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(MATH203)[2010](f)final~cs_zxxab^_10029.pdf
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Math 203 Analysis I
Final Examination, Fall 2010
16:30 C 19:30, December 20, 2010
Instruction: This is an open book exam. You can use the textbook "Principles of Mathematical Analysis" by Walter Rudin, but you cannot use any other materials, including the solutions of the exercises.
1.
(20 Marks) If f 2 and f 3 are differentiable, is f differentiable? Justify your answer.

2.
(20 Marks) Suppose f is differentiable on .a,.... Suppose both


limf .x. and lim f ..x.
x... x...
exist. Prove that lim f ..x.. 0.
x...
3. (20 Marks) Suppose f is a real continuous function on .such that
f .r .1
n.. f .r.
for any rational number r and any positive integer n . Prove that f is a constant.
4. (20 Marks) Suppose f is a real function on ., with f .. . 0 . Prove that
for any x . 0 , we have
2 x . 3 .

f ..t dt . xf . x.
.
x
. 2 .
5. (20 Marks) Suppose g is a real function in a neighborhood J of 0 such
that g..0 . 0 and g..0.. 0 . Prove that if g has bounded second derivative in J , then the function
x
2
.... g.xtfx .dt
0
has a local minimum at 0.