Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Question

Chapter 3.5, Problem 37E

To determine

If there is a value of $b$ that will make the given function differentiable and continuous at $x=0$ and then to give reasons for the answer.

Expert Solution & Answer

Answer to Problem 37E

Function is continuous at $x=0$ when $b=1$ , there is no value of $b$ that will make the function differentiable at $x=0$ .

Explanation of Solution

Given information: Function is $g(x)={x+b, x<0cosx, x≥0$ .

Calculation:

$g(x)={x+b, x<0cosx, x≥0$

For making the function continuous at $x=0$ ,

Ascertaining left-hand and right-hand limits,

$limx→0+g(x)=limx→0+cosx = cos(0) =1limx→0−g(x)=limx→0−(x+b) =b$

For $g(x)$ to be continuous at $x=0$ , it is needed that $g(0)=limx→0+g(x)=limx→0−g(x)$ ,

$⇒b=1$

Differentiable:

For $b=1$ ,

Calculating left hand and right hand derivative,

Calculus: Graphical, Numerical, Algebraic, Chapter 3.5, Problem 37E

As $L.H.D.≠R.H.D.$ , function is not differentiable, for other values of $b$ , function is discontinuous at $x=0$ and left-hand derivative doesn’t exist.

Thus, function is continuous at $x=0$ when $b=1$ , there is no value of $b$ that will make the function differentiable at $x=0$ .

Chapter 3.5, Problem 36E

Chapter 3.5, Problem 38E

Chapter 3 Solutions

Calculus: Graphical, Numerical, Algebraic

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If there is a value of b that will make the given function differentiable and continuous at x = 0 and then to give reasons for the answer.