Challenge problems are longer, more interesting homework problems. They will often be broken into parts, in a way that guides you through an argument, so the important part for you will be making a clear argument, using clear sentences and logic, rather than just computing some answer. You will be allowed one rewrite if your answers seem promising but not sufficiently clear or detailed, or if substantial points are not covered. The writeup should be more extensive than a normal problem, and should be readable, like an essay, with words filling in the connections between one piece of a computation and the next, with the logic clear. Challenge problems are to be done by you alone, although you may work with others in figuring out what the problem is asking, or in learning helpful techniques of a more general kind (like proof by induction in Problem 1 below).
1. p. 454, Problem 25, "Consider the series ..." Here is a case where clear exposition of your idea is the main thing. The best method to use for part (b) would be a proof by induction. Ask around (you can ask me) if you don't know how to do this. Due by Friday, Oct. 21.
2. p. 475, Project 1. Here is another case where clear exposition is the main thing. The text itself guides the argument, but figuring out what it is saying will require thought. Due by Friday, Oct. 21.
3. This problem begins from p. 443, Problem 56 (a)-(c) but then takes off in a slightly different direction. Due by Friday Oct. 21.
(a) find the first 12 terms of the Fibonacci sequence Fn defined by the Fibonnaci relationship
(b) Show that the ratio of successive F's appears to converge to a number satisfying r2=r+1.
(c) Let r satisfy r2=r+1. Show that the sequence sn=Arn, where A is any constant, satisfies the Fibonnaci relationship. Is the Fibonnaci sequence that you found in (a) given by this formula for some A?
(d) Notice that the quadratic equation satisfied by r above has two roots. Let them be r1 and r2, and show that Ar1n+Br2n satisfies the Fibonnaci relationship, for any choice of constants A and B.
(e) Use the observation in (d) to find a formula for the nth term of the Fibonnaci sequence, and prove that it works.
4. Consider the function f(x)=x/(ex-1)+x/2.
(a) f has a so-called "removeable singularity" at x=0, where it is (so far) undefined. What value should we assign to f(0) to make f continuous at x=0?
(b) With this taken care of, f actually has a Taylor series about x=0. Find the first 10 terms or so of this Taylor series (use MAPLE!).
(c) What pattern do you notice in the degrees of the terms in the Taylor polynomials?
(d) Prove the property that you noticed in (c). Due by Friday, Oct. 21.
5. Let two long circular cylinders, of diameter D, intersect in such a way that their symmetry axes meet perpendicularly. Let each of these axes be horizontal, and consider the "room" above the plane that contains these axes, common to both cylinders. (In architecture this room is called a "cross vault".) The floor of the cross vault is a square of side D, and the ceiling consists of four curvilinear triangles, meeting at the top, and intersecting in arcs that come down from the topmost point to the vertices of the square. Problem: Calculate the volume of the cross vault. Amazingly, the painter Piero della Francesca managed to do this in the 1400s, long before the invention of calculus! Due by Monday, November 28.
6. Find exact values for Riemann sums approximating the integral of the function f(x)=x2 on the interval [0,1]. Split up the interval into N equal segments, and find the upper sum (taking the maximum function value in each segment) and the lower sum (taking the minimum function value in each segment). You will need a mysterious looking result for the sum of the first N squares,
1+4+9+...+N2=N(N+1)(2N+1)/6
which you should prove by induction. What is the limit of your Riemann sums as N approaches infinity? Due by Monday, November 28.
7. This problem generalizes the factorial function, as in n!=n(n-1)(n-2)...(2)(1), to more general arguments than just the positive integers.
(a) Use integration by parts to show that for any positive integer n, the integral with respect to x from 0 to infinity of xne-x is n!
(b) Make a clear case that the integral exists for any n>-1, whether n is an integer or not. Thus we could define the factorial function by this integral.
(c) Use this new definition to compute 0!
(d) Use spline interpolation on the values of 0!, 1! and 2! to estimate a reasonable value for (1/2)! This means you should find a polynomial An2+Bn+C that agrees with the factorial function at n=0,1,2, and evaluate it at n=1/2.
(e) The exact value of (1/2)! is sqrt(pi)/2. Check that your estimate above was reasonably accurate, and use this fact to find (-1/2)! Make a sketch of the graph of the factorial function on the interval [-1/2,2] based on these values, assuming that factorial is a reasonably smooth function. Due by Monday, Nov. 28.
8. What fraction of the area of a square is closer to the center of the square than to any edge of the square? This one is harder than it looks! I am looking for an exact answer, but you can get a rough numerical estimate by a Monte Carlo method , and this might help you check your answer. Due by Monday, Nov. 28.
9. Use problems 8 and 9 on p. 348 as an outline to write a clear explanation why Simpson's rule is a good way to approximate definite integrals over a finite interval. Due by Monday, Nov. 28.
10. Numerical approximations to integrals typically get better -- i.e., their error goes down -- proportional to a power of N, the number of subintervals in the interval of integration. For the upper and lower sums, the error typically goes down like 1/N as N increases. For the midpoint and trapezoidal rules, the error typically goes down like 1/N^2. For Simpson's rule, the error typically goes down like 1/N^4. (a) Demonstrate this behavior numerically, using the integral of x^3 on [0,1] as a typical integral. (b) Demonstrate that this normal behavior is NOT seen in the integral of sqrt(x) on [0,1]! Apparently the slightly bad behavior of sqrt(x) at 0 (it is not differentiable there) is to blame.
You may find the following "diary" of a Matlab session helpful. It shows commands that are sufficient to explore this question about numerical integration. The function being integrated here was sin(x) from 0 to pi/2, but if you make obvious changes you can do another function on another interval. One thing that isn't illustrated there: if x is a list of x-values, as in the diary, then x.^3 is a list of the cubes, and sqrt(x) is the list of square roots. Note in particular the command .^3 ! That "dot" preceding "^" means the cubing operation is applied to EACH x. Due by Monday, Nov. 28.
11. The existence and uniqueness theorem for ordinary differential equations (ODE) says that the solution of a 1st order ODE with given initial value exists and is unique. It is discussed briefly on p. 528 of the text.
Let y=f(x) be the solution of dy/dx=y that satisfies f(0)=1. It is thus uniquely specified. Of course we know that this function is the usual exponential function y=exp(x), but suppose we didn't know that. In this problem we will show the main properties of the exponential function from the differential equation alone, together with the existence and uniqueness theorem.
(a) Let A be any constant. Write down a differential equation satisfied by g(x)=f(A)f(x), and also give the value of g(0). Do the same for the function h(x)=f(x+A). Conclude, with a clear argument, that g and h are the same functions.
(b) Let r be any constant. Write down a differential equation satisfied by g(x)=[f(x)]^r, and also the value of g(0). Do the same for the function h(x)=f(rx). Conclude, with a clear argument, that g and h are the same functions.
(c) How do we usually state the properties proved in (a) and (b) above? Due by the last day of classes, Dec. 12.
12. The Bessel function of order 0 is defined in problem 34 after section 9.5. Verify that it is the solution to the differential equation x^2 y'' + x y' + _x^2 y = 0, satisfying y(0)=1, y'(0)=0. Here y' means the first derivative of y(x) and y'' means the second derivative. Due by the last day of classes, Dec. 12.