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Chapter 1: Q. 1 (page 134)

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

Short Answer

Expert verified

The value of each expression is calculated.

Step by step solution

01

Step 1. Given information

We have to find the value of the given functions.

02

Step 2. Values of the function.

When

$\begin{matrix} f (x) = \csc x \\ = \frac{1}{\sin x} \\ F o r x = π \\ = \frac{1}{0} \\ = undefined \end{matrix}$

For $x = \frac{π}{2}$

$\begin{aligned} f (π / 2) = \csc (π / 2) \\ = \frac{1}{\sin (π / 2)} \\ = \frac{1}{1} \\ = 1 \end{aligned}$

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

For $x = π$

localid="1668092908461" $\begin{aligned} f (π) = \tan^{2} π \\ = 0 \end{aligned}$

For $x = \frac{π}{2}$

$\begin{matrix} f (π / 2) = \tan^{2} (π / 2) \\ = undefine \end{matrix}$

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

For x=-1

$\begin{aligned} f (- 1) = \sin^{- 1} (- 1) \\ = \frac{- π}{2} \end{aligned}$

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

$\begin{aligned} f (\frac{1}{2}) = \sin^{- 1} (\frac{1}{2}) \\ = \frac{π}{6} \end{aligned}$

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

For x=1

$\begin{aligned} f (1) = \tan^{- 1} \sqrt{1} \\ = \frac{π}{4} \end{aligned}$

For x=3

$\begin{aligned} f (3) = \tan^{- 1} \sqrt{3} \\ = \frac{π}{3} \end{aligned}$

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Most popular questions from this chapter

Use algebra to find the largest possible value of δ or smallest possible value of N that makes each implication true. Then verify and support your answers with labeled graphs.
$If x > N, then 1 - 2 x < - 500$

For each limit in Exercises 33–38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

$\lim_{x \to 0} x^{- 3} .$

For each limit $\underset{x \to c}{l i m} f (x) = L$ in Exercises 43–54, use graphs and algebra to approximate the largest value of $δ$ such that if $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$

$\underset{x \to - 1}{l i m} \frac{x^{2} - 2 x - 3}{x + 1} = - 4, ε = 0.1$

Use the delta-epsilon definition of continuity to argue that f is or is not continuous at the indicated point $x = c$ .

$f (x) = \{\begin{aligned} 2 - x, if x rational \\ x^{2}, if x irrational, \end{aligned} c = 1$

For each limit in Exercises 33–38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

$\lim_{x \to - 1} 6 .$

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