MAC 2312 Final Exam Review - UF Teaching Center · 1. Investigate the convergence of the series X1...
Transcript of MAC 2312 Final Exam Review - UF Teaching Center · 1. Investigate the convergence of the series X1...
MAC 2312 Final Exam Review
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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
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