Advanced Calculus Assignment 3

1. Advanced Calculus, Assignment 3
1. Let
f(x) =
x2
if x ∈ Q
0 otherwise.
Prove that f is differentiable at 0, but discontinuous at all real numbers
except 0.
2. Let I1, I2 be intervals. Let f : I1 → I2 be a bijective function and
g : I2 → I1 be the inverse.Suppose that both f is differentiable at c ∈ I1
and f (c) = 0 and g is differentiable at f(c). Use the Chain Rule to find
a formula for g (f(c)) (in terms of f (c)).
3. (a) Prove the Power Rule.
(b) Let f : I → R be differentiable. Given n ∈ Z, define fn
be the
function defined by fn
(x) = (f(x))n
. If n < 0, assume also that
f(x) = 0. Prove that
(fn
) (x) = n(f(x))n−1
f (x).
4. Suppose f : I → R is a bounded function and g : I → R is a function
differentiable at c ∈ I. Suppose that g(c) = g (c) = 0. Show that h(x) =
f(x)g(x) is differentiable at c.
Hint: Note that you cannot apply the product rule.
5. Suppose f : R → R is a differentiable function such that f is a bounded
function. Prove that f is a Lipschitz continuous function.
6. Suppose f : (a, b) → R and g : (a, b) → R are differentiable functions
such that f (x) = g (x) for all x ∈ (a, b), then show that there exists a
constant C such that f(x) = g(x) + C.
7. Prove the following statements:
(a) sin 1
x is not continuous at 0.
(b) x sin 1
x is continuous at 0, but not differentiable at 0.
(c) x2
sin 1
x is differentiable at 0, but its derivative is not continuous at
0.
(d) x3
sin 1
x has continuous derivative at 0.
8. By using the definition of derivatives find the derivative of given f at given
c:
(a) f(x) = 4
3−x and c = 2.
(b) f(x) = x3
− 4x2
and c = 0.
9. Calculate the derivatives of the following functions:
1

2. (a) f(x) = x2
+3x−2 tan x√
x
.
(b) g(x) = sin e(3x+5)2
+cos(2x)
.
(c) h(x) = x sin(x)
ecos(x) .
10. Applications of Mean Value Theorem:
(a) Show that | cos x − cos y| ≤ |x − y| for all x, y ∈ R.
(b) Suppose that f is differentiable on R, and f (x) is a constant. Then
show that f is of the form f(x) = mx + b for some m, b ∈ R.
11. Suppose that f : R → R is a function such that
|f(x) − f(y)| ≤ |x − y|2
for all x, y ∈ R.
(a) Show that f is differentiable at all points.
(b) Compute the derivative f .
(c) Show that f(x) = C for some constant C.
12. Let f : (a, b) → R be an unbounded differentiable function. Show that
f : (a, b) → R is unbounded.
13. (Applications of L’Hˆospital’s Rule) Evaluate the following limits:
(a)
lim
x→0+
xx
(b)
lim
x→0
1
x2
−
cot x
x
(c)
lim
x→0
sin x
x
1/x2
14. Suppose p is a polynomial of degree d. Given any x0 ∈ R, show that the
(d + 1)th Taylor polynomial for p at x0 is equal to p.
15. If f : [a, b] → R has n + 1 continuous derivatives and x0 ∈ [a, b] prove that
lim
x→x0
Rx0
n (x)
xn
= 0.
2