Spring 2010: MATH 3210-001
Foundations of Analysis I
Wed. 21st April
Midterm Examination
Name:
Student ID:
Instructions.
1. Attempt 4 out of the 5 questions.
2. Circle the 4 question numbers which you want to be graded.
3. Give the full statement of a theorem when asked to state it.
4. If you need a theorem to justify your answer, then you must give the full statement.
5. Scratch paper is on the last page. Ask the instructor if you need some more.
Question Points Your Score
Q1
10
Q2
10
Q3
10
Q4
10
Q5
10
TOTAL
40
Dr. R. Lodh
50 minutes
Q1]. . . [10 points] Let I be an open interval, f a function on I and c ∈ I.
(i) Define lim f (x) = L, lim+ f (x) = L, lim− f (x) = L.
x→c
x→c
x→c
(ii) Prove that lim f (x) = L if and only if lim+ f (x) = L and lim− f (x) = L.
x→c
x→c
x→c
OVER
Q2]. . . [10 points] Let I ⊂ R be an open interval, f : I → R a function, and a, L ∈ I.
(i) Write down what it means to say that f is continuous at a in terms of limits.
(ii) Suppose that g is a function on an open interval J with g(J) ⊂ I, u ∈ J, and lim g(x) = L ∈ I.
x→u
Prove that if f is continuous at L, then lim (f ◦ g)(x) = f (L).
x→u
Q3]. . . [10 points] 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.
Q4]. . . [10 points] 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].
Q5]. . . [10 points] 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].
(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
0
Z
f (x)g(x)dx = f (b)g(b) − f (a)g(a) −
a
a
b
g 0 (x)f (x)dx.
Scratch Paper