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Chapter 2: Q. 44 (page 166)

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

f(x)=x2−4,c=1

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)=x2−4,c=1
02

Step 2. Use sequence of approximation

Let,

h=2,1.5,1.25,1.1

Consider the expressions,

f(2)−f(1)2−1=[0]−[3]1=−3f(1.5)−f(1)1.5−1=(1.5)2−4−[3]0.5=−2.5

and,

f(1.25)−f(1)1.25−1=(1.25)2−4−[3]0.25=−2.25f(1.1)−f(1)1.1−1=(1.1)2−4−[3]0.1=−2.1

The slope of the tangent will be :

f′(1)=−2

The graph is ;

03

Step 3. First secant graph

Take c=1 , c+h=2, then the corresponding values are:

f(1)=3,f(2)=0

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=1 and c+h=1.5, then the corresponding values are :

f(1)=3,f(1.5)=1.75

The secant graph is :

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