Homework

Assigned Aug. 21

• Prove that Property I4 (see notes) holds for the standard inner product on R^n.
• P. 63, exercise 1. Steps:
  1. Obviously true if y = 0 so assume y not 0.
  2. Show: If ||x+y|| = ||x|| + ||y||, then < x,y > = ||x|| ||y||. (See proof of Triangle Inequality for norms.)
  3. Show: If < x,y > = ||x|| ||y|| and p = ( < x,y > / < y,y > ) y, then ||p|| = ||x||.
  4. Show: In the situation of the Projection Lemma, if ||p|| = ||x||, then q = 0.
  5. Conclude that x is a positive multiple of y.
• P. 98, exercise 12a
• Turn in Friday, Aug. 30.

Assigned Aug. 23

P. 108, exercises 1, 3, 6. In exercise 6, if you find the taxicab metric confusing, you can use instead the metric rho(x,y) = max |x_i - y_i| that we discussed in class. Turn in Friday, Aug. 30.

Assigned Aug. 26

• Let A be a subset of R^n. Show that Int(A) is an open set.
• Let A be a subset of R^n. Let B be the set of all accumulation points of A. Let x be an accumulation point of B. Show that x is an accumulation point of A.
• Turn in Friday, Aug. 30.

Assigned Aug. 30

• Look at but do not turn in: Sec. 2.3, exercises 1, 2, 5; sec. 2.4, exercises 1, 2 (answer is yes, give proof), 3ab; sec. 2.6, exercises 1, 3; sec. 2.7, exercise 5.
• Turn in Friday, Sept. 6: Sec. 2.5, exercise 5; sec. 2.6, exercise 2; sec. 2.7, exercises 2, 3; p. 144, exercises 4c, 9a.
  • On sec. 2.6, exercise 2, second part: If yes, give proof; if no, give example.
  • On sec. 2.7, exercise 3: Don't just quote Prop. 2.7.6 (ii). This problem is part of the proof of that proposition.

Assigned Sept. 6

Click here. Turn in Friday, Sept. 13.

Assigned Sept. 13

Click here. Turn in Friday, Sept. 20.

Assigned Sept. 20 and Sept. 23

Click here. Additional problems were added on Sept. 23. Turn in Friday, Sept. 27.

Assigned Oct. 8

Sec. 4.6, problems 2 and 6b. Suggestion for 6b: x_n=n.
Sec. 5.1, problems 1, 2, 3. On problems 1 and 2, prove your answer. Problem 3: The answer is yes. Prove by adapting the proof of Prop. 5.1.4.
Turn in Friday, Oct. 11.

Sec. 6.1, problems 2, 4.
Sec. 6.2, problems 1, 2.
P. 384, problem 2.
Turn in Friday, Oct. 18.


Assigned Oct. 11

Sec. 6.4, problem 1. Do not use Theorem 6.4.1 as the book suggests. Instead just use the definition of derivative to show that the derivative at the point (0,0) is the matrix [0,0].
Sec. 6.4, problem 2. Investigate this function in the following steps:
(1) Show that f is continuous at (0,0).
(2) Calculate all directional derivatives of f at (0,0).
(3) Is f differentiable at (0,0)? Justify your answer.
Turn in Friday, Oct. 18.


Assigned Oct. 18

Click here. Turn in Friday, Oct. 25.

Assigned Oct. 25

Sec. 6.8, problem 5; sec. 6.9, problems 3, 6; p. 386, problem 16; p. 388, problem 32.
On p. 386, problem 16, I suggest writing Df(x) = A (for all x), and writing f(x)-f(0) as an integral from 0 to 1, like we did in the proof of the degree 2 Taylor's formula.
Turn in Friday, Nov. 1.

Assigned Nov. 1

Sec. 7.1, problems 1, 2, 5. Turn in Friday, Nov. 8.

Assigned Nov. 5

P. 442, problem 25. Turn in Friday, Nov. 8.

Assigned Nov. 6

Pp. 439 - 440, problems 4, 6, 8. Turn in Friday, Nov. 8.

Assigned Nov. 19

Click here. Turn in Wednesday, Nov. 27.

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