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Chapter 2: Q. 38 (page 166)

For each function f and value x=cin 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)=x+x3,c=1

Short Answer

Expert verified

The value is f'(1)=4.

Step by step solution

01

Step 1. Given Information  

We are given a function f(x)=x+x3,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=2+23-1+131=10-21=8h=1.5,f(1.5)-f(1)1.5-1=1.5+1.53-1+130.5=4.875-20.5=5.75h=1.1,f(1.1)-f(1)1.1-1=1.1+1.13-1+130.1=2.431-20.1=4.31h=0.9,f(0.9)-f(1)0.9-1=0.9+0.93-1+13-0.1=1.629-2-0.1=3.71

We might guess that the slope of the tangent line is f'(1)=4.

The graph of the function is

03

Step 3. Graphs of secant lines  

When c=1,c+h=2with h=1we have f(1)=2,f(2)=10the secant line can be drawn as follows:

When c=1,c+h=1.5with h=0.5we have f(1)=2,f(1.5)=4.875the secant line can be drawn as follows:

When c=1,c+h=1.1with h=0.1we have f(1)=2,f(1.1)=2.431the secant line can be drawn as follows:

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