assignments
due Wed Sep 9
Assignment 1
From Priestley:
Chapter 1: 5.2, 9.3(c), 9.3(h), Problems 16, 22(1-c)
Chapter 2: 3.1, 6.7, Problems 12, 15 (Bonus points: can you prove 15(c)?)
Prove that 
is irrational.
due Fri Sep 25
Assignment 2
Prove the identity
| n(n + 1)(2n + 1) |
| 12 + 22 + ⋅⋅⋅ + n2 = |
|
| 6 |
Find the mistake in the following proof by induction:
Theorem: All cars are the same color.
Proof. Let P(n) be the statement that every set of n cars is the same color
Base Case: Given a set of a single car, every car in the set is the same color, so P(1) is true).
Ind hypo.: Assume P(k) is true; that is every set of k cars is monochromatic (i.e. the same color).
Take any set of k+1 cars. The first k cars are monochromatic by the indcutive hypothesis
The last k cars are monochromatic by the inductive hypothesis.
Since these two lists of k cars overlap, every car in the set must be the same color.
Hence the inductive step holds and every set of n cars is monochromatic. QED.
From Priestley:
Chapter 4: 4.6 (use the definition of derivative, not the rules we've developed) and 6.5.
Prove that if f(x) = g(x) - h(x), for "well beahved" functions g and h, then f'(x) = g'(x) - h'(x).
Let f(x) = 1/x. Find f'(x) from first principles.
due Fri Oct 2
Assignment 3
Prove the power rule for negative integer values of n. (Hint: use induction. You proved the base case last week.)
Chapter 5 Problem Set: 10, 16, 19, 31, 43.
due Wed Oct 7
Quiz 1
due Fri Oct 16
Assignment 4
Prove the power rule for fractional powers. (Hint: take y = xp / q
, do some sneaky algebra, differentiate implicitly and then tidy it all up.)
Sketch the graph of the "lemniscate" (x2 + y2)2 = x2 − y2
Sketch the graph of y = x / (x − 2)
Chapter 6 Problems: 5, 8, 11.
due Wed Nov 4
Assignment 5
From Priestley, Chapter 7 Problem Set: 2, 8, 10, 12.
Evaluate the following integrals:
-
-
| 1 | |
| ∫ | (x + 1) / (x2 + 2x + 6)2 dx |
| 0 | |
due Fri Nov 13
Assignment 6
- Prove two of the identities on the trig sheet (link in resources).
- What is the largest possible area of a rectangle inscribed in a semicircle of radius 1?
- Evaluate ∫ sin2(x) dx.
.
- Evlauate ∫ sin3(2x)cos(2x) dx
- Sketch the graph of f(x) = sin(1 / x)
due Wed Nov 25
Assignment 7
From Silverman: p58: 12; p66: 4, 5; p73: 4, 6; p151: 7.
due Wed Nov 25
Quiz 2
due Fri Dec 11
Assignment 8
From Silverman: p158: 7, 13; pp163-4: 3, 8; p172: 15.
