18.014 Exam 3
Page 1 of 4
Nov 20, 2015
All four problems are weighted equally. You may use any theorems or results
that have appeared in class, problem sets, or sections of the textbook covered
thus far. Just make it clear what you are using!
Name:
1. Find real numbers a, b, c such that
2
1 + cx
lim
−
= 3.
x→0
x
a sin x + b cos x
18.014 Exam 3
2. (a) Compute the integral
Page 2 of 4
Z
Nov 20, 2015
2
xex dx.
(b) Compute the integral
Z
0
1
2
x3 ex dx.
18.014 Exam 3
Page 3 of 4
Nov 20, 2015
3. Circle either T or F to indicate whether each statement is true or false. You do not need
to justify your answers.
T
F
1. If p(x) is a polynomial of degree 3, then there exist real numbers A1 , A2 , A3
such that
Z
p(x)dx
A2
=
A
x
+
+ A3 log(x − 2) + C.
1
(x − 1)2 (x − 2)
x−1
T
F
2. If f : R → R is twice-differentiable and satisfies f (0) = 0 and
f 00 (x) = −f (x) for all x ∈ R, then there exists a ∈ R such that
f (x) = a sin x for all x ∈ R.
T
F
3. If f (x) = o(x2 ) as x → 0, then f (x) = o(x sin x) as x → 0.
T
F
4.
xx
2 = +∞
x→+∞ ex
lim
T
F
5. If f (x) is a polynomial of degree n, then the nth Taylor polynomial of f
at the point 1 is equal to f .
T
F
6. Let f : R → R be the inverse function to the polynomial p(x) = 31 x3 + x.
Then f is differentiable and
f 0 (x) =
1
for all x ∈ R.
1 + f (x)2
18.014 Exam 3
Page 4 of 4
4. Suppose that f : R → R is infinitely differentiable and satisfies
f (n) (0) = 1
for every integer n ≥ 0, where f (n) is the nth derivative of f . Prove that
f (x) − ex
=0
x→0
xn
lim
for any integer n ≥ 0.
Nov 20, 2015