Calculus | Custom PHD Thesis

Problem Set 6.

Due in tutorial on February 1–2

Instructions:

• Print this cover page, fill it out entirely, sign at the bottom, and STAPLE

it to the front of your problem set solutions. (You do not need to print

the questions.)

Doing this correctly is worth 1 mark.

• Submit your problem set ONLY in the tutorial in which you are enrolled.

• Before you attempt this problem set read the notes we posted online about

the definition of the integral and do all the practice problems from section

12.1, 5.2, 5.3 (see course website).

PLEASE NOTE that so far over 20 students have been penalized for academic misconduct and now have a record with OSAI. Do not be the next one. Re-read “Important

notes on collaboration” on the cover page for Problem Set 1.

Last name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

First name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Student number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Tutorial code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

TA name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Please, double-check your tutorial code on blackboard, and double-check your TA name

on the course website. Remember that if there is a discrepancy between Blackboard and

ROSI/ACORN, then your correct tutorial is the one on Blackboard, not on ROSI/ACORN.

See http://uoft.me/137tutorials

Note: Question 0 is a warm-up question to make sure you understand the definition of

lower/upper integral and the definition of supremum and infimum. Do not submit it.

Only submit the other five questions.

0. Let f be a bounded function on the real interval [a, b].

(a) Prove that I b a(f) satisfies the following two properties:

i. L

f(P) ≤ I b a(f) for every partition P of [a, b].

ii. For every ε > 0, there exists a partition P of [a, b] such that I b a(f) − ε <

f(P) ≤ I b a(f).

(b) Let J be a real number. Assume J satisfies the following two properties:

i. L

f(P) ≤ J for every partition P of [a, b].

ii. For every ε > 0, there exists a partition P of [a, b] such that J − ε <

f(P) ≤ J.

Prove that J = I b

a(f)

Note: This problem is very short once you are very comfortable with the definition

of supremum and of lower integral.

1. Write the statement (without proof) of the results equivalent to Question 0 for I b a(f)

instead of I b

a(f).

2. Let f be a bounded function on the interval [a, b].

(a) Assume that f satisfies the following property:

∀ε > 0, ∃ partition P of [a, b], such that Uf(P) − Lf(P) < ε.

Prove that f is integrable on [a, b].

(b) Assume that f is integrable on [a, b]. Prove that f satisfies the following property:

∀ε > 0, ∃ partition P of [a, b], such that Uf(P) − Lf(P) < ε.

Hint: Use the definition of integrability: f is integrable on [a, b] if and only if

I b

a(f) − I b a(f) = 0. Also use the definitions of I b a(f) and I b a(f). Finally, remember

that we always know that I b a(f) − I b a(f) ≥ 0, whether f is integrable or not.

3. Let f and g be two bounded functions on the interval [a, b].

(a) Let P be a partition of [a, b]. Only one of the following two inequalities is always

true:

f+g(P) ≤ Lf(P) + Lg(P), Lf+g(P) ≥ Lf(P) + Lg(P)

Determine which one is always true, prove it, and then show the other one is

not always true with an example.

(b) Repeat Question 3a with upper sums instead of lower sums.

on [a, b].

Hint: Use Problem 2 repeatedly.

4. Give an example of two bounded functions f and g on an interval [a, b] such that

I b

a(f + g) 6= I b a(f) + I b a(g).

5. In this question, you are going to compute the exact value of Z2 5(5x − x2) dx using

Riemann sums. Let us call f(x) = 5x − x2. Since f is continuous on [1,3], we know

it is integrable. Hence, its value can be computed using any Riemann sums via

equation (5.2.7) in the book.

For every natural number n, let us call Pn the partition that splits [2,5] into n equal

sub-intervals. Notice that lim

n→∞

||Pn|| = 0. Hence, we can write

Z2 5 5x − x2 dx = lim n→∞ S(Pn),

where S(Pn) is any Riemann sum for f and Pn. In particular, to make things simpler,

we are going to choose the Riemann sum S(Pn) where at every subinterval we use

the righ-endpoint to evaluate f.

(a) Let us write Pn = {x0, x1, . . . , xn}. Find a formula for xi in terms of i and n.

(b) What is the length of each sub-interval in Pn?

your above answers to obtain an expression for S(Pn) in the form of a sum with

sigma notation.

(d) Using the formulas

NX i

i =

N(N + 1)

NX i

i2 = N(N + 1)(2N + 1)

NX i

i3 = N 2(N + 1)2

if needed, add up the expression you got to obtain a nice, compact formula for

S(Pn) without any sums or sigma symbols.

(e) Calculate lim

n→∞

S(Pn). This number will be the exact value of Z2 5(5x − x2)dx.

Hint: Your final answer should be 27

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