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Chapter 2: Q. 35 (page 166)

For each function f and value x=cin 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)=4-x2,c=1

Short Answer

Expert verified

The value isf'(1)=-2.

Step by step solution

01

Step 1. Given Information

We are given a functionf(x)=4-x2,c=1 and using a sequence of approximations we have to estimate f'(c).

02

Step 2. Using a sequence of approximations  

h=2,f(2)-f(1)2-1=4-22-4-121=-31=-3h=1.5,f(1.5)-f(1)1.5-1=4-1.52-4-120.5=1.75-30.5=-2.5h=1.1,f(1.1)-f(1)1.1-1=4-1.12-4-120.1=2.79-30.1=-2.1h=0.9,f(0.9)-f(1)0.9-1=4-0.92-4-12-0.1=3.19-3-0.1=-1.9

We might guess that the slope of the tangent line is f'(1)=-2.

The graph of the function is

03

Step 3. Graphs of secant lines

When c=1,c+h=2with h=1we have f(1)=3,f(2)=0the secant line can be drawn as follows:

When c=1,c+h=1.5with h=0.5we have f(1)=3,f(1.5)=1.75the secant line can be drawn as follows:

When c=1,c+h=1.1with h=0.1we have f(1)=3,f(1.1)=2.79the secant line can be drawn as follows:

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