Question

If f is a continuous, positive function defined on the interval (0, 1] such that limx→0+ = ∞ we have seen how to make sense of the area of the infinite region bounded by the graph of f, the x-axis and the vertical lines x = 0 and x = 1 with the definition of the improper integral.

Consider the function f(x) = x sin(1/x) defined on (0, 1] and note that f is not defined at 0.

• Would it make sense to argue similarly and define the length of the infinite curve y = f(x) on the interval (0, 1]?

• If yes, what would be a reasonable suggestion for defining such a notion?

• For bonus marks find a differentiable (or, at least, continuous) function f such that limx→0+ f(x) = 0 and the graph of f on [0, 1] has infinite length.

Answer #1

*#Could you please leave a THUMBS UP for my
work...*

We know that any continuous function f : [a, b] → R is uniformly
continuous on the finite closed interval [a, b]. (i) What is the
definition of f being uniformly continuous on its domain? (This
definition is meaningful for functions f : J → R defined on any
interval J ⊂ R.) (ii) Given a differentiable function f : R → R,
prove that if the derivative f ′ is a bounded function on R, then f
is uniformly...

Consider the function f : R → R defined by f(x) = ( 5 + sin x if
x < 0, x + cos x + 4 if x ≥ 0. Show that the function f is
differentiable for all x ∈ R. Compute the derivative f' . Show that
f ' is continuous at x = 0. Show that f ' is not differentiable at
x = 0. (In this question you may assume that all polynomial and
trigonometric...

Given that a function F is differentiable.
a
f(a)
f1(a)
0
0
2
1
2
4
2
0
4
Find 'a' such that limx-->a(f(x)/2(x−a)) = 2.
Provide with hypothesis and any results used.

Let f: [0, 1] --> R be defined by f(x) := x. Show that f is
in Riemann integration interval [0, 1] and compute the integral
from 0 to 1 of the function f using both the definition of the
integral and Riemann (Darboux) sums.

Sketch the graph of a function f(x) that satisfies all of the
conditions listed below. Be sure to clearly label the axes.
f(x) is continuous and differentiable on its entire domain,
which is (−5,∞)
limx→-5^+ f(x)=∞
limx→∞f(x)=0limx→∞f(x)=0
f(−2)=−4,f′(−2)=0f(−2)=−4,f′(−2)=0
f′′(x)>0f″(x)>0 for −5<x<1−5<x<1
f′′(x)<0f″(x)<0 for x>1x>1

4. Please work each part. (a) Discuss the existence or
non-existence of limx→0 2 sin 1 x − x 2 cos 1 x using the limit
theorems. (b) Let I be an open interval with a ∈ I and suppose that
f is a function defined on I\{a}. Suppose that limx→a (f(x) + D(x))
exists, where D(x) = χQ(x) is the Dirichlet function. Show that
limx→a f(x) does not exist.

Does the function satisfy the hypotheses of the Mean Value
Theorem on the given interval?
f(x) = x3 + x − 9, [0, 2]
Yes, f is continuous on [0, 2] and differentiable on
(0, 2) since polynomials are continuous and differentiable on .No,
f is not continuous on [0, 2]. No,
f is continuous on [0, 2] but not differentiable on (0,
2).Yes, it does not matter if f is continuous or
differentiable; every function satisfies the Mean Value
Theorem.There is...

Consider the piecewise defined function
f(x) = xa− xb if 0<x<1. and
f(x) = lnxc if x≥1.
where a, b, c are positive numbers chosen in such a way that
f(x) is differentiable for all 0<x<∞. What can be said about
a, b, and c?

1. Does the function satisfy the hypotheses of the Mean Value
Theorem on the given interval?
f(x) = x3 + x − 5, [0, 2]
a) No, f is continuous on [0, 2] but not differentiable
on (0, 2).
b) Yes, it does not matter if f is continuous or
differentiable; every function satisfies the Mean Value
Theorem.
c) There is not enough information to verify if this function
satisfies the Mean Value Theorem.
d) Yes, f is continuous on [0,...

Consider the function f defined on R by f(x) = ?0 if x ≤ 0, f(x)
= e^(−1/x^2) if x > 0.
Prove that f is indefinitely differentiable on R, and that
f(n)(0) = 0 for all n ≥ 1. Conclude that f does not have a
converging power series expansion En=0 to ∞[an*x^n] for x near the
origin. [Note: This problem illustrates an enormous difference
between the notions of real-differentiability and
complex-differentiability.]

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