MIDTERM FOR ADVANCED CALCULUS
Time: 13:10–15:00, Monday, Apr 22, 2002 Instructor: Shu-Yen Pan
No credit will be given for an answer without reasoning.
1. Brief explanation is required for this problem.
(1) [5%] Give an example of a bounded function f such that |f | is (Riemann) integrable on a bounded set A but f is not integrable on A.
(2) [5%] Give an example of a subset of Rnwhich has measure zero but whose boundary has positive measure.
2. [10%] Let A be an open set in Rn and f, g are differentiable function from A to Rm. Prove that f + g is also differentiable and
D(f + g) = Df + Dg.
3. [10%] A function f : Rn → R is called homogeneous of degree m if f (tx) = tmf (x) for all x ∈ Rn and t ∈ R. If f is also differentiable, show that for x = (x1, . . . , xn) ∈ Rn,
Xn
i=1
xi∂f
∂xi = mf (x).
(Hint: the chain rule)
4. [10%] Compute the second-order Taylor formula for f (x, y) = excos y around (0, 0).
5. [10%] Let f (x) = x + 2x2sinx1 for x 6= 0 and f (0) = 0. Show that f0(0) 6= 0 but that f is not locally invertible near 0. Why does this not contradict the inverse function theorem?
6. [10%] Compute the index of 2x2+ 6xy − y2− y4 at (0, 0).
7. [10%] Find the extrema of f (x, y, z) = x − y subject to the condition x2− y2= 2.
8. [10%] Evaluate limn→∞
R1
0 1−e√−nx x dx.
9. [10%] Suppose that A is a bounded set in R and f is a bounded function on A. By the Lebesgue’s theorem we know that if f is (Riemann) integrable on A, then the discontinuities of f have measure. Now we assume B is a subset of R (i.e., not necessarily bounded) and the function g is improper integrable on B. Show that the discontinuities of g also have measure zero.
10.
(1) [8%] Show thatR1
0 ln x dx converges.
(2) [7%] Show thatR∞
1 1
ln xdx diverges.
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