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Chapter 2: Q. 43 (page 166)

For each function f and value x = c in Exercises 35鈥�44, use a sequence of approximations to estimate f'(c). Illustrate your work with an appropriate sequence of graphs of secant lines.

f(x)=|x鈭�1|,c=3

Short Answer

Expert verified

We have approximated the slope by using the concept of the secant line.

Step by step solution

01

Step 1. Given information.

We have to use a sequence of approximations to estimatef'(c)

f(x)=|x鈭�1|,c=3
02

Step 2. Use a sequence of approximation.

Let,

h=4,3.7,3.2,3.1

Consider the expressions,

f(4)鈭�f(3)4鈭�3=[3]鈭�[2]1=1f(3.7)鈭�f(3)3.7鈭�3=[2.7]鈭�[2]0.7=1

And,

f(3.2)鈭�f(3)3.2鈭�3=[2.2]鈭�[2]0.2=1f(3.1)鈭�f(3)3.1鈭�3=[2.1]鈭�[2]0.1=1

From these,

f鈥�(3)=1

The graph is :

03

Step 3. First secant graph 

Take c=3 , c+h=4, then the corresponding values are:

f(3)=2,f(4)=3

The secant line can be drawn as :

04

Step 4. Second secant graph 

Take c=3 and c+h = 3.5 then the corresponding values are :

f(3)=2,f(3.5)=2.5

The secant graph is :

05

Step 5. Third secant graph.

Take c=3 and c+h=3.1, then the corresponding values are :

f(3)=3,f(3.1)=2.1

The secant graph is :

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