MAT125A Final and Solutions March 17, 2010
1. Show that if f : [0, ) R is continuous and
lim f (x)
is finite, then f is bounded on [0, ). Solution: Let L = limx f (x). Then there exists > 0 such that x > implies f (x) < L + 1. Since f is continuous and
Homework one additional problem 1. Let cfw_xk and cfw_yk be convergent sequences and set L1 = lim xk
L2 = lim yk .
Show that if xk yk then L1 L2 . Give an example to show that xk < yk does not imply L1 < L2 .
Below are a few practice problems for the first midterm. They are intended to be slightly longer and/or harder than the exam problems (the theory being if you can master these problems then the exam should be a piece of cake). 1. Show that f (x) = is not
Practice Midterm II
1. Show that the function f (x) = x sin(1/x) x = 0 0 x=0
is continuous but not dierentiable at 0. 2. Find all the values of x for which the series xk k k=1 converges. 3. Suppose that f : (a, b) R is dierentiable on (a, b) and f (x) is
MAT125A Midterm One and Solutions Feb. 1, 2010
1. Show that if f : R R is bounded, then the function g ( x) = x2 f ( x) x = 0 0 x=0
is dierentiable at 0. What is the value of g (0)? There exists C > 0 such that |f (x)| C for all x since f is bounded. We h
TBB exercises: 2.8.2, 2.9.2, 2.11.3, 2.11.6, 2.12.3, 5.1.16
TBB exercises: 5.4.8, 5.5.1, 5.5.2, 5.6.3, 5.6.9
Show that the image of a compact set under a continuous function is compact.
TBB exercises: 7.2.2,7.2.3,7.
MAT125A Midterm II with solutions
1. Find all x for which the series
k2 k x 2k
converges. Justify your answer. Solution: We apply the root test to the terms of the series. We have k2 k x 2k as k since k 2/k = eln(2)/k 1 as k . So the series diverges f