Determine differientiable and graph. See Images Determine whether the function is15.X=1$ then sketchdifferentiable.ata graph of ysf (x).fis continuous at X=landK righ sidelim%3Drleft sidef'-() =limf(i+Ax)-f(ムX→0AX la f (x) =(x if x #1(2it x=1%3D

Name: Determine differientiable and graph. See Images Determine whether the
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Description: Determine differientiable and graph. See Images Determine whether the function is15.X=1$ then sketchdifferentiable.ata graph of ysf (x).fis continuous at X=landK righ sidelim%3Drleft sidef'-() =limf(i+Ax)-f(ムX→0AX la f (x) =(x if x #1(2it x=1%3D

Step:-1 Given that fx = xif x≠12if x=1 And Right , left derivatives are respectively, f+' 1…

Answered: Determine whether the function 1s…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.6: Exponential And Logarithmic Equations

Problem 40E

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Determine differientiable and graph. See Images

Determine whether the function is
15.
X=1$ then sketch
differentiable.at
a graph of ysf (x).
fis continuous at X=l
and
K righ side
lim
%3D
rleft side
f'-() =
lim
f(i+Ax)-f(
ムX→0
AX

Expert Solution

Step 1

Step:-1

Given that $f (x) = \{\begin{cases} x & \begin{matrix} if & x & \neq 1 \end{matrix} \\ 2 & \begin{matrix} if & x & = 1 \end{matrix} \end{cases}$

And Right , left derivatives are respectively,

$f_{+}^{'} (1) = {lim}_{∆ x \to 0^{+}} \frac{f (1 + ∆ x) - f (1)}{∆ x}, f_{-}^{'} (1) = {lim}_{∆ x \to 0^{-}} \frac{f (1 + ∆ x) - f (1)}{∆ x} Here, ∆ x is difference.$

Step:-2

$f_{+}^{'} (1) = {lim}_{∆ x \to 0^{+}} \frac{f (1 + ∆ x) - f (1)}{∆ x} f_{+}^{'} (1) = {lim}_{∆ x \to 0^{+}} \frac{1 + ∆ x - 2}{∆ x} = {lim}_{∆ x \to 0^{+}} \frac{∆ x - 1}{∆ x} f_{+}^{'} (1) = {lim}_{∆ x \to 0^{+}} (1 - \frac{1}{∆ x}) = - \infty f_{+}^{'} (1) = - \infty$

Step:-3

$f_{-}^{'} (1) = {lim}_{∆ x \to 0^{-}} \frac{f (1 + ∆ x) - f (1)}{∆ x} f_{-}^{'} (1) = {lim}_{∆ x \to 0^{-}} \frac{1 + ∆ x - 2}{∆ x} = 1 - \frac{2}{∆ x} f_{-}^{'} (1) = + \infty H e r e, f_{-}^{'} (1) \neq f_{+}^{'} (1)$

so, function is not differentiable at x=1.

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