Homework assignments, Real Analysis II (Math 447)

Prof. Janeba, Spring 2012

 Week #1:

• Jan. 17:
  
• Show that given a sequence of real numbers {a_n}, if a_nL, then |a_n||L|.
• Show how the prior problem proved that the absolute value function is continuous for all x.
• Prove that if a is a limit point of set A, then there exists a sequence {x_n} of points in A such that x_na for all n, and x_na.
• Jan. 19: Read chapter 5.2 and do 5.2 #1ce, 3ab (note the example cited), 5a, 13,15.
  

Week #2:

• Jan. 24:
• do 5.2 #12 (go ahead and read the hint in the "answers", I did, #16, 19 (add to #19:  Prove that f '(0) exists),20
• Use the Inverse Function Theorem to prove that the derivative of x^1/n is (1/n)x^{1/n -1}, for n and x>0.
• Use the chain rule and the previous problem to prove that the derivative of x^p is px^p-1 for p rational and x>0.
• Show that if f is continuous and strictly increasing on an interval I, and a,bI, then f( [a,b] ) = [f(a),f(b)].  Note one must both prove the set in question actually is an interval, and that it has endpoints as shown.
• As a corollary to the previous, show that if f is continuous and strictly increasing on an interval I, and a,bI, then f( (a,b) ) = ( f(a),f(b) ).
• Jan. 26:
• For next Tuesday,  Turn in the 1/19,1/24 homework, and rewrite 1/17 homework as needed to get the rest of the points.
• Group oral presentations for Tuesday 1/31: Consider the function L:(0,) such that L'(x)=1/x for all x(0,) and L(1)=0.
• a) Show that a,b>0, L(ab)=L(a)+L(b).  Hint: Consider the derivative of L(ax) for a fixed a>0, and use some of our theorems.
• b) Show that b>0, L(1/b) = -L(b).  Hint:  Consider the derivative of L(1/x).
• b') Show that a,b>0, L(a/b)=L(a)-L(b).
  
• c) Show that b>0,r rational, L(b^r) = rL(b).  Hint:  First prove L(bⁿ) = nL(b) for n (induction?), then find L(b^1/n) for n by considering L((b^1/n)ⁿ).  Note the rational restriction.  In fact, this holds for all real r, but you needn't prove that.
• d) Show that L(e) = 1, where e is the limit as n goes to infinity of (1+1/n)ⁿ.  Hint:  (1+1/n)ⁿe so L((1+1/n)ⁿ)?  (and why?).  But this limit is tricky to evaluate.  Further hint:  L'(1) exists, write out its definition in the x0 form and simplify.  Finally, 1/n0, apply the continuity of a function built from L.
  
• e) Show Range(L)=.  Hint:  We know L is continuous and strictly increasing (why?), and it includes 1.  What's the alternative to the range being ?

Week #3:

• Jan. 31: (Jumping back to finish continuity): Read 4.3 Uniform Continuity, and do 4.3 #2ab,3a,4a.  Lots of hints in the back, here's another for 4a:  You can do cases: xy, x>y, or explain why, WOLOG, xy.
• Feb. 2:

Week #4:

• Feb. 7:
  
• Feb. 9:

Week #5:

• Feb. 14:
• Feb. 16:

Week #6:

• Feb. 21:
• Feb. 23:

Week #7:

• Feb. 28:
• Mar. 1:  Do the integration problems (NOT ALL OF THEM)! #1,2,3,5, and 6.2.  Start to think about 6.3.
  

Week #8:

• Mar. 6:
• Mar. 8:

Week #9:

• Mar. 13:
• Mar. 15:

Week #10:

• Mar. 20:
• Mar. 22:

Week #11:

• Apr. 3: Read 8.1, 8.2, do 8.1 #1-3,5
  
• Apr. 5: Read 8.3, do 8.2 #2,5,8a,c,11,13 and 8.3 #4  (use the results from class when they help, even if they are out of sequence!)
  

Week #12:

• Apr. 10: Turn in Thursday:
• 8.1 #5, 8.2 #2(ab at least, try for c.  Hint:  at least one of f,g must be unbounded on E by part (b)),
  
• 8.2 #11,13
• New:  Prove that if A,B are subsets of , and f_nf uniformly on A and f_nf uniformly on B then f_nf uniformly on AB.
• Use the previous problem to do 8.2 #3 as a mostly-easy corollary
• New: Do 8.3 #3
  
• Apr. 12: Do 8.5 #3,6bcd
  

Week #13:

• Apr. 17:Read 8.7, especially the examples, do 8.7 #1ab,3,5,11,15 (#15 through n=3) and
• Prove that if the power series a_n(x-x_o)ⁿ converges to f(x) with radius of convergence R, then a_n/(n+1)*(x-x_o)ⁿ⁺¹ converges to F(x) for some function F on an interval with the same radius of convergence (and same center, yes) such that F'(x)=f(x) for all x in (x_o-R,x_o+R).
  
• Apr. 19: Do 9.1 #1,2 (use theorem 9.1.4), and #4,6a.  Note with glee that you've already proven #13.  Read 9.1, 9.2
  

Week #14:

• Apr. 24:Do 9.1 #5, 9.2 #1 (almost off-topic, that one is), 9.2 #2, 3 (use!! 9.1.7 & the fact that this is given to be an orthogonal system)  9.3 #1,3b,c (use #1 for 3c, and use Maple for all of #3)
  
• Apr. 26:

Week #15:

• May 1 (Last day of class)
  

Week #16:

• Friday May 4, 2-5 p.m..  Final Exam
  

Last modified April 24, 2012.
Prof. Janeba's Home Page | Send comments or questions to: mjanebawillamette.edu