Due: Friday, April 13, 2007
Math 360 - Advanced Calculus / Problem Set 10 (two pages)
1) Prove the following assertions made in the class:
a) If f:D→
R
is differentiable at x0∈D, then fis continuous at x0.
b) If f , g are differentiable at x0, then so is fg and f/g, provided g(x)6= 0 on Din the last case.
2) Complete the proof of the Inverse Function Theorem along the following lines: Let f:I= (a, b)→
R
be
differentiable on (a, b), and f0(x)6= 0 for all x∈(a, b). Then one has:
a) f0is either strictly positive, or strictly negative on I.
b) fis either strictly increasing, or strictly decreasing on I.
c) J:= f(I)⊆
R
is an open interval in
R
.
d) f−1:J→Iis continuous.
e) f−1is differentiable and (f−1)0(y) = 1/f0(x), provided y=f(x).
[Hint: To d): Let [c, d]⊂Ibe a closed interval. Then [c, d] is compact, hence f: [c, d]→f([c, d]) is a
continuous bijection of compact topological spaces, hence a homeomorphism (WHY?). Deduce from this that
fis a homeomorphism, hence f−1is continuous. To e): Take yn→y0in J, and xn→x0the preimages in I.
Then yn=f(xn), y0=f(x0), and proceed as in the proof of the “chain rule”.]
3) Complete the proof of the fact expaand logaare differentiable along the following lines: First recall that
e:= lim(1 + 1
n)n. Show the following:
a) lim
x→∞(1 + 1
x)x= lim
x→0(1 + x)1
x= lim
x→1x1
x−1.
b) log0
a(x0) := lim
x→x0
loga(x)−loga(x0)
x−x0= lim
x→x0
loga(x/x0)
x0(x/x0−1) =1
x0lim
u→1
1
u−1·loga(u) = 1
x0loga(e) = 1
ln(a)
1
x0
(WHY?).
c) Deduce from this exp0
ausing the Inverse Function Theorem.
4) Prove that for all α∈
R
, the map f: (0,∞)→
R
,x7→ xαis differentiable and one has f0(x) = α xα−1.
[Hint: xα= explog(xα)= expαlog(x), and use the chain rule, etc.]
5) Prove that the trigonometric functions are differentiable, along the following lines: First recall that
lim
u→0
sin u
u= 1 (WHY?), and then show:
a) sin0(x0) := lim
x→x0
sin(x)−sin(x0)
x−x0= lim
u→0
sin(x0+u)−sin(x0)
u= cos(x0) (WHY?).
b) Deduce from this that cos0(x) = −sin(x) using cos(x) = sin( π
2−x) and the chain rule.
c) Deduce that tan and cot are differentiable using the definitions and a), b), above.
Taylor polynomials and Maclaurin series. Let I⊂
R
be an open interval, and let f:I→
R
be a function
which is at least (n+ 1)-times differentiable on I. Let a < b in Ibe fixed. The polynomial
Pf,n(X) := f(a) + f0(a)
1! (X−a) + . . . +f(n)(a)
n!(a)(X−a)n=
n
X
k=0
f(k)(a)
k!(a)(X−a)k
is called the Taylor polynomial of degree nattached to fabout x=a. If fis k-times differentiable for all k,
and a= 0, then the (formal) power series
Σf(X) := f(0) + f0(0)
1! X+. . . +f(n)(0)
n!(a)Xn+. . . =
∞
X
k=0
f(k)(0)
k!(0)Xk
