MATH 5B Pasadena City College Radius of Convergence and Interval Worksheet: Calculus Answers 2021

MATH 5B Pasadena City College Radius of Convergence and Interval Worksheet: Calculus Answers 2021

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MATH 5B Pasadena City College Radius of Convergence and Interval Worksheet

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L.Piterbarg , Math 5b, Final
Student name:
Instructions: Remember that an answer without a solution, even if correct, will get little or no credit.
In Maclaurin/Taylor series you should show both, the general term depending on n, and at least 3
?rst terms, n = 0, 1, 2. This is an open book/notes exam, but you are bound by the Honor Code:
nobody can help you during the exam. If you have a question, ask me in chat.
x ? sin x
x?0 x ? tan x
1. (6 pts) Find the limit lim
2. (24 pts) Evaluate the integral if it does exist
(a)
(c)
?4
1
?
x3/2 ln xdx
(b)
?
??
dx
?2 x2 +2x+2
1
(d)
dy
y 2 ? 4y ? 12
?
0
t
dt
1?t
3. (10 pts) Sketch the curve using the standard guidelines (intercepts, symmetries, asymptotes, increasing/decreasing,
2
max, min, concavity, in?ection points) y = e2x?x
4. (16 pts) Determine whether the series is convergent or divergent.
(a)
??
n?1
ln
n=1 (?1)
(
n+1
n
)
(b)
??
n2
n=1 (?1)
5. (8 pts) Find the radius of convergence and interval of convergence of the series
6. (10 pts) Find the Maclaurin series for f (x) =
??
n=1
? 4 ? 6 ? ? ? (2n)
2n n!
(?1)n (x + 2)n
n3n
1
. Determine the radius and interval of convergence.
4 + 3?2
7. (10 pts) Find the family orthogonal to y = kx2 . Sketch at least 4 curves of each family.
8. (8 pts) Sketch the parametric curve x = cos(2t), y = sin(2t),
revolution when the curve is rotated around the x -axis.
?
4
?t?
3?
8
and ?nd the area of the surface of the
9. (8 pts) Find an equation of the tangent line to the given curve at t = 1: x = ln t, y = 1 + t2 . Evaluate the second
d2 y
derivative dx
2 at t = 1
10. (10 pts) Sketch the polar curve r = 2 sin ? and ?nd the points on the curve where the tangent line is horizontal
or vertical. You should give the polar coordinates of that points.
11. (8 pts) Sketch the polar curve r = 2 ? cos ? and ?nd the area enclosed by the curve
12. (12 pts) Sketch and ?nd the length of the polar curve r = 1 + cos ?,
? ? ? ? 2?

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