Calculus 1
NAME:
Final
St.#:
Instructions: The T /F questions should indicate the number clearly so
that we may grade without ambiguity. You must show all necessary work
to get credit. Partial credit will only be given for progress toward a correct
solution.
1. Use a separate paper, as you did in homework, to solve the questions
by hand.
2. Convert your answers into a pdf file using a smart phone or scanner.
3. Upload it on icampus on the due time.
I. (10 pt each) Mark ”T” for the true statement and mark ”F” for the false statement.
There are −5 penalty points for the wrong answer.
1. ( ) The length of one arch of the cycloid x = θ −sin θ, y = 1 cos θ is L = 8.
P∞ ee
P
xn
n
2. ( ) The following equality holds true: ∞
n=0 n! (x − e) .
n=0 n! =
√ √
√ √
√
P
3( 3−1) 2
3
n
3. (√ √
) If |x|√ < 1, then (1 + x) 3 = ∞
x +
3x
+
x
=
1
+
n=0 n
2!
3( 3−1)( 3−2) 3
x + · · ·.
3!
R P
P∞ R
n
4. ( ) The following equality holds true: [ ∞
n=0 [cn (x−
n=0 cn (x−a) ]dx =
n
a) ]dx for all x ∈ IR.
R∞ 1
5. ( ) The integral 1 x1/44
dx is divergent.
n
6. ( ) The sequence an = nn! is divergent.
7. ( ) A monotone increasing sequence {an }, with an > 44 for all n ≥ 1, is
convergent.
R∞
8. ( ) The integral 0 xe−x dx diverges.
9. ( ) There does not exist the antiderivative of the function f (x) = 2x sin(x2 ).
R 44 dx
10. ( ) The following are true: 0 x−1
= ln |x − 1|]44
0 = ln 43 − ln 1 = ln 44.
II. (10 pt) For what values of x does the series
P∞
n=1
(x−1)n
n
converge?
III. (10 pt. each) Test the following series for convergence or divergence
1
1.
P∞
2.
P∞
3.
P∞
4.
P∞
n
n=1 2n2 −4n−1 .
n=1
(−1)n 3n
.
2n−1
(−1)n+1 n2
n=1 n3 +10n+1 .
2n+1
n=3 n2 ln n .
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IV. (10 pt each) Recall that f (x) = e−x is not an elementary function in the
R sense
2
that no matter how hard we try. we will never succeed in evaluating e−x dx
in terms of the functions we know.
2
1. Prove that f (x) = e−x is continuous on [0, ∞) by finding its derivative.
2
2. Show that f (x) = e−x is integrable on [0, 44].
3. Prove that
R ∞ −x
e dx = e−44 .
44
4. Prove that
R 44 −x2
e dx is convergent.
0
5. Prove that
R ∞ −x2
e dx is convergent.
0
End of Exam!
2