑n=0

e6(x− 6)n

n!

(c) e12∞∑n=0

(x− 6)n

n!

(d) e12∞∑n=0

2n(x− 6)n

n!

5. Find a power series for f (x), centered at 0.

f (x) =x

(1 + x)2

6. Find the sumof the series

∞∑n=2

(−22n

(−9)n−1+

(−1)n

(2n)!

)

(a)16

13+ cos 1

(b) −16

11− cos 1

(c)16

11+ e

(d) −16

13− sin 1

(e)11

9+ cos 1

7. Find the value of t for which the series

∞∑n=0

(−1)ntn+1

32n(2n + 1)

converges, and find the function it convergesto for these values of t.

(ans. f (t) = t arctan

(t

3

), t ∈ [−3.3])

8. (TRUE/FALSE)Let

f (0) = 4, f ′(0) = 3,

f”(0) = 3, f′′′

(0) = 2.

The first 4 terms of the Maclaurin Series of fis

f (x) = 4 + 3x + 3x2 + 2x3

(a) TRUE

(b) FALSE

Parametric and Polar:

9. Find the length of the pathover the given interval.

c(t) = (t3 + 8, t2 + 3), 0 ≤ t ≤ 1

(a) (100 + 49)

3/2 − (25 + 49)

3/2

(b) (100 + 94)

3/2 − (25 + 94)

3/2

(c) (100 + 49)

3/2 + (25 + 49)

3/2

(d) 127(133/2 − 43/2)

10. Which integral below gives thelength of the parametric curve?

c(t) = (2et − 2t, 8et/2), t ∈ [0, 3]

(a)

∫ 3

0

(2et + 2)2 dt

(b)

∫ 3

0

(2et + 2) dt

(c)

∫ 3

0

(et + 1) dt

(d)

∫ 3

0

(4et + 4) dt

11. Find the TL to the PE,

x = tet, y = e2t

at the point (e, e2).

(a) y = ex− e(b) y = ex

(c) y = ex + e

(d) y = x + e

12. At which angle θ does the polar curve inter-sect the origin?

r = 4 cos(3θ)

(a) π/4.

(b) π/6.

(c) 4π.

(d) 6π.

13. Find the area inside the r1 curve, outside ther2 curve in the 1st and 4th quadrants.

r21 = 9 cos(2θ), r2 =√

6 cos(θ)

(a) 2

(1

2

∫ π/4

0

r21 − r22 dθ

)

(b) 2

(1

2

∫ π/4

0

r22 − r21 dθ

)

(c) 2

(1

2

∫ π/6

0

r21 − r22 dθ

)

(d)1

2

∫ π/2

0

r22 − r21 dθ

14.

Volume:

15. Find the volume of the solid obtainedby rotating the region bounded by

y = x3, y = 0, x = 1

about the y = −1.

(a)

∫ 1

0

π[12 − ( 3√y)2] dy

(b)

∫ 1

0

π[( 3√y)2 − 12] dy

(c)

∫ 1

0

π[12 − (x3)2] dx

(d)

∫ 1

0

π[(x3 + 1)2 − 12] dx

16. Find the volume of the solid obtained by ro-tating the region bounded byy = sinx, y = 0, 0 ≤ x ≤ π

2 .

a. About the line y = 2 b. About the line x = −1.

c. rotate the region y = sinx, y = 0, 0 ≤ x ≤ π.

about the line x = −1

17. Find the volume of the solid generated by ro-tating the region bounded by

y =√x− 1, y = 0, x = 5

and revolved around the line y = 3.

(a)

∫ 5

1

π[(

3−√x− 1

)2 − 32]dx

(b)

∫ 5

1

π[32 −

(√x− 1

)2]dx

(c)

∫ 5

1

π[(√

x− 1)2 − 32

]dx

(d)

∫ 5

1

π[32 −

(3−√x− 1

)2]dx

18. Find the volume of the solid if the base ofthe solid is the region between the curve y =2 sinx and the x−axis on the interval [0, π]and the cross-sections perpendicular to thex−axis are squares with bases running fromthe x−axis to the curve.

ex. Find the volume of the solid if the base ofthe solid is the region between the curves y =ex, y = e−x and x = 1 and the cross-sectionsperpendicular to the x−axis are squares.

19. Find the volume of the solid if the base of thesolid is the region between the curvey = 2 sinx and the x−axis on the interval[0, π] and the cross-sections perpendicular tothe x−axis are semi-circles with bases run-ning from the x−axis to the curve.

(a)1

2π

(b)1

2π2

(c)1

4π

(d)1

4π2

20. Find the volume of the solid obtainedby rotating the region bounded by

y = ex, y = e−x, x = 1about the y−axis.

(a)2

e

(b)4

e

(c)4π

e

(d)2π

e

21. Find the volume of the solid obtainedby rotating the region bounded by

y = lnx, y = 0, x = 2about the y−axis.

(a) 4 ln 2− 3

2

(b) π

(4 ln 2− 3

2

)(c) 2π

(4 ln 2− 3

2

)(d) 4 ln 2− 2

22. Consider the region bounded by the curvesy = −x2 + x and y = 0. Using the ShellMethod to find the volume of revolution gen-erated by rotating this region about the linex = 5. Which integral below is the correctset up?

(a) V =

∫ 1

0

2π(y)

(√1

4− y +

1

2

)dy

(b) V =

∫ 1

0

2π(x)(−x2 + x) dx

(c) V =

∫ 5

0

2π(5− x)(−x2 + x) dx

(d) V =

∫ 1

0

2π(5− x)(−x2 + x) dx

23. Consider the region bounded by the curvesy = −x2 + x and y = 0. Using the WasherMethod to find the volume of revolution gen-erated by rotating this region about the liney = 3. Which choice below gives correct ORand IR?

(a) OR = 3, IR = 3 + (−x2 + x)

(b) OR = 3, IR = (−x2 + x)− 3

(c) OR = 3, IR = 3− (−x2 + x)

Convergence tests:

24. Consider the series

∞∑n=1

(−1)n

3 + 2n.

Let R2 be the error made in estimating thesum of the series after summing the first 2terms.

According to the Alternating Serieserror estimation theorem,

|R2| ≤ .

(a) −1/11

(b) 1/11

(c) −1/7

(d) 1/7

(e) 1/19

25. Choose the letter of the column whoserows give the statement’s truth value.

(a) (b)∑ 2 + (−1)n

10nconv by DCT T T∑ 2 + (−1)n

nconv by DCT T F∑ n3

2n(n + 1)conv abs. by LCT F T

26. Determine if the series converge. Choose theletter of the column in the following tablewhose rows give each statement’s truth value.

(a) (b) (c) (d)∑sin(

nπ

2) cond. conv. T F F F

∑(sin( 1

n2)

( 1n2

)

)conv. F T F F∑ cosn

n2abs. conv F T T F∑ 1

2conv. F F F T∑ en

n!conv. T F T T∑

n√

71 conv. F F F T

27. Use Direct Comparison test to determine if

∞∑3

10− cosn

2n

is convergent?

(a) convergent

(b) divergent

(c) Can’t use comparison tests

28. IBP: Evaluate

∫lnx dx,

∫x2 cosx dx,∫

arctanx dx,

∫e2x sin(3x) dx,

∫x3ex

2dx

29. Trig Integrals:

∫sin3 x cos2 x dx,

∫cos2(2x) dx,∫

sec4 x tan4 x dx,

∫tan3 x secx dx,

∫secx dx

30. Trig-sub:

∫1√

1− x2dx,

∫1

x2√x2 + 4

dx∫x√

3− 2x− x2dx,

∫x√

−x2 + 2xdx.

31. Use an appropriate trig-sub to transform theintegral∫

x2√9− 25x2

dx

into a trig integral for some constant k.

(a) I = k

∫sin2 θ dθ

(b) I = k

∫cos2 θ dθ

(c) I = k

∫sin 2θ dθ

(d) I = k

∫cos 2θ dθ

(e) I = k

∫sin θ cos2 θ dθ

Partial Fractions

32. Split

∫x + 4

x2 + 2x + 5dx into two integrals

which can be easily evaluated using the 2-stepprocess (u−sub and arctangent formula.)

(a)

∫x

x2 + 2x + 5dx−

∫4

(x + 1)2 + 22dx

(b)

∫x + 1

x2 + 2x + 5dx +

∫3

(x + 1)2 + 22dx

(c)

∫x + 2

x2 + 2x + 5dx−

∫2

(x + 1)2 + 22dx

(d)

∫x− 1

x2 + 2x + 5dx +

∫5

(x + 1)2 + 22dx

(e)

∫x− 2

x2 + 2x + 5dx +

∫6

(x + 1)2 + 22dx

33.

∫6x− 2

(x− 3)(x + 2)dx