1. Homework 10
f
(3h)
−
f
(−2h)
(1) If f 0 (0) = −1, find lim
.
h→0
h
(2) Define

 1 − cos x if x ∈ R \ {0}
f (x) =
x
0,
if x = 0.
Find f 0 (0) if it exists.
(3) Let d, α > 0 and f be a real value function on (−d, d). Suppose there exists C > 0 such that
|f (x)| ≤ C|x|1+α ,
for all |x| < d.
0
Show that f is differentiable at 0 and f (0) = 0. Show that

x1+α sin 1 if x ∈ R \ {0}
f (x) =
x
0,
if x = 0.
is differentiable at all x ∈ R and find the derivative f 0 of f.
(4) Let g, h be differentiable functions on R and f is continuous on R.1 Define2
Z h(x)
S(x) =
f (t)dt, x ∈ R.
g(x)
Z
0
(a) Find a formulae for S (x). (Hint: consider F (x) =
x
f (t)dt. Then
0
S(x) = F (h(x)) − F (g(x)).
Use chain rule.)
Z x2
(b) Find f (4) if
f (t)dt = x cos πx, x > 0.
0
(c) Find all continuous functions f and a ∈ R such that
Z x
√
f (t)
6+
dt = 2 x.
2
t
a
(5) Let f (x) = x + x2 + · · · + xn , x ∈ R. Then f 0 (x) = 1 + 2x + 3x2 + · · · + nxn−1 . Use f 0 to
∞
∞
X
X
n
n
and
evaluate
.
determine the n-th partial sum of
n−1
n−1
2
2
n=1
n=1
(6) Let f (x) be a polynomial of degree n of real coefficients and g(x) be a polynomial of degree
≤ n of real coefficients. Suppose that all the roots of f (x) are real and
a1 < · · · < an
are roots of f (x). Assume that for x 6∈ {a1 , · · · , an }, we have
g(x)
A1
An
=
+ ··· +
.
f (x)
x − a1
x − an
g(ak )
, 1 ≤ k ≤ n.
f 0 (ak )
(b) Suppose that Ak Ak+1 > 0 for 1 ≤ k ≤ n − 1. Show that deg g = n − 1 and g has n − 1
distinct real roots.
(7) Let P (x) = a0 + a1 x + · · · + an xn with ai ∈ R.
(a) Calculate the polynomial F (x) from the equation
(a) Show that Ak =
F (x) − F 0 (x) = P (x).
1The domain of f, g, h does not have to be whole R. f, g, h can be only defined on an interval.
2
The domain of S does not have to be R.
1
2
(b) Calculate the polynomial F (x) from the equation
c0 F (x) + c1 F 0 (x) + c2 F 00 (x) = P (x),
where c0 , 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
n X
n (k) (n−k)
dn
(f g) =
f g
.
n
dx
k
k=0
(9) Let f (x) be a continuous functions on [a, b]. Define
Z x
F (x) =
f (t)dt.
a
Suppose that g : [c, d] → [a, b] is a C 1 -functions, increasing on [c, d] such that g(c) = a and
g(b) = d. Let h(u) = F (g(u)) for u ∈ [c, d]. By chain rule, h is also a C 1 -function and
h0 (x) = f (g(x))g 0 (x)
for a ≤ x ≤ b.
(a) Prove the change of variable formula for integrals:
Z b
Z d
f (x)dx =
f (g(u))g 0 (u)du.
a
c
(b) Use the above formula and the integral for cosine function to show that
Z
Z b
1
1 λb
cos udt = (sin λb − sin λa).
cos λxdx =
λ
λ
λa
a
(10) Suppose f, g are C 1 -functions on [a, b]. By fundamental theorem of calculus, we know for
any C 1 -function h on [a, b], one has
Z b
(1.1)
h(b) − h(a) =
h0 (x)dx.
a
1
Let h = f (x)g(x). Then h is a C -function on [a, b].
(a) Use (1.1) to prove the integration by parts formula:
Z b
Z b
0
f (x)g (x)dx = (f (b)g(b) − f (a)g(a)) −
f 0 (x)g(x)dx.
a
a
(Fact: h0 (x) = f (x)g 0 (x) + f 0 (x)g(x).)
Z
π
(b) Use the above formula to evaluate
x sin xdx. Hint: write x sin x = x(cos x)0 .
0
We usually denote
f (x)g(x)|ba = (f (b)g(b) − f (a)g(a)).