Problems for the Final Exam
MATH 6102 - Spring 2007
1. Define a function fon the real line by f(x) = x+⌊x2⌋−⌊x⌋, where ⌊x⌋is the greatest integer
less than or equal to x. Where is fnot continuous? Explain your answer.
2. Find the radius of convergence for the power series
∞
X
n=1
xn
nn.
3. A function fsatisfies the two conditions
f′(x) = 1 + (f(x))10 and f(0) = 1.
Find the first four terms in the Taylor series expansion of fabout x= 0.
4. Find the Taylor series and the radius of convergence for
f(x) = Zx
0
dt
√1 + t2about x= 0.
5. If f(1) = 1 and f′(1) = 2, compute
lim
x→1
[f(x)]2−1
x2−1.
6. If a, b, c, d ∈Rfind
lim
x→0
sin ax sin bx
sin cx sin dx.
7. Suppose that fis differentiable with derivative f′(x) = (1 + x3)−1/2. Show that g=f−1
satisfies g′′(x) = 3
2[g(x)]2.DO NOT FIND f(x).
8. Find f−1for each of the following:
(a) f(x) = x+⌊x⌋.
(b) f(0.a1a2a3...) = 0.a2a1a3... for all numbers between 0 and 1.
9. Suppose that fand gare two differentiable functions which satisfy f g′−f′g= 0. Prove that
if aand bare adjacent zeros of f1, and g(a) and g(b) are not both 0, then g(x) = 0 for some
xbetween aand b.
Hint: Derive a contradiction from the assumption that g(x)6= 0 for all xb etween aand b.
1This means that f(a) = f(b) = 0 and f(x)6= 0 for all x∈(a, b).
