Homework due Thursday, Sept. 10 (at recitation):
p. 27-28: 6, 8, 20, 21, 24 plus
(i) Use "set algebra" rules to show why C[CB \ (A \cap B)] =B, where A and B are sets, C denotes complement, and \cap denotes intersection. Briefly identify each particular set algebra rule when you use it.
Homework due Thursday, Sept. 17:
p. 29: 32, 44, 45
PLUS (1) Let A and B be bounded sets of real numbers such that a <= b for every a in A and every b in B. Show that sup A <= inf B. (Remember that sup A may not belong to A, and inf B may not belong to B.)
(2) Let A be a bounded set of real numbers. Define -A = {x: -x \in A}. Show that -A is bounded and that sup (-A) = - inf A.
(3) Use the triangle inequality and the reverse triangle inequality to find an upper bound for the set of all numbers of the form |(x^2-3)/(x-2)| as x ranges over the interval defined by |x-1| < 2/3.
(4) Fix a real number x and a positive number epsilon. If |x-1| <= epsilon, show that |2-x| >= 1 - epsilon.
Homework due Thursday, Sept. 24:
Use the appropriate definitions of limits to show that
(1) lim_{n \to \infty} (5n^2 + 2n)/(n^2 -3) = 5
(2) lim_{n \to \infty} 10 - sqrt{n} = -\infty
(3) lim_{n \to \infty} n - 2 sqrt{n} = + \infty
In (2) and (3), the notation sqrt{n} means the square root of n.
p.55:10, 16
Homework due Thursday, Oct. 1:
p. 55: 9 (This is the Squeeze Theorem.) You should use the epsilon, N definition of limit to prove it.
p. 55: 19, 22
Our FIRST EXAM is scheduled for Monday, Oct. 12. The exam will cover Chapters 0 and 1.
Homework due Thursday, Oct. 8:
Give an epsilon, N type proof of part 2 of the simple limit theorem, namely, that if {a_n} converges to a finite limit A and if c is a real number, then the sequence {ca_n} converges to limit cA. [Hint: Consider the case c=0 separately.]
p.56-57: 32abef (You don't have to use the definition of limit for these. Use other ways, like the simple limit theorems, the Squeeze theorem, etc.)
PLUS
(1) Let 0 < a < b. Find the value of lim_{n \to infinity} (a^n + b^n)^{1/n}. (Hint: Think Squeeze theorem.)
(2) Suppose that {a_n} and {b_n} are two sequences with the properties lim_{n \to infinity} a_n = + infinity and lim_{n \to infinity} b_n = B, where B is finite. Prove that lim_{n \to infinity}(a_n - b_n) = + infinity. Warning: You can't use the Simple Limit Theorems for infinite limits, so you have to prove this by using the definitions.
Homework due Thursday, Oct. 15:
(1) Let E be a set of real numbers. If x = lim_{n \to \infty} x_n and if each x_n is a limit point of E, show that x is also a limit point of E.
(2) Use the epsilon, delta definition of limit to verify that lim_{x \to 1} (x^2 + x)/(3x - 2) = 2.
Homework due Thursday, Oct. 22:
(1) Let A and x_0 be real numbers, and let f(x) be a real-valued function defined in a deleted neighborhood of x_0. Use the definition of limit to prove that lim_{x \to x_0} f(x) = A if and only if lim_{x \to 0} f(x_0 + x) = A.
(2) Use an appropriate definition of limit to verify that
(a) lim_{x \to 1+} (x+1)/(x-1) = + infinity
(b) lim_{x \to - infinity} (1-5x)^{1/3} = + infinity
Book p.79-80: 8, 14, 22(just for the sum f+g); for 8 and 22, you don't have to use the epsilon, delta definition if you can otherwise justify your steps.
Homework due Thursday, Oct. 29:
(1) Use the appropriate definition to show that lim_{x \to + infinity} x^3/(x^2 +10) = + infinity.
Book p.80:19
Book p. 104: 2,6,7,15 (For number 6, when x =0, the question is to show continuity only from the right side since the function sqrt{x} is not defined when x<0.)
Book p. 105: 21
Homework due Thursday, Nov. 5:
Book p. 105: 19 (do only the first part about f+g)
Plus,
(1) Show that the function 1/(x^2) is continuous on (0,1) but NOT uniformly continuous on (0,1). (For the first part, you may use the simple continuous function theorems, but for the second part, use the definition of uniform continuity.)
(2) Show that the irrational numbers are neither an open set nor a closed set.
(3) Let E be a set which is bounded and closed. Show that sup E belongs to E.
Book p. 105: 30
Book p. 105-106: 31,36
EXAM 2 IS SCHEDULED FOR MONDAY, NOV. 23
Homework due Thursday, Nov. 12:
Book p. 106:38
PLUS
As you work on the 5 examples below, try to view them in the context of the Continuous Function Theorems that we are now studying. Each of the examples shows that the conclusion of one of our theorems fails if the hypothesis of the theorem is not met.
Let a and b be real numbers with a < b.
(1) Give an example of a continuous f on [a,b) which is not bounded.
(2) Give an example of a continuous f on [a,\infty) which is not bounded.
(3) Give an example of a bounded f on [a,b] for which sup_[a,b] f is not achieved.
(4) Give an example of a bounded, continuous f on [a,b) for which sup_[a,b) f is not achieved.
(5) Give an example of a bounded, continuous f on [a,\infty) for which sup_[a,\infty) f is not achieved.
PLUS these two applications of some of our Continuous Function Theorems:
(A) If E is a bounded set and p(x) is a polynomial that has no real roots, explain why 1/p(x) is uniformly continuous on E.
(B) Let E be a compact set and let f be continuous on E. Suppose there is a sequence {x_n}_{n=1}^\infty of points of E such that |f(x_n)| < 1/n for each n. Explain why there is a point y of E such that f(y)=0.
Homework due Thursday, Nov. 19:
Book p. 106: 42,44 (Use the Intermediate Value Theorem. For 44, note that there is a solution of f(x) = x if and only if there is a solution of f(x) - x = 0.)
PLUS (1) Suppose that f is a nonconstant continuous function on [0,2] and satisfies f(0) = f(2) = 0. Show that f cannot be one-to-one on the open interval (0,2).
(2) Let f be monotone increasing and continuous on an open interval (a,b). Let A = lim_{x \to a+} f(x) and B = lim_{x \to b-} f(x). Prove that f maps (a,b) onto the open interval (A,B).
Homework due Thursday, Dec. 3:
(1) Fix a function f and a point x_0, and suppose that f is differentiable at x_0. For all real numbers b, consider the family of lines L_b(x) defined by L_b(x) = f(x_0) +b (x-x_0). Note that L_b(0) = f(x_0) for every b and that L_b(x) has slope b. Find all values of b so that lim_{x \to x_0} [(f(x)- L_b(x))/(x-x_0)] =0. There should be only one value, and for this value we call L_b(x) the tangent line to f(x) at the point (x_0,f(x_0) of the plane.
(2) Suppose that f is an odd function which is differentiable for all x. Show that f' is even.
(3) Suppose that f'(2)=3, f'(5)=4, and let h(x) be the composite function h(x) = f(x^2+1). Find h'(2).
BOOK p.129-130: 6, 12
FINAL EXAM DATE IS FRIDAY, DEC. 18, 8:00-11:00 AM, IN OUR USUAL ROOM FOR LECTURE (TIL 252).
Homework due Thursday, Dec. 10
Book p. 129-130: 3,11
Book p. 130-131: 18,20,23,25,29,30
Here are a few extra problems related to our most recent lecures. They are just for your practice and not to be handed-in or graded.
(1) Let p be a fixed real number satisfying 0 < p <= 1. Show that (1+x)^p <= 1+ x^p for all x>=0. You may assume that the derivative of x^p is p x^{p-1} if x>0.
(2) Show that the polynomial x^4+10x -12 has one and only one root in the interval (0,2).
(3) Use the Mean Value Theorem to prove that sin x >= x - (x^3)/6 for all x >= 0. (You may need to use the MVT more than once!)
(4) Use L'Hospital's rule to find lim_{x\to 0} (x^2 sin x)/(sin x - x - cos x + 1) and lim_{x \to 0+} (e^{-1/x})/x. (You may assume that (d/dx)(e^x) = e^x for all x. Also, remember that L'Hospital's rule works for both 0/0 and infinity/infinity.