Homework page for Math 435; Spring 2004

 

Homework problems will be posted here; the most recent on top. If in a problem set, there are two exercises marked (G) and (U), then students taking the course for graduate credit should submit the exercise marked (G) while the rest can submit either one of the two.

  • Wednesday, 4/21/04: Read Section 4.4 until the end of Theorem 4.27 and do Exercises # 2, 6, 7, and 4 (Extra Credit Problem); due Monday 4/26/04.

Hint for Exercise # 2: Use the Inverse Function Theorem.

Hint for Exercise # 4:  (This exercise is meant to be done without using the starred theorem 4.30).

a)      Prove that f is 1-1 using proof by contradiction. Let c=inf{f(x): x in (a,b)} and d=sup{f(x): x in (a,b)}; then c and d are extended real numbers and c<d. Show that f maps (a,b) onto (c,d): Let y_0 in (c,d) be given and show that y_0 = f(x_0) for some x_0 in (a,b).

b)      Use the Inverse Function Theorem and its proof. To show that (f^{-1})^prime is continuous on (c,d), let y_0 in (c,d) be given and show that (f^{-1})^prime is continuous at y_0.

c)      Consider the function f(x)= x^3 on (-1,1). Find (c,d) and find f^{-1} on (c,d); then show that (f^{-1})^prime does not exist at one point.

Hint for Exercise #6: Use the Inverse Function Theorem, and your “Calculus knowledge” of antiderivatives to get f(x) on (a,b). Then use the fact that f is continuous on [a,b] to get f(x) on [a,b].

Hint for Exercise # 7: Use the Inverse Function Theorem, and the fact that arcsin x is in (-pi/2,pi/2) for x in (-1,1).

  • Wednesday, 4/14/04: Finish reading Section 4.3 and do Exercises 1 (b,e) (U), 1(d,e) (G); due Monday 4/19/04. Hint: Use L’Hopital’s Rule.
  • Friday, 4/9/04: Read Example 4.16 and Theorem 4.17 (Section 4.3) and do Exercises 2 (a,b) (U), 2(a-c) (G), 5 and 6 (Extra Credit Problem); due Monday 4/19/04. Hint for Exercises 2 b) and 2 c): Apply the Mean Value Theorem as in Example 4.16. Hint for Exercise 5: Apply the Mean Value Theorem in each part (a-c). Hint for Exercise 6: Apply the Mean Value Theorem, use the fact that |f^prime|<M on (a,b) for some M>0 in R and use the definition of uniform continuity.
  • Monday, 4/5/04: Read pages 93-95 (Section 4.3).
  • Friday, 4/2/04: Finish reading Section 4.1.
  • Wednesday, 3/31/04: Read Theorem 4.4 and Example 4.5 in Section 4.1, read Section 4.2, and do Exercise # 8 (Section 4.2). Extra Credit Problem: Exercise # 4 (Section 4.1); due Monday 4/5/04. Also read and understand (but do not submit) Exercise # 5 (Section 4.1) which states some properties of the logarithm function.

Hint for Exercise # 4 (Section 4.1): Submit only parts a, d, e; but think about b) and c) until I give you the solution next week and assume they are true if you need them in parts d) and e). Also read the notations in the second paragraph of page 85 (needed for parts d and e).

Part a) To show that sin x is continuous at 0, use the inequalities in  (vi) and similar ones that hold for –pi/2<x<0 to show that lim_{xà0}sin x = 0 = sin 0. Then use that and Equation (iii) to show that cos x is continuous at 0.

Part d) Let a in R be given. Consider the difference quotient: [sin(a+h)-sin a]/h; then use Equation (iv) to rewrite sin(a+h); then use part c) to show that the limit of the difference quotient (as hà0) exists and is equal to cos a.

Part e)  To show that cos x is differentiable on R (i.e. at a for all a in R) with (cos x)’=-sin x, use part d), Equation (v), and the Chain Rule. Use the Quotient Rule to say where tan x is differentiable (not all of R, as stated in the book!) and show that for all such x, (tan x)’=sec^2 x=1/cos^2 x.

       Hint for Exrcise # 8 (Section 4.2):

Part a) First, you need to show that y=f(x) is differentiable on (0,infinity) (for that you may use without proof the result of Exercise 6b) in Sec 4.1). To show that  ny^{n-1}y^prime =mx^{m-1}, note that (f(x))^n = x^m. Use the result of Exercise # 6 a) of Section 4.1 and the Chain Rule to write the derivatives of both sides of the equation (f(x))^n = x^m.

Part b) Consider the three different cases: q>0, q=0, and q<0. For the case q>0, use part a). The case q=0 is easy. For the case q<0 (then –q>0), use the fact that x^q=1/ x^{-q} for all x>0 and use that to show that x^q is differentiable on (0, infinity) with derivative qx^{q-1}.

  • Wednesday, 3/24/04: Read Section 3.4 and do Exercises # 1(b), 2(a,c) (U), 2(b,c) (G), and 5; due Monday 3/29/04.

Hint for Exercise # 2: Use Theorem 3.40.

Hint for Exercise # 5 a):  You may choose either one of the following two approaches (the first approach is much easier and shorter).

                                            I.      Using Theorem 3.40.

                                         II.      Using the definition of uniform continuity: Choose a delta>0 that corresponds to epsilon=1 in the definition of uniform continuity and use the boundedness of the interval  I to divide it into finitely many (say m) subintervals, each of length< delta. Fix an x_0 in I and show that for every x in I, |f(x)-f(x_0)|<m, from which then show that f(x) is bounded on I.

 

Hint for Exercise # 1:  Use the Intermediate Value Theorem.

Hint for Exercise #8:

Part a) First show that f(nx)=nf(x) for all x in R and for all positive integers n using induction on n; then show that f(0)=0 to conclude that f(0x)=0f(x) for all x in R. Make use of the previous work to show that f(nx)=nf(x) for all x in R and for all negative integers n.

Part b) For q in Q, write q=m/n where n,m in Z, n>0; then use part a).

Part d) Make use of parts b) and c) to show that f(yx)=yf(x) for all x,y in R; then use that to show that f(x)=mx for all x in R, where m=f(1).

Extra Credit Problem: Let a<b be given in R, and let f: [a,b]à R be continuous. Prove that f([a,b])=[m,M] for some m less than or equal to M in R. Hint: Use the Extreme Value Theorem and the Intermediate Value Theorem.

  • Wednesday, 3/17/04: Read pages 70-72 (Section 3.3) and do Exercises # 2 (G), 2 (a,b,d) (U), 6 (a) (Extra Credit Problem); due Monday 3/22/04.  Hint: You may use the fact that sin x, cos x and e^x are continuous on R.
  • Monday, 3/15/04: Finish reading Section 3.2 including the handout and do Exercises # 1 (b), 2 (a,c,e), 3 (a,c,e,f); due Monday 3/22/04. 
  • Friday, 3/05/04: Read pages 65, 66 (until Theorem 3.14) in Section 3.2 and do Exercises # 1 (a), 2(b,d), 6; due Monday 3/15/04.  Hint for Exercise #6: You may use the fact that sin(y)à0 and cos(y)à1 as yà0 as well as the trigonometric identities: sin(2y)=2sin(y)cos(y) and cos(2y)=1-2sin^2(y).
  • Monday, 2/23/04: Read Section 3.1 and do Exercises 1 (a, c), 2 (a, b), 3, 4 (G), 8 (U); due Monday 3/15/04. Extra Credit Problem: Let P(x) be a polynomial and let a be given in R. Prove that P(x)àP(a) as x à a; hint: Use the limit theorems in Section 3.1 and Example 3.2 to first show (by induction) that x^nàa^n as xàa for all positive integers n. Then show that P(x)àP(a) as x à a (again using induction over the degree n of  the polynomial P(x)).
  • Monday, 2/16/04: Read Sec 2.4 and do Exercises 1-9 (not to be handed in; I will go over as many of them as possible on Wednesday!)
  • Friday, 2/13/04: Finish reading Sec 2.3 and do Exercise # 1; due Friday 2/20/04.
  • Wednesday, 2/11/04: Read Sec 2.3 until the end of Example 2.21 and do Exercises # 2, 4 (U), 8 (G); due Friday 2/20/04. Hint for Exercise # 2. Hint for Exercise # 4: First show by induction that 0<x_{n+1}<x_n<1 for all n. Hint for Exercise # 8: When showing that the limit exists, consider the three separate cases: x_0<1, x_0=1 and x_0>1; then prove that the limit is equal to 1.
  • Friday, 2/6/04: Finish reading Sec 2.2 and do Exercises # 1 (a, c), 2 (c, d), 3 (G), 4 (U), 5 (Extra Credit Problem); due Friday 2/20/04. Also do (but do not submit) Exercises 6, 7, 8.
  • Wednesday, 2/4/04: Start reading Sec 2.2.
  • Monday, 2/2/04: Read Section 2.1 and do Exercises # 1-8 (# 8 is an extra credit problem), due Monday 2/9/04.
  • Friday, 1/30/04: Read the handout (Sec 1.4) and do Exercise # 3 in Sec 1.3 (using countability arguments from the handout), due Monday 2/9/04.
  • Wednesday, 1/28/04: Finish reading Section 1.3 and do Exercises (Sec 1.3) # 1, 4 (due Friday, 01/30/04), and 5, 6, 7, 10 (due Monday 2/9/04.) Hint for Exercise # 4: Use Theorem 1.24 on page 20.
  • Monday, 1/26/04: Read pages 18 and 19 (Sec 1.3).
  • Friday, 1/23/04: Read Section 1.2 and do Exercises # 1, 2, 7, 9 on page 17 and the exercise from the lecture (namely, to show that Z is closed under addition and multiplication- An extra credit problem.) Due Friday, 01/30/04 at the beginning of the class.
  • Friday, 1/16/04: Finish reading Section 1.1 and do exercises 2 (a), (c); 3 and 4 using only the axioms for fields and order (and existence of roots in #4, proof of (8)). Also do Exercise 6 and read Appendices A.1-A.6 in the book. (All homework problems from Sec 1.1 are due on Friday, 1/23/04).
  • Wednesday, 1/14/04: Read pages 4-7 in the textbook, finish reading the handout and work on Exercise 3 and the Extra Credit problem in the handout (due Friday, 1/23/04.) Be sure to pay close attention to the text between Equation (9) on page 7 and Definition 1.4 on page 8.
  • Monday, 1/12/04: Read pages 1-3 in the textbook, read the first two pages of the handout and work on Exercises 1 and 2 in the handout (due Friday, 1/23/04.)

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