Q. 43 - Calculus

Question

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

$f\left(x\right)=|x-1|,c=3$

Step-by-Step Solution

Answer

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

Step 1. Given information.

We have to use a sequence of approximations to estimate $f^{\text{'}}\left(c\right)$

$f\left(x\right)=|x-1|,c=3$

Step 2. Use a sequence of approximation.

Let,

$h=4,3.7,3.2,3.1$

Consider the expressions, 

$\begin{array}{r}\frac{f\left(4\right)-f\left(3\right)}{4-3}=\frac{\left[3\right]-\left[2\right]}{1}\\ =1\\ \frac{f\left(3.7\right)-f\left(3\right)}{3.7-3}=\frac{\left[2.7\right]-\left[2\right]}{0.7}\\ =1\end{array}$

And,

$\begin{array}{r}\frac{f\left(3.2\right)-f\left(3\right)}{3.2-3}=\frac{\left[2.2\right]-\left[2\right]}{0.2}\\ =1\\ \frac{f\left(3.1\right)-f\left(3\right)}{3.1-3}=\frac{\left[2.1\right]-\left[2\right]}{0.1}\\ =1\end{array}$

From these,

$f^{\mathrm{\prime }}\left(3\right)=1$

The graph is :

Step 3. First secant graph

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

$f\left(3\right)=2,f\left(4\right)=3$

The secant line can be drawn as :

Step 4. Second secant graph

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

$f\left(3\right)=2,f\left(3.5\right)=2.5$

The secant graph is :

Step 5. Third secant graph.

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

$f\left(3\right)=3,f\left(3.1\right)=2.1$

The secant graph is :