Homework Help is Here – Start Your Trial Now!
Math
Calculus
Sx, if xe Q if x ER – Q. f(x) = Using only the definition of derivatives and Lemma 4.2.2, determine whether f is differentiable at 0. If it is, find f'(0); if it is not, show why not.

Sx, if xe Q if x ER – Q. f(x) = Using only the definition of derivatives and Lemma 4.2.2, determine whether f is differentiable at 0. If it is, find f'(0); if it is not, show why not.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
See similar textbooks
icon
Related questions
Question

4.2.2

Please include a formal proof. 

Definition 4.2.1. Let I CR be an open interval, let c E I and let f:1¬R be a function. 1. The function f is differentiable at e if lim f(x) – f(c) x-c exists; if this limit exists, it is called the derivative of f at c, and it is denoted f'(c). 2. The function f is differentiable if it is differentiable at every number in 1. If f is differentiable, the derivative of f is the function f' : I → R whose value at x is f'(x) for all x €I. Lemma 4.2.2. Let ICR be an open interval, let c e I and let f : 1¬R be a function. Then f is differentiable at e if and only if lim exists, and if this limit exists it equals f'(c). Proof. Suppose that f is differentiable at c. Let F:1- {c} →R be defined by F(x) = L1) – f(c) X-e for all x €I- {c}. Then lim F(x) exists and equals f'(c). Let J= {x-c|x€ I}, and let g: J – {0} → I – {c} be defined by g(x) = x+c for all x €J- {0}. By Exercise 3.2.1 and Exercise 3.2.5 we know that lim g(h) =c. It is struightforward to verify that (Fog)(h) = Lc+h) – (c) for all h EJ-{0}. Because lim F(x) exists, then Theorem 3.2.12 implies that lim (Fo 8)(h) exists and lim (Fog)(h) = lim F(x), which is equivalent to saying that S(c+h) – {(c) lim exists and equals S'(c). The other implication is similar, and we omit the details.
Transcribed Image Text:Definition 4.2.1. Let I CR be an open interval, let c E I and let f:1¬R be a function. 1. The function f is differentiable at e if lim f(x) – f(c) x-c exists; if this limit exists, it is called the derivative of f at c, and it is denoted f'(c). 2. The function f is differentiable if it is differentiable at every number in 1. If f is differentiable, the derivative of f is the function f' : I → R whose value at x is f'(x) for all x €I. Lemma 4.2.2. Let ICR be an open interval, let c e I and let f : 1¬R be a function. Then f is differentiable at e if and only if lim exists, and if this limit exists it equals f'(c). Proof. Suppose that f is differentiable at c. Let F:1- {c} →R be defined by F(x) = L1) – f(c) X-e for all x €I- {c}. Then lim F(x) exists and equals f'(c). Let J= {x-c|x€ I}, and let g: J – {0} → I – {c} be defined by g(x) = x+c for all x €J- {0}. By Exercise 3.2.1 and Exercise 3.2.5 we know that lim g(h) =c. It is struightforward to verify that (Fog)(h) = Lc+h) – (c) for all h EJ-{0}. Because lim F(x) exists, then Theorem 3.2.12 implies that lim (Fo 8)(h) exists and lim (Fog)(h) = lim F(x), which is equivalent to saying that S(c+h) – {(c) lim exists and equals S'(c). The other implication is similar, and we omit the details.
Exercise 4.2.2. Let f: R → R be defined by if x E Q f(x) = |x², ifx€R- Q. Using only the definition of derivatives and Lemma 4.2.2, determine whether f is differentiable at 0. If it is, find f'(0); if it is not, show why not.
Transcribed Image Text:Exercise 4.2.2. Let f: R → R be defined by if x E Q f(x) = |x², ifx€R- Q. Using only the definition of derivatives and Lemma 4.2.2, determine whether f is differentiable at 0. If it is, find f'(0); if it is not, show why not.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS