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ASSIGNMENT #1. DUE THURSDAY, SEPTEMBER 7
(i) Read Chapter
1 in Rosenlicht's book and get started on reading Chapter 2.
(ii) Do the following problems from Chapter 1 (pages 12-13): 1(c), 10(c), and all parts of 4, 5, 7, 8, and 9.
(iii) Use mathematical induction to prove that, for each positive integer n, the sum of the third powers (cubes) of the first n positive integers is [n(n+1)/2]^2. Thus, for n = 3, 1 +8+27=36=6^2 with 6 = 3(4)/2. Done efficiently, the induction proof takes only a few lines.
(iv) Use mathematical induction and the product rule for derivatives to prove that the derivative of the monomial x^n is nx^(n-1). Again, only a few lines are needed. Don't use any properties of derivatives other than the product rule.
Example of an inductive proof: For each positive
integer n, we assert that
(Hn) 1 + 2 + ...+n = n(n+1)/2 [arithmetic progression formula]
The assertion (H1) is true since 1=1(2)/2. If (Hn) is true, then the sum of the first n+1 integers being the sum of the first n plus n+1, we infer that the sum of the first n+1 integers is equal to
n(n+1)/2 + (n+1) =(n+1)(n/2 +1) = (n+1)(n+2)/2. Since n+2 = (n+1)+1, this means (H(n+1)) is true.
By the principle of mathematical induction, (Hn) is true for every positive integer n. [Using a few shorthand symbols to eliminate the words I used above, the same proof could be written down in two lines.]
ASSIGNMENT #2. DUE THURSDAY, SEPTEMBER 14
Do the following problems from Chapter 2 (pages 29-31): 3, 5b, 5c , 7, 10c , 12, 13, 14, 15
5c. Look separately at the cases x > 0 and x < 0.
10c. What's going on here is that a sequence is being defined inductively by saying that the square root of 2 is the initial member and that the (n+1)st member is the square root of (2 + nth member). Start off by showing inductively that the sequence is bounded, then note that it's an increasing sequence, and deduce an equation for the l.u.b.
12. In class, we mentioned that one way to construct the real number field is by the method of Dedekind cuts. This problem is showing that, no matter how one constructs the real numbers, any Dedekind cut defines a real number.
15. The digits of the fractional part of a decimal number are values of a function defined on the indices 1, 2, 3, 4,... In general, a function is called periodic with period N if f(k+N) =f(k) for all sufficiently large k.
ASSIGNMENT #3. DUE THURSDAY, SEPTEMBER 21
Do the following 4 problems from Chapter 3 (pages 61-63): #1(c), 11, 16, 17. Also do the following 3 additional problems:
(5) Let f(x)
= 1/(x+2) for x a real number not equal to -2. We can use f to inductively
define sequences by choosing a starting value x_0 and defining x_(n+1)
= f(x_n). Here you should read x_n
as xsubn, i.e. the nth term of the sequence. Check that f has two equilibrium values, one positive and one negative. If x_0 > -2, show that the sequence converges to the positive equilibrium value.
As mentioned in #13, the key thing is to check that the subsequences with n even and n odd are each monotonic, one monotone increasing and the other monotone decreasing. For this, you'll want to study f(f(x))= (x+2)/(2x+5).
For extra credit, discuss what happens when x_0 < -2. WARNING: this gets tricky so, if your time is limited, don't consider it.
that the arc length metric on the unit sphere in R^3 satisfies the triangle
Why is it O.K. to limit attention to points p, q, r, one of which is the north pole of the sphere? You may want to use spherical coordinates since the arc length along a line of longitude from the north pole to some other point is measured by the change in the angle denoted by phi in most calculus books. Don't mix up arc length with chord length. But comparing chord lengths to are lengths is one of the MANY ways to approach this problem.
(7) In class,
we defined the lim inf and lim sup of any bounded sequence
of real numbers. Use these definitions to do the following:
(a) Show that we can choose a subsequence of any bounded sequence converging to the lim sup and can also choose a subsequence converging to the lim inf. Why must the limiting value of any converging subsequence lie in the closed interval whose endpoints are the lim sup and lim inf?
(b) Use (a) to prove that the lim sup of a bounded sequence is the unique number U for which at most finitely members of the sequence are >U* for each U* >U but infinitely members of the sequence are >U** for each U**<U. Think through for yourself the analagous statement for the lim inf of the sequence but don't bother to write down the proof since it comes down to the usual business of changing all of the signs of the original sequence with inequalities reversed by sign changes.
NOTE: Since there will be an in-class Exam on Tuesday, Sept. 26, there won't be any homework due during the week of the exam. The above problems, some of them decidedly not easy, will be good practice for the exam. But don't get discouraged if you have trouble with these problems. Most of the exam questions will be much easier with only a few at the end approaching the difficulty of the tougher problems above.
ASSIGNMENT #4. DUE THURSDAY, OCT. 5
Do the following problems from Chapter 3, pp. 63-64: #23 (first read
over #22 and notice that we've been using metrics coming from norms in
all of our examples except for the arc length metric example), #26-30,
33, 35. On the cluster point business, notice that there are
two kinds of limit points for any set S: cluster points where
any open ball about the point contains infinitely many distinct points
in S and isolated points p in S where there is some open ball
about p containing no points in S other than p.
You should start reading Chapter 4. We'll start on it Tuesday, coming back to the somewhat nasty business of connectivity for arbitrary subsets of metric spaces after we've discussed continuous functions and the much easier notion of path connectivity.
ASSIGNMENT #5. DUE THURSDAY, OCT. 12
Do the following problems from Chapter 4, pp. 90-95: 1(b), 1(d), 4, 9(b),
10(b), 14,15, 17,
Comments and hints: 1(b) is easy--use continuity of F(x) =x to get continuity of f. But 1(d) is not easy--you need to separately investigate continuity at rational and irrational points and use the properties of rational numbers we discussed earlier. The key point in #4 is the assumption that f is onto--without this, the conclusions wouldn't follow. #9(b) and 10(b) are "old chestnuts" which likely appeared in your Calculus III textbook--just a line or two ought to suffice. #14 and #15 are "intuitively
clear" but you'll need to think through how to concisely justify the indicated conclusions. For #17, you use the same trick as in 9(b), namely that when a and b are positive, a - b is the product of the sum and difference of the postive square roots of a and b. #21 isn't particularly hard but is very useful in applications. #22 and #23 will very likely cause you trouble, especially if your linear algebra background is weak. Again, these problems are very useful in applications. The key to number 23 is to take m to be the minimum and M the maximum of the positive continuous function || . ||_1 (subscript 1) on the set of elements x in V for which || x||_2 =1. To establish the existence of m and M, you may want to first consider the case V=R^n with one of the two given norms being one of our standard norms. Note that in the special case when V is R^n, #23 proves that any pair of metrics on R^n arising from norms are automatically equivalent. This expands enormously our previous family of equivalent R^n-metrics d_p, 1<= p <= infinity, all of which arise from norms. No surprise, nothing like #23 works for infinite dimensional vector spaces, the problem being that the unit sphere is never compact in a normed infinite dimensional vector space. This is the "basic" reason why analogs of the d_p metrics aren't equivalent on infinite dimensional spaces.
ASSIGNMENT #6. DUE THURSDAY, OCT. 19
Do the following problems from Chapter 4, pp. 90-95: 30, 31, 32, 34, 37, 39, 41, 45.
Comments and hints:
#30. When Rosenlicht says "closed interval in E^2", he means "closed rectangle in R^2", i.e. the Cartesian product S = I x J of two closed intervals I=[a,b] and J=[c,d] in R. If f is a continuous R-valued function on S, then, for each y in J, g(x) = f(x,y) is continuous from I to R. Use the intermediate value theorem for each such g to show f can't be one-to-one.
#31. As this problem shows, contrary to intuition,
one can construct continuous functions mapping the interval [0,1] onto
the unit cube in R^3. The same idea leads to continuous maps
from [0,1] onto the unit cube in R^n for any n and one can go on to construct
continuous maps from R onto R^n. These space-filling curves are used
as counter-examples for various assertions about continuous curves in
#32. Note that we are dealing with a sequence of
functions defined inductively by initial function =
square root of x, (n+1)st function at x = square root of (x+(nth function at x)). This is the same kind of thing we encountered earlier with dynamical systems problems so the same techniques apply. Thus, if the sequence is going to converge at some x, the limit F(x) must be >= 0 and satisfy the equation
F(x)^2 = x + F(x). The point of the problem is to show that the sequence does converge to F(x). Is the convergence uniform or only pointwise?
#34. These are a little tricky. In each case, it's not hard to calculate pointwise limits but determining whether the converge is or isn't uniform takes a little work.
#37. It's easy to see that the sum functions converge uniformly but not so clear what happens for the product functions. First consider the case where the limit functions for each of the two sequences are bounded and use the triangle inequality to see that the product sequence converges uniformly to the product of the limits. Then take E = R (real numbers) and look at examples of situations where one or both of the limit functions isn't bounded.
#39. There are many ways to go about constructing
a sequence of functions with the stated properties.
You may want to try to find a pointwise convergent sequence continuous functions where the limit function is 1 at the points 1, 1/2, 1/3, 1/4,..., 1/n,... and 0 elsewhere.
#41. This one is admittedly not easy but it isn't as bad as you may initially think. It's important that the functions are defined on a compact metric space. Try to come up with an argument first showing that any point p lies in an open set where the sequence converges uniformly, then use compactness to obtain E as the union of finitely many such open sets and deduce uniform convergence on all of E.
#45. In contrast to the last three problems, this one is very easy. Just unravel the definitions. So, if you're having trouble with some of the earlier problems, you may want to do this one first.
ASSIGNMENT #7. Do the following problems from Chapter 5, pp. 108-110: # 4, 6, 7, 12. Also do the following problems from Chapter 9, pp. 212-213: #1,3,4,12.
Comments: The four Chapter 5 problems plus #3 and
#4 from Chapter 9 rest on the Mean Value Theorem (click on the course homepage
link called Proofs of Basic Theorems ... to see the statement and
the easy proof). For #12, you'll need to use the Mean Value Theorem
several times for both f and f '. Unraveling the somewhat confusing
wording in the problem, for any a and b in the interval with a<b and
any points c and x in (a,b), there are 3 (x,y) points be considered:
the point (x,y_chord(x)) on the chord line through (a,f(a)) and (b,f(b));
the point (x,f(x) on the graph of f: and the point (x,y_tan(x))
on the line tangent to the graph of f at (c,f(c)). Convexity of f means that
y_tan(x) <= f(x) <= y_chord(x) for all a,b,c, and x. All you'll need is the point-slope formula for straight lines and the Mean Value Theorem.
For #1 in Chapter 9, you'll want to first show that the indicated function is differentiable at points (a,b) where both a and b are non-zero, then go on to investigate whether or not the function is differentiable at (a,b) when either a or b or both are zero. Note that if f is differentiable at (a,b), then at (a,b) the differential df must be the linear map taking (x-a,y-b) to (x-a)A +(y-b)B where A and B are the partial derivatives of f at (a,b) with respect to x and y. So, to investigate differentiability at (a,b), first see whether or not A and B exist, then go on to see whether the error function (defined as in the notes on the proof of the chain rule) goes to zero as (x-a,y-b) approaches (0,0). We'll prove next week that if all partial derivatives of a function exist and are continuous on some open set, then the function is differentiable at each point of the open set. You can just quote this result as a shortcut to proving that the error function goes to zero provided you can show that the partial derivatives exist and are continuous in an open ball around your chosen point.
For #12 in Chapter 9, first check that the two partial
derivatives of f at (0,0) exist and both are 0.
Then compute the (obviously continuous) partial derivatives of f at points (x,y) other than (0,0) and check that these go to 0 as x and y go to 0. This is enough to show that f is continuously differentiable
on R^2. You need to use the definition of derivatives (not the Calc I quotient rule recipe) to show that the two mixed partials of f at (0,0) exist but aren't equal.
SOLUTION TO PROBLEM #12, CHAPTER 5 FROM ASSIGNMENT #7. Fix two points (a,f(a)
and (b,f(b) on the graph of f and consider only values of x between a and b. By the Mean Value Theorem, the slope m=(f(b) - f(a))/(b-a) of the chord line is also the slope f '(c) of the tangent line to the graph of f at (c,f(c)) for some c between a and b. If we take any number d between a and b,
the tangent line at (d,f(d)) has the equation y_tan(x) = f(d) +f '(d)(x-d) while the chord line has the equation y_chord(x) = f(a) + m(x-a) = f(b) + m(x-b). We want to show that
y_tan(x) <= f(x) <= y_chord(x) holds for each x and each d if and only in f '' is everywhere non-negative. By the Mean Value Theorem, f '' is everwhere non-negative if and only in f ' is monotonically increasing, i.e., non-decreasing. We can write y_chord(x) - f(x) either in the form
f '(c)(x-a) - (f (x) - f(a)) =((f ' (c) - f '(t))(x-a) with t between a and x or in the form
f '(c)(x-b) - (f(x) - f(b)) = (f '(c) - f '(s))(x-b) with s between x and b. When f ' is increasing, by separately looking at the cases a<x<=c (so t < c) and c<= x < b
(so c<s), we deduce that y_chord - f(x) >= 0. Similarly
f(x) - y_tan(x) = f(x)-f(d) - f '(d)(x-d)=(f '(u)-f '(d))(x-d) with u between x and d is >= 0 when f ' is increasing. For the converse, it's easiest to argue by contradiction. Thus, if f ' '(d) <0 for some d, then
f ' is strictly decreasing in an interval about d and by taking x to be in this interval, we obtain
f(x) < y_tan(x).
ASSIGNMENT #8, DUE THURSDAY, NOV. 2.
Do the following problems from Chapter 9, pp. 212-214: # 2, 5, 9, 10, 11, 16, 20
Comments: In #2, the only uncertainty is differentiability
of f at (0,0) since elsewhere the partial derivatives of f obviously exist
and are continuous. In #5, you can (and should!) use the Fundamental
Theorem of Calculus to write f(x) - f(y) as the integral from 0 to 1 of
g '(t) where g(t) =
f(x + t(y-x)). For #9 and #11, you can use without proof the basic result on differentiating under the integral sign described in #6. For #16, the definition of a positive (respectively, negative) definite symmetric n x n matrix is a symmetric matrix A for which x' A x is >0 (resp, <0) for each n by 1
column vector x with x' denoting the transpose of x (so x' is a 1 by n row vector with ith entry of
the row x ' being the same as the ith entry of the column x). Positive semi-definite replaces > by
>= and similarly for negative semi-definite. The point here is that when all of the partial derivatives of f at a point a are zero and all of the 2nd order partial derivatives of f exist and are continuous in an open set about a, then the 2nd order Taylor expansion of f at a becomes
T(x) = f(a) + (x-a)' H (x-a) where H is the n by n symmetric matrix consisting of the 2nd order partial derivatives of a. Also f(x) - T(x) =||x-a||^2 E(x) where E(x) goes to 0 as x approaches a. Hint: T(x) has maximum and minimum values on the compact set of points where ||x-a|| =1.
For #20, all you need use is first order Taylor expansions of f, i.e., the definition of differentiability, and the Cauchy-Schwartz inequality.
ASSIGNMENT #9, DUE THURSDAY, NOV. 30
Do the following problems from Chapter 6, pp. 133-134: #8, 9, 10, 13, 20, 22. In the context of #8, show that any monotone function has at most countably many points where it isn't continuous.
Also do #7 and #14 from Chapter 8, pp. 191-2.
Comments: As mentioned in class, a hard theorem to be proved next semester is that a bounded function is Riemann integrable on closed, bounded intervals if and only if its set of discontinuous points has measure zero. This characterization makes integrability of monotonic functions (#8) automatic and also makes #9 and #10 trivial but please don't use it. Doing #8-10 the hard way may help cure you of any lingering fondness for Riemann integrals of functions not known to be continuous.