Description
Answer all questions. your solutions neatly, legibly, and in a well-organized manner. They must be clear to understand.Your solution must include your work and the final answer. Always show your work. Your work is more important than your final answer. Any results without showing steps might not be counted. makemake sure to use formulas in the Calculus 2 level.don’t miss any step.
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Name:
3
Z
1. (10pts) Evaluate
sin2 x cos3 xdx. Show your work.
Answer:
Z
2. (15pts) Evaluate
Answer:
√
ex
dx. Show your work.
3 + e2x
Name:
4
Z ∞
1
dx convergent. Also, evaluate the integral when it is
xp
1
convergent. You MUST explain how you derive your conclusion.
3. (15pts) Find p values that make
Answer:
Name:
4. (15pts) Find the centroid of the region bounded by y = x2 and x = y 2 .
Answer:
5
Name:
6
5. (15pts) Find the solution of y ′ = 2xy(1 + 2y) with y(0) = −1. Show your work and simplify your
answer. You need to write your final answer as a function form, i.e., y as a function of x.
Answer:
6. (15pts) Solve xy ′ = y − 3×3 . Show your work and simplify your answer. You need to write your
final answer as a function form, i.e., y as a function of x.
Answer:
Name:
7
7. (15pts) Find the area of the region common to r = 2 sin θ and r = 2 cos θ. You must graph the
functions first and evaluate your integral. Simplify your answer.
Answer:
Name:
8
8. (10pts) Find the eccentricity, identify the conic, give an equation of the directrix, and sketch the
conic with the Cartesian coordinates of the vertex and focus (or vertices and foci) for the
following polar equation:
2
r=
2 − 3 sin θ
Your graph and answers need to be clear to understand, and your answers must be easily identified.
Answer:
Name:
9
9. (10pts) Find the orthogonal trajectories of the family of the curves x2 + y 2 = k where k is an
arbitrary constant.
Answer:
Name:
10
10. (10pts) Determine the limit of an =
If it is converges, find its limit.
1− n1
n
or show that the SEQUENCE (not a series) diverges.
Answer:
√
4
n+1
√
11. (10pts) Determine whether
converges. Show your work.
3
n4 − n
n=1
∞
X
Answer:
Name:
12. (10pts) Determine whether
11
∞
X
n − 1 n
n=1
2n + 3
converges. Show your work.
Answer:
13. (10pts) Find the interval of convergence of
∞
X
5n xn
n=1
Answer:
n!
. Show your work.
Name:
12
14. (10pts) Find the power series for f (x) =
your work.
x
and identify its interval of convergence. Show
1 + x2
Answer:
15. (10pts) Find the sum of the series. Simplify your answers.
∞
X
π 2n+1
(−1)n 2n+1
2
(2n + 1)!
n=0
Answer:
Name:
13
16. (10pts) Find the sum of the series. Simplify your answers.
(ln 3)2 (ln 3)3
−
+ ···
1 − ln 3 +
2!
3!
Answer:
17. (10pts) Evaluate the indefinite integral as an infinite series.
Z
cos x − 1
dx
x
No credit for other methods without an infinite series.
Answer:
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