Question

If f is a continuous, positive function defined on the interval (0, 1] such that limx→0+...

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.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

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

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
We know that any continuous function f : [a, b] → R is uniformly continuous on...
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...
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...
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...
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...
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...
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)...
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...
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?...
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)...
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.]
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT