# Calculus II

Winter 2013

Math 152, Section 32Instructor: Laurence Field

## Course information

Please read the course information handout for information about classes, homework and exams.

The **office hours** have been updated and moved since the handout.
They are now Monday 4:00, Tuesday 1:00, Thursday 1:00 (1 hour each), always in Eck 14 (in the Eckhart basement). You can also email me to request an appointment.

## Exams

The best practice for midterms and finals is to do problems from the relevant sections of the book, and then try a past paper under timed conditions. You can find past papers on last year's page.Note that Midterm 2 this year will cover only sections 5.9, 6.1–6.4, and 7.1. That means there will be no questions involving logarithms (in particular, no integrals of 1/x, tan x or cot x).

First midterm and solutions.

Second midterm and solutions.

Final exam and solutions.

## Homework

The exercises marked with an asterisk are to be handed in. The rest are extra exercises to check your understanding, and should not be handed in. You are welcome to ask about them or other questions at office hours or problem session.

*Tuesday, January 8:*

§4.6 ex 1, *2, 3, *4, 5, *6, 27, 35, 39, *40;

§4.7 ex 1, *2, 7, *20, *26, 29, *34, 43, 45.

*Thursday, January 10:*

§4.8 ex *4, 9, 16, *20, 24, 27, *30, 35, *46, 50, *54, *55, *57, *58(b).

*Tuesday, January 15 and Thursday, January 17:*

§4.10 ex 2, *6, 7, *12, 13, *19, 22, 29, *40;

§5.2 ex 4, *10, 11, *18, *20, 21, 23, 25—30, *31, *41 (use a calculator; answer correct to 4 decimal places)

§5.3 ex *2, 4, *6, *8, 10, *14 (give reasons), *20, 25, *28, 35, *36.

*Tuesday, January 22:*

§5.4 ex 4, *7, 11, *14, 17, 31, 32, *34, *38, *40, 48, *51, *54, 56, 58, *60—64;

§5.5 ex 5, *10, 16, *17, 26, 27, *34, 35.

*Thursday, January 24:*

§5.6 ex 4, *6, 12, *16, *21, *33, 34, 37, *45, 47;

§5.7 ex 2, 5, 6, *10, 13, *14, 19, 20, 21, 24, *26, 35, *38, 39, *40, 43, 48, 54, 60, 62, 63, *64, *72, 79, *82, 85.

*Tuesday, January 29:*

§5.8 ex *2, *4, *6, *8, *10, *12, *14, *16, 20, *24, *30, *33, *34, 36, *38.

*Extra problem.

*Thursday, January 31:*

The first midterm was given. There is no homework.

*Tuesday, February 5 and Thursday, February 7:*

§5.9 ex 15, *16, 17, *18, 23, *24, 25, *26, 27, *28, *30, *35.

§6.1 ex 5, *12, *20, *28, *33, *37, 39, *40, *46.

*Extra problem: Suppose that *f* is a continuous function
that has integral zero on every interval symmetric about zero.
Show that *f* is odd.

§6.2 ex 27, *28, *30, 33, *35, *36, 39.

*Tuesday, February 12:*

§6.2 ex *2, 7, *14, 21, 24, *40, 41, 42, *44, 45, *46, 47, *52, *54, 57, *60.

§6.3 ex *6, *12, 23, *27 (use shells), *34, *38, 41, *43, *46, 47.

*Thursday, February 14:*

§6.4 ex 3, *12, 19, *20, *24, *25, *27, *30, 33.

Extra problems:

*1. Find the volume of the solid formed by intersecting a cylinder of radius √3 with
a prism whose cross-sections are equilateral triangles of side length 2, where one of the edges of the prism passes through the center line of the cylinder at right angles, and the opposite prism face just touches the top of the cylinder.
(Assume the cylinder and the prism are infinite in both directions.)
For extra credit: what happens if the prism's edge and the cylinder's center line meet at an angle θ?

*2. Write down an integral expressing the volume of the solid formed by two cylinders of radii a and b respectively (where a < b), whose center lines meet at right angles. (I don't expect you'll be able to evaluate this integral).

*
Tuesday, February 19:*

§7.1 ex 3, *8, 12, *16, 17, 27, 29, *30, *32, *33, *34, 37, 39, 42, *44, *46, *48, *49, *50, *52, *53, *54, 55, *56.

*Thursday, February 21:*

The second midterm was given. There is no homework.

*
Tuesday, February 26:*

§7.2 ex *12, 17–19, *20, *22–3. Public service announcement: exercises 24–5 are far too difficult and the book's claimed solution is naïve and wrong. (See if you can figure out what the issue is.)

§7.3 ex 4, *8, 9, *10, 12, 39, *46.

*Extra problem: Let h(n) = 1 + (1/2) + (1/3) + ... + (1/n). (This is known as “the nth partial sum of the harmonic series”.)

(a) Prove that h(n) - 1 < log(n) < h(n-1) for all integers n>1.
Hint: use lower and upper sums for the function 1/x.
I strongly advise you to draw the picture.

(b) Prove that f(n) = log(n) - h(n) + 1 is an increasing function of n. (It's enough to show that f(n+1) > f(n).)

(c) Prove that f is bounded below by 0 and above by 1.

*Side remark: this proves that the limit as n → ∞ of h(n) - log(n) exists and is between 0 and 1.
The limit is called Euler's constant and is denoted by γ.*

(d) (optional; harder) Show that γ > 1/2. Hint: use the concavity of the logarithm function. (In fact, γ = 0.577... . It is still unknown even whether γ is rational.)

*Thursday, February 28:*

§7.3 ex 19, *22, 24, *28, 29, 31, *32, 33, *34, *36, *42, *47, 48, *50, 51, *56, *60, 63, *65, *66, *74.

*Tuesday, March 5:*

§7.4 ex *34, 35, *38, *42, 49, *52, 53, *54, 55, 58, 65, *67, *68, 70, *72, 73.

§7.5 ex 13, 19, *22, 24, 27, 31, *32, *44, 50, 51, *54, 55.

*Thursday, March 7:*

§7.5 ex 12, *16, *34, 58.

§7.6 ex *3, *6, 8, 13, *14, 15, *16, 23, *24, *26, 30, 33.