Sample Midterm Problems Math 3210
Spring 2010
Instructor: Rémi Lodh
Apr. 12th 2010
NB: When you are asked to state a theorem, then you must give the full statement
of the theorem. Similarly, if you need a theorem to justify your answer, then you must
give the full statement of the theorem.
1. Let I ⊂ R be an open interval, f : I → R a function, and a ∈ I.
(i) Define lim f (x) = L, lim+ f (x) = L, and lim− f (x) = L.
x→a
x→a
x→a
(ii) Prove that lim f (x) = L if and only if lim+ f (x) = L and lim− f (x) = L.
x→a
x→a
x→a
2. Let I ⊂ R be an open interval, f : I → R a function, and a, L ∈ I.
(i) State a criterion for lim f (x) = L in terms of sequences.
x→a
(ii) Write down what it means to say that f is continuous at a in terms of limits.
(iii) Suppose that g is a function on an open interval J with g(J) ⊂ I, u ∈ J, and
lim g(x) = L ∈ I. Prove that if f is continuous at L, then lim (f ◦g)(x) = f (L).
x→u
x→u
3. Let f : (a, b) → R be a function.
(i) Define lim+ f (x) = +∞.
x→a
(ii) Assume that f is a positive function. Prove that lim f (x) = +∞ if and only
x→a+
1
= 0.
if lim+
x→a f (x)
4. Let I be an open interval and let f, g be functions on I.
(i) Define what it means to say that f is differentiable at a ∈ I.
(ii) Prove that if f is differentiable at a ∈ I, then f is continuous at a.
(iii) Prove that if f and g are differentiable at a ∈ I, then so is f g.
5. Let [a, b] be a closed bounded interval and let f : [a, b] → R be continuous.
(i) Define what a critical point for f on [a, b] is.
(ii) Prove that if f attains its maximum at c ∈ [a, b], then c is a critical point for f
on [a, b].
1
6. Let [a, b] be a closed bounded interval, f a continuous function on [a, b], differentiable on (a, b).
(i) State the Mean Value Theorem for f on [a, b].
(ii) Prove that if f 0 (x) = 0 for all x ∈ (a, b), then f is a constant function.
(iii) Prove that if f 0 (x) > 0 for all x ∈ (a, b), then f is a strictly increasing function.
7. Let f be a differentiable function on an open interval (a, b).
(i) State the Mean Value Theorem for f on a closed bounded interval [x, y] ⊂ (a, b).
(ii) Prove that if f 0 is bounded on (a, b), then f is uniformly continuous on (a, b).
8. Let f be a bounded function on a closed bounded interval [a, b].
(i) For a given partition P of [a, b], define the upper and lower sums for f and P .
Define the numbers Uab (f ) and Lba (f ). Define what it means to say that f is
integrable on [a, b].
(ii) Prove that f is integrable on [a, b] if and only if for each > 0 there is a partition
P of [a, b] such that U (f, P ) − L(f, P ) < 9. Let f be an integrable function on a closed bounded interval [a, b].
Rb
(i) Write a f (x)dx as a limit, carefully defining any symbols you may require.
Ra
Pn
a2
n(n + 1)
(ii) Prove that 0 xdx = . You might need to use the formula k=1 k =
2
2
at some point.
10. Let f be a continuous function on a closed bounded interval [a, b] with m =
min(f ) and M = max(f ).
[a,b]
[a,b]
(i) Prove that f is integrable on [a, b].
(ii) Prove the inequalities
Z
m(b − a) ≤
b
f (x)dx ≤ M (b − a).
a
1 Rb
f (x)dx.
b−a a
11. Let f, g be continuous functions on [a, b] which are differentiable on (a, b) and
whose derivatives f 0 , g 0 are integrable on [a, b].
Deduce that there is c ∈ [a, b] such that f (c) =
(i) State and prove the 1st Fundamental Theorem of Calculus.
(ii) Assuming that f 0 g and g 0 f are integrable, prove the integration by parts formula:
Z b
Z b
f 0 (x)g(x)dx = f (b)g(b) − f (a)g(a) −
g 0 (x)f (x)dx.
a
a
12.
(i) State the 2nd Fundamental Theorem of Calculus.
d R x2
sin(t)dt.
dx x
http://www.math.utah.edu/~remi/teaching/3210Spr2010/3210Spr2010.html
(ii) Evaluate
2