Q. 41 - 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.

$f\left(x\right)=\sin x,c=\frac{\pi }{2}$

Step-by-Step Solution

Answer

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

Step 1. Given information

$f\left(x\right)=\sin x,c=\frac{\pi }{2}$

Step 2. Use sequence of approximation

Let,

$h=\frac{\pi }{6},\frac{\pi }{4},\frac{\pi }{3}$

Consider the expression,

$\begin{array}{r}\frac{f\left(\frac{\pi }{6}\right)-f\left(\frac{\pi }{2}\right)}{\frac{\pi }{6}-\frac{\pi }{2}}=\frac{\left[\sin \left(\frac{\pi }{6}\right)\right]-\left[\sin \left(\frac{\pi }{2}\right)\right]}{\frac{\pi }{6}-\frac{\pi }{2}}\\ \frac{f\left(\frac{\pi }{4}\right)-f\left(\frac{\pi }{2}\right)}{\frac{\pi }{4}-\frac{\pi }{2}}=\frac{\left[\sin \left(\frac{\pi }{4}\right)\right]-\left[\sin \left(\frac{\pi }{2}\right)\right]}{\frac{\pi }{4}-\frac{\pi }{2}}\\ =0.3729\end{array}$

and,

$\begin{array}{r}\frac{f\left(\frac{\pi }{3}\right)-f\left(\frac{\pi }{2}\right)}{\frac{\pi }{3}-\frac{\pi }{2}}=\frac{\left[\sin \left(\frac{\pi }{3}\right)\right]-\left[\sin \left(\frac{\pi }{2}\right)\right]}{\frac{\pi }{3}-\frac{\pi }{2}}\\ =0.2558\end{array}$

From these values, the resulting value approach to 0,

This implies $f^{\mathrm{\prime }}\left(\frac{\pi }{2}\right)=0$

The graph is :

Step 3. First secant graph

Take c=$\frac{\mathrm{\pi }}{2}$ , c+h=$\mathrm{\pi }$, then the corresponding values are:

$f\left(\frac{\pi }{2}\right)=1,f\left(\pi \right)=0$

The secant line can be drawn as:

Step 4. Second secant graph

Take c=$\frac{\mathrm{\pi }}{2}$ and c+h = $\frac{2\mathrm{\pi }}{3}$ then the corresponding values are :

$f\left(\frac{\pi }{2}\right)=1,f\left(\frac{2\pi }{3}\right)=\frac{\sqrt{3}}{2}$

The secant graph is :

Step 5. Third secant graph

Take c=$\frac{\mathrm{\pi }}{2}$ and c+h=$\frac{5\mathrm{\pi }}{9}$, then the corresponding values are :

$f\left(\frac{\pi }{2}\right)=1,f\left(\frac{5\pi }{9}\right)=0.9848$

The secant graph is :