advanced calculus questions

FINAL EXAM, ADVANCED CALCULUS I, FALL 2020
Name:
1. (15 pts) Let E be the set of real numbers so that δ ∈ E if and only
if for every real number x satisfying |x+2| < δ we have |x
2+2x| < 0.21.
Find sup E.
1
2
2. (15 pts) Which numbers a ∈ R have the property that, for any
sequence xn converging to a, there exists the limit
limn→∞
xn+1
xn
?
3
3. (15 pts) The function
f(x) = 1 − cos x
x
2
is defined on R {0}. How can one define f(0) so that the function
becomes continuous at x = 0? Is this new function Lipschits on R?
Hint for the second question: compute the derivative.
4
4. (15 pts) Suppose that f is continuously differentiable on (a, b).
Is it necessarily true that for any point c ∈ (a, b) there exist x1, x2 ∈
(a, b), x1 6= x2, so that
f(x2) − f(x1) = f
0
(c)(x2 − x1) ?
5
5. (15 pts) Let n ∈ N. Find local maximal and minimal points of
the function
f(x) = x
2n + 2x
n
.
How does the answer depend on n?
6
6. (15 pts) Compute the limit
lim x→+∞
R x
0
e
t
2
dt2
R x
0
e
2t
2
dt
.
7
7. (15 pts) Prove that if f is a continuous function on [0, 1], then
(i)
Z π
2
0
f(sin x)dx =
Z π
2
0
f(cos x)dx.
(ii)
Z π
0
xf(sin x)dx =
π
2
Z π
0
f(sin x)dx.
8
8. (15 pts) Suppose that the function f is integrable on [0, 3]. Prove
that
lim
h→0
Z 2
1
f(x + h)dx =
Z 2
1
f(x)dx.
9
9. (15 pts) Suppose that f is a continuous function on [0, ∞) and
limx→∞ f(x) = A. Find
limx→∞
1
x
Z x
0
f(t)dt.
10
10. (15 pts) Suppose that K is a compact set in R, and F is an open
set in R. Prove that the difference
K F = {x ∈ K : x /∈ F}
is a compact set.