Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
19.C. Show that the function defined for r 0 by f(x) sin (1/x) is differentiable at each non-zero real number. Show that its derivative is not 0. (You may make use of trigonometric bounded on a neighborhood of x = identities, the continuity of the sine and cosine functions, and the elementary limiting relation sin u as u → 0.)

19.C. Show that the function defined for r 0 by f(x) sin (1/x) is differentiable at each non-zero real number. Show that its derivative is not 0. (You may make use of trigonometric bounded on a neighborhood of x = identities, the continuity of the sine and cosine functions, and the elementary limiting relation sin u as u → 0.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Topic Video
Question

Help with exercise 19.C
Prove by definition

19.1 DEFINITION. If c is a cluster point of D and belongs to D, we say that a real number L is the derivative of f at c if for every positive number e there is a positive number 8(e) such that if x belongs to D and if 0 < |æ – cl < ô(e), then f(x) – f(c) (19.1) L< x - c In this case we write f' (c) for L. Alternatively, we could define f' (c) as the limit (19.2) f(x) - f(c) lim (x € D). x - c It is to be noted that if c is an interior point of D, then in (19.1) we consider the points x both to the left and the right of the point c. On the other hand, if D is an interval and c is the left end point of D, then in relation (19.1) we can only take x to the right of c. In this case we sometimes say that "L is the right-hand derivative of f at r = c." How- ever, for our purposes it is not necessary to introduce such terminology. Whenever the derivative of f at c exists, we denote its value by f' (c). In this way we obtain a function f' whose domain is a subset of the domain of f. We now show that continuity of f at c is a necessary con- dition for the existence of the derivative at c.
Transcribed Image Text:19.1 DEFINITION. If c is a cluster point of D and belongs to D, we say that a real number L is the derivative of f at c if for every positive number e there is a positive number 8(e) such that if x belongs to D and if 0 < |æ – cl < ô(e), then f(x) – f(c) (19.1) L< x - c In this case we write f' (c) for L. Alternatively, we could define f' (c) as the limit (19.2) f(x) - f(c) lim (x € D). x - c It is to be noted that if c is an interior point of D, then in (19.1) we consider the points x both to the left and the right of the point c. On the other hand, if D is an interval and c is the left end point of D, then in relation (19.1) we can only take x to the right of c. In this case we sometimes say that "L is the right-hand derivative of f at r = c." How- ever, for our purposes it is not necessary to introduce such terminology. Whenever the derivative of f at c exists, we denote its value by f' (c). In this way we obtain a function f' whose domain is a subset of the domain of f. We now show that continuity of f at c is a necessary con- dition for the existence of the derivative at c.
19.B. If f and g are real-valued functions defined on an interval J, and if they are differentiable at a point e, show that their product h, defined by h(x) f(x)g(x), x € J, is differentiable at c and h' (c) = f'(c)g(c) + f(c)gʻ(c). 19.C. Show that the function defined for x # 0 by f(x) sin (1/x) is differentiable at each non-zero real number. Show that its derivative is not bounded on a neighborhood of x = identities, the continuity of the sine and cosine functions, and the elementary limiting relation 0. (You may make use of trigonometric sin u as u → 0.)
Transcribed Image Text:19.B. If f and g are real-valued functions defined on an interval J, and if they are differentiable at a point e, show that their product h, defined by h(x) f(x)g(x), x € J, is differentiable at c and h' (c) = f'(c)g(c) + f(c)gʻ(c). 19.C. Show that the function defined for x # 0 by f(x) sin (1/x) is differentiable at each non-zero real number. Show that its derivative is not bounded on a neighborhood of x = identities, the continuity of the sine and cosine functions, and the elementary limiting relation 0. (You may make use of trigonometric sin u as u → 0.)
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Permutation and Combination
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS