4. Even when a function f(x) is differentiable at x = a, its derivative function f(x) can behave oddly near x = a, in the sense that lim f(x) does not exist. Thus, in general, we cannot compute fº(a) by putting x→a into fº(x), which is calculated by formulas for x 6= a. Let f(x) { x² cos Bo X x→a (a) Show that f(x) is differentiable at x = 0. (b) Write down f(x) explicitly for each x. f'(0) # lim f'(x). (c) Show that x-0 if x #0 if x = 0.
4. Even when a function f(x) is differentiable at x = a, its derivative function f(x) can behave oddly near x = a, in the sense that lim f(x) does not exist. Thus, in general, we cannot compute fº(a) by putting x→a into fº(x), which is calculated by formulas for x 6= a. Let f(x) { x² cos Bo X x→a (a) Show that f(x) is differentiable at x = 0. (b) Write down f(x) explicitly for each x. f'(0) # lim f'(x). (c) Show that x-0 if x #0 if x = 0.
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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Transcribed Image Text:4. Even when a function f(x) is differentiable at x = a, its derivative function f(x) can
behave oddly near x = a, in the sense that lim f(x) does not exist. Thus, in
general, we cannot compute fo(a) by putting x→a into f(x), which is calculated by
formulas for x 6= a.
Let
f(x)
x² cos
{ (1)
x→a
?? 0
(a) Show that f(x) is differentiable at x = 0.
(b) Write down f(x) explicitly for each x.
f'(0) ‡ lim f'(x).
x→0
(c) Show that
if x #0
if x = 0.
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