1. Homework 10 (1) Iff0(0) =−1,find lim
h→0
f(3h)−f(−2h)
h .
(2) Define
f(x) =
1−cosx
x ifx∈R\ {0}
0, ifx= 0.
Findf0(0) if it exists.
(3) Letd, α >0 andf be a real value function on (−d, d).Suppose there existsC >0 such that
|f(x)| ≤C|x|1+α, for all|x|< d.
Show thatf is differentiable at 0 andf0(0) = 0.Show that f(x) =
x1+αsin1
x ifx∈R\ {0}
0, ifx= 0.
is differentiable at allx∈Rand find the derivative f0 off.
(4) Let g, hbe differentiable functions onRandf is continuous onR.1Define2 S(x) =
Z h(x) g(x)
f(t)dt, x∈R.
(a) Find a formulae forS0(x).(Hint: considerF(x) = Z x
0
f(t)dt.Then S(x) =F(h(x))−F(g(x)).
Use chain rule.) (b) Findf(4) if
Z x2 0
f(t)dt=xcosπx, x >0.
(c) Find all continuous functionsf anda∈Rsuch that 6 +
Z x a
f(t)
t2 dt= 2√ x.
(5) Let f(x) =x+x2+· · ·+xn, x ∈R. Thenf0(x) = 1 + 2x+ 3x2+· · ·+nxn−1. Usef0 to determine then-th partial sum of
∞
X
n=1
n
2n−1 and evaluate
∞
X
n=1
n 2n−1.
(6) Letf(x) be a polynomial of degreenof real coefficients andg(x) be a polynomial of degree
≤nof real coefficients. Suppose that all the roots off(x) are real and a1<· · ·< an
are roots off(x).Assume that forx6∈ {a1,· · · , an},we have g(x)
f(x) = A1
x−a1 +· · ·+ An
x−an. (a) Show thatAk= g(ak)
f0(ak), 1≤k≤n.
(b) Suppose thatAkAk+1>0 for 1≤k≤n−1.Show that degg=n−1 andg hasn−1 distinct real roots.
(7) Let P(x) =a0+a1x+· · ·+anxn withai∈R. (a) Calculate the polynomialF(x) from the equation
F(x)−F0(x) =P(x).
1The domain off, g, hdoes not have to be wholeR. f, g, hcan be only defined on an interval.
2The domain ofSdoes not have to beR.
1
2
(b) Calculate the polynomialF(x) from the equation c0F(x) +c1F0(x) +c2F00(x) =P(x), wherec0, c1, c2∈R.
(8) Let f, g be differentiable functions on (a, b). Denote f(k) be the k-th derivative of f. Set f(0)=f.Show that by induction
dn
dxn(f g) =
n
X
k=0
n k
f(k)g(n−k).
(9) Let f(x) be a continuous functions on [a, b].Define F(x) =
Z x a
f(t)dt.
Suppose thatg : [c, d]→[a, b] is aC1-functions, increasing on [c, d] such thatg(c) =aand g(b) =d.Leth(u) =F(g(u)) foru∈[c, d].By chain rule,his also aC1-function and
h0(x) =f(g(x))g0(x) fora≤x≤b.
(a) Prove the change of variable formula for integrals:
Z b a
f(x)dx= Z d
c
f(g(u))g0(u)du.
(b) Use the above formula and the integral for cosine function to show that Z b
a
cosλxdx= 1 λ
Z λb λa
cosudt= 1
λ(sinλb−sinλa).
(10) Suppose f, g are C1-functions on [a, b]. By fundamental theorem of calculus, we know for anyC1-functionhon [a, b], one has
(1.1) h(b)−h(a) =
Z b a
h0(x)dx.
Leth=f(x)g(x).Thenhis aC1-function on [a, b].
(a) Use (1.1) to prove the integration by parts formula:
Z b a
f(x)g0(x)dx= (f(b)g(b)−f(a)g(a))− Z b
a
f0(x)g(x)dx.
(Fact: h0(x) =f(x)g0(x) +f0(x)g(x).) (b) Use the above formula to evaluate
Z π 0
xsinxdx.Hint: writexsinx=x(cosx)0. We usually denote
f(x)g(x)|ba= (f(b)g(b)−f(a)g(a)).
