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Chapter 2: Q. 41 (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)=sin鈦�x,c=蟺2

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

f(x)=sin鈦�x,c=蟺2

02

Step 2. Use sequence of approximation

Let,

h=蟺6,蟺4,蟺3

Consider the expression,

f蟺6鈭�f蟺2蟺6鈭�蟺2=sin鈦�蟺6鈭�sin鈦�蟺2蟺6鈭�蟺2f蟺4鈭�f蟺2蟺4鈭�蟺2=sin鈦�蟺4鈭�sin鈦�蟺2蟺4鈭�蟺2=0.3729

and,

f蟺3鈭�f蟺2蟺3鈭�蟺2=sin鈦�蟺3鈭�sin鈦�蟺2蟺3鈭�蟺2=0.2558

From these values, the resulting value approach to 0,

This implies f鈥�蟺2=0

The graph is :

03

Step 3. First secant graph

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

f蟺2=1,f(蟺)=0

The secant line can be drawn as:

04

Step 4. Second secant graph

Take c=蟺2and c+h = 2蟺3then the corresponding values are :

f蟺2=1,f2蟺3=32

The secant graph is :

05

Step 5. Third secant graph

Take c=蟺2and c+h=5蟺9, then the corresponding values are :

f蟺2=1,f5蟺9=0.9848

The secant graph is :

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