Q. 1 - Calculus

Question

Values of transcendental functions: Without a calculator, find each of the function values that follow. For some values the answer may be undefined. 

Step-by-Step Solution

Answer

The value of each expression is calculated. 

Step 1. Given information

We have to find the value of the given functions.

Step 2. Values of the function.

When 

$\begin{array}{c}f\left(x\right)=\csc x\\ =\frac{1}{\sin x}\\ For x =\mathrm{\pi }\\ =\frac{1}{0}\\ =\text{ undefined }\end{array}$

For $x=\frac{\mathrm{\pi }}{2}$

$\begin{array}{r}f(\pi /2)=\csc (\pi /2)\\ =\frac{1}{\sin (\pi /2)}\\ =\frac{1}{1}\\ =1\end{array}$

When $f\left(x\right)={\tan }^{2}x$

For $x=\mathrm{\pi }$

$\begin{array}{r}f\left(\pi \right)={\tan }^2\pi \\ =0\end{array}$

For $x=\frac{\mathrm{\pi }}{2}$

$\begin{array}{c}f(\pi /2)={\tan }^2(\pi /2)\\ =\text{ undefine }\end{array}$

When $f\left(x\right)={\sin }^{-1}x$

For x=-1

$\begin{array}{r}f(-1)={\sin }^{-1}(-1)\\ =\frac{-\pi }{2}\end{array}$

For $x=\frac{1}{2}$

$\begin{array}{r}f\left(\frac{1}{2}\right)={\sin }^{-1}\left(\frac{1}{2}\right)\\ =\frac{\pi }{6}\end{array}$

When $f\left(x\right)={\tan }^{-1}\sqrt{x}$

For x=1

$\begin{array}{r}f\left(1\right)={\tan }^{-1}\sqrt{1}\\ =\frac{\pi }{4}\end{array}$

For x=3

$\begin{array}{r}f\left(3\right)={\tan }^{-1}\sqrt{3}\\ =\frac{\pi }{3}\end{array}$