Q. 42 - Calculus

Question

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

$f\left(x\right)=\arctan x ,c=0$

Step-by-Step Solution

Answer

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

Step 1. Given information.

We have been given a function and value of c:

$f\left(x\right)=\arctan x , c=0$

We have to estimate f '(c) using a sequence of approximations. 

Step 2. Use sequence of approximation

Let,

$h=1,0.5,0.1,0.01$

Consider the expression,

$\begin{array}{r}\frac{f\left(1\right)-f\left(0\right)}{1-0}=\frac{[\arctan (1\left)\right]-[\arctan (0\left)\right]}{1}\\ =0.78\\ \frac{f\left(0.5\right)-f\left(0\right)}{0.5-0}=\frac{[\arctan (0.5\left)\right]-[\arctan (0\left)\right]}{0.5}\\ =0.9272\end{array}$

Consider further,

$\begin{array}{r}\frac{f\left(0.1\right)-f\left(0\right)}{1-0}=\frac{[\arctan (0.1\left)\right]-[\arctan (0\left)\right]}{0.1}\\ =0.9966\\ \frac{f\left(0.01\right)-f\left(0\right)}{0.01-0}=\frac{[\arctan (0.01\left)\right]-[\arctan (0\left)\right]}{0.01}\\ =0.9999\end{array}$

The slope of tangent will be :

$f^{\mathrm{\prime }}\left(0\right)=1$

The graph is ;

Step 3. First secant graph

Take c=0 , c+h=1, then the corresponding values are:

$f\left(0\right)=0,f\left(1\right)=\frac{\pi }{4}$

The secant line can be drawn as:

Step 4. Second secant graph

Take c=0 and c+h = 0.5 then the corresponding values are :

$f\left(0\right)=0,f\left(0.5\right)=0.4636$

The secant graph is :

Step 5. Third secant graph

Take c=0 and c+h=0.1, then the corresponding values are : 

$f\left(0\right)=0,f\left(0.1\right)=0.0996$

The secant graph is :