MATH5301 Elementary Analysis. Homework 10.
10.1
Compute the derivatives of the following functions
(a) x
2
sin
1
x
(b) e
x + e
−x
2
(c) e
x − e
−x
2
(d) e
x + e
e
x
+ e
e
e
x
(e) x
x
x
x
10.2
(a) Prove the following
Theorem 1. If f : (−1, 1) → R is differentiable unbounded function, then f
0
is also unbounded on [−1, 1].
(b) Provide an example of bounded differentiable function on [−1, 1] with unbounded derivative.
(c) Prove the following
Theorem 2. If f : (−1, 1) → R is differentiable function, such that f
0
is bounded on [−1, 1], then f is
uniformly continuous.
10.3
Find f
(n)
(0) for the functions
(a) sin(ax) cos(bx)
(b) x
k
sin
1
x
(c) f(x) = (
e
− 1
x2 , x > 0
0, x 6 0
10.4
Construct an example of infinitely many times differentiable function f(x) such that f(x) = 0 for x 6 0,
f(x) = 1 for x > 1 and f(x) is strictly monotone on the interval (0, 1).
Using such function you could construct for example a monotone function g(x) such that lim x→+∞
g(x) = 0
but lim x→+∞
g
0
(x) 6= 0. (How?)
10.5
Find the limit
(a) limx→0
tan x − x
x
3
(b) limx→0
arctan(arcsin x) − arcsin(arctan x)
sin x − tan x
(c) lim x→+∞
x
ln x
(ln x)
x
10.6
Find the example of a function f(x) which is continuous at every point of the interval (0, 1), but is not differentiable at every point of (0, 1).
Read about the construction of the function, which is differentiable at every point of (0, 1) but whose
derivative is discontinuous at every point of (0, 1).
