7. We define the one-sided derivative from the right and the left by the formulas (respectively) D+ f(x) = lim h→0+ D¯ f(xo) = lim h→0- f(xo + h)-f(xo) h f(xo + h) -f(xo) h Show that a function f has a derivative at xo if and only if D* f(xo) and D¯f(xo) exist and are equal.
7. We define the one-sided derivative from the right and the left by the formulas (respectively) D+ f(x) = lim h→0+ D¯ f(xo) = lim h→0- f(xo + h)-f(xo) h f(xo + h) -f(xo) h Show that a function f has a derivative at xo if and only if D* f(xo) and D¯f(xo) exist and are equal.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
7 pls
![5. State hypotheses which assure the validity of the formula (the extended Chain rule)
F'(x) = f'{u[v(xo)]}u'[v(x)]v'(xo)
where F(x) = f{u[v(x)]}. Prove the result.
6. Show that if f has a derivative at a point, then it is continuous at that point
(Theorem 4.4).
7. We define the one-sided derivative from the right and the left by the formulas
(respectively)
D+ f(x) = lim
h→0+
f(xo +h)-f(xo)
h
f(xo + h)-f(xo)
h
Show that a function f has a derivative at xo if and only if D* f(x₁) and D¯f(x)
exist and are equal.
D f(x) = lim
h→0-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77cfc5ac-076f-4cb7-b69f-6b7f1cfee42f%2F1aee410d-ce76-4f7e-904b-87d5fcfa256f%2Fp9o0gk_processed.png&w=3840&q=75)
Transcribed Image Text:5. State hypotheses which assure the validity of the formula (the extended Chain rule)
F'(x) = f'{u[v(xo)]}u'[v(x)]v'(xo)
where F(x) = f{u[v(x)]}. Prove the result.
6. Show that if f has a derivative at a point, then it is continuous at that point
(Theorem 4.4).
7. We define the one-sided derivative from the right and the left by the formulas
(respectively)
D+ f(x) = lim
h→0+
f(xo +h)-f(xo)
h
f(xo + h)-f(xo)
h
Show that a function f has a derivative at xo if and only if D* f(x₁) and D¯f(x)
exist and are equal.
D f(x) = lim
h→0-
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