MAA 4103/5105 HOMEWORK

HOMEWORK POLICY

It is ok to obtain help on the homework but any help must be acknowledged. It is NOT OK to copy anothers work including work from books or websites. Any person, book or website used must be acknowledged.

• Homework 13 is due Wednesday, April 16.
  Do Ex 8.4 (Textbook p.357) Problems 1(a),(b),(f),(g), 5*, 6.
  See SOLUTION.
• Homework 12 is due Monday, April 14.
  Do Ex 8.3 (Textbook pp.350-351) Problems 1, 11.
  See SOLUTION.
• Homework 11 is due Friday, April 4.
  Do Ex 8.1 (Textbook p.340) Problems (1)(a),(c),(f),(j) and Ex 8.2 (Textbook p.346) Problems (1)(b),(c),(d).
  See SOLUTION.
• Homework 10 is due Friday, March 28. Do Ex 7.4 (Textbook pp.322-323) Problems 2, 6 and 11.
  See SOLUTION.
• Homework 9 is due Friday, March 14. Do Problems 7.2, 7.3 and 7.4.
  See SOLUTION.
• Homework 8 is due Friday, February 28. Do Problem 6.8. This is the problem on Integration by Substitution.
  See SOLUTION.
• Homework 7 is due Friday, February 21.
  See SOLUTION.
• Homework 6 is now due Monday, February 17.
  Do the following textbook problems: (1) #3 (Ex.6.1, p245), (2) #7 (Ex.6.2., p.249), (3) #12 (Ex.6.2., p.250).
  [A grade for two correct problems, A+ for three correct problems.]
  See SOLUTION.
• Homework 5 is due Monday, February 3.
  (1) Let (a), (b) be real numbers with (a < b). Prove that
  [ a < frac{a+b}{2} < b].
  (2) In Example 6.3 prove that
  [ {{int}_{_}}^1_0 g,dx le frac{1}{2}.]
  Note that the integral above is a lower Riemann integral.
  (3) Prove that
  [ frac{1}{2} – frac{1}{2n} le {{int}_{_}}^1_0 g,dx,]
  for all (n ge 1), and hence
  [{{int}_{_}}^1_0 g,dx = frac{1}{2}.]
  (4) Do Problem 6.1 (p.55 of online notes).
  See SOLUTION.
• Homework 4 is due Wednesday, January 29.
  Let (f,:, mathbb{R} longrightarrow mathbb{R}) be differentiable and suppose
  (f’) is bounded (i.e. there is a positive constant such that (|f'(x)| le M) for all (x)).
  Prove that (f) is uniformly continuous. [HINT: Use the Mean Value Theorem]
  See SOLUTION.
• Homework 3 due Friday, January 24.
  Let (f,:, (u,v) longrightarrow mathbb{R}) be differentiable.
  Prove using the Mean Value Theorem that if (f’ > 0) (i.e. (f'(x) > 0) for all (x in (u,v))) then (f) is strictly increasing. For ONE BONUS POINT show that the converse is not true.
  See SOLUTION.
• Homework 2 due Wednesday, January 22.
  (a) Let (nin mathbb N^{+}). Let (f,:, mathbb{R} longrightarrow mathbb{R}) by
  (f(x) = x^n). Use the product rule and mathematical induction to prove that
  $$
  f'(x) = n x^{n-1}.
  $$
  (b) Again let (nin mathbb N^{+}). Define (g,:, (0,infty) longrightarrow (0,infty)) by (g(x) = x^{frac{1}{n}}). Prove that
  $$
  g'(x) = frac{1}{n} x^{frac{1}{n}-1}.
  $$
  See SOLUTION.
• Homework 1 due Wednesday, January 15.
  Do Problem 4.4 (p.37 of the online notes).

Last update made Mon Jan 27 11:53:21 EST 2014.