71-72. One-sided derivatives The right-sided and left-sided derivatives of a f(a+h) − f(a) h and function at a point a are given by f'+(a) = lim h→0+ f(a+h)-f(a), respectively, provided these limits exist. The f'_(a) = lim h→0- h derivative f'(a) exists if and only if f'+(a) = f'_(a). a. Sketch the following functions. b. Compute f'(a) and f'_(a) at the given point a. c. Is f continuous at a? Is f differentiable at a? 71. f(x) = x - 2|; a = 2 (4-x² 72. f(x) = 2x + 1 if x ≤ 1 ; a = 1 if x > 1'
71-72. One-sided derivatives The right-sided and left-sided derivatives of a f(a+h) − f(a) h and function at a point a are given by f'+(a) = lim h→0+ f(a+h)-f(a), respectively, provided these limits exist. The f'_(a) = lim h→0- h derivative f'(a) exists if and only if f'+(a) = f'_(a). a. Sketch the following functions. b. Compute f'(a) and f'_(a) at the given point a. c. Is f continuous at a? Is f differentiable at a? 71. f(x) = x - 2|; a = 2 (4-x² 72. f(x) = 2x + 1 if x ≤ 1 ; a = 1 if x > 1'
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Question 71? (Show work) answer is given.
Transcribed Image Text:71-72. One-sided derivatives The right-sided and left-sided derivatives of a
f(a+h)-f(a) and
h
function at a point a are given by f'+(a) = lim
h→0+
f(a+h) − f(a)
-
f'_(a) = lim
h→0-
h
derivative f'(a) exists if and only if f'+(a) = f'_(a).
9
71. f(x)= x - 2|; a = 2
4- x²
72. f(x) =
2x + 1
respectively, provided these limits exist. The
a. Sketch the following functions.
b. Compute f'(a) and f'_(a) at the given point a.
c. Is f continuous at a? Is f differentiable at a?
if x < 1
2
if x > 1
; a = 1
Transcribed Image Text:71.10
b. f' (2) = 1; f'_(2) = -1
c. f is continuous but not differentiable at x = 2.
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