   Chapter 10.5, Problem 103E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

In which, if any, of the following cases is f differentiable at a? (There may be none or more than one.)(A) a = 2 ;  domain of  f   :  all real numbers; lim h → 0 + f ( 2 + h ) − f ( 2 ) h = 3 , lim h → 0 − f ( 2 + h ) − f ( 2 ) h = 5 (B) a = 0 ;  domain of  f   :   ( 0 , 2 ) ; lim h → 0 f ( 1 + h ) − f ( 1 ) h = 3 (C) a = 3 ;  domain of  f   : [ 0 , 3 ] ; lim h → 0 f ( 3 + h ) − f ( 3 ) h = 5 (D) a = 0 ;  domain of  f   :  all real numbers; lim h → 0 f ( h ) − f ( 0 ) h = + ∞

To determine

The correct option(s) where the function is differentiable at point x=a from the provided choices.

Explanation

Given information:

A function f and any point a.

The options are:

(A) a=2; domain of f: all real numbers;

limh0+f(2+h)f(2)h=3limh0f(2+h)f(2)h=5

(B) a=0; domain of f: (0,2);

limh0f(1+h)f(1)h=3

(C) a=3; domain of f: [0,3];

limh0f(3+h)f(3)h=5

(D) a=0; domain of f: all real numbers;

limh0f(h)f(0)h=+

The function can be differentiable at any point if the following conditions may exist.

First the left and right hand side derivative must exist and equal.

Then the point must be in domain and not on the extremes as it will be not possible to calculate the either left or the right hand derivative

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