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Chapter 12: Q 26. (page 916)

In Exercise, evaluate the given function at the specified points in the domain, and then find the domain and range of the function.
$f (x, y) = \frac{\sin (x - y)}{x - y}, (0, 蟺), (\frac{蟺}{2}, \frac{蟺}{3})$

Short Answer

Expert verified

$f (0, 蟺) = 0 f (\frac{蟺}{2}, \frac{蟺}{3}) = \frac{3}{蟺}$

Domain is $R^{2} - \{(x, y) | x - y = 0\}$

Range is R.

Step by step solution

01

Step 1. Given information

Function is $f (x, y) = \frac{\sin (x - y)}{x - y}$

02

Step 2. Explanation

$\begin{array}{rcl} f (0, 蟺) & = & \frac{\sin (0 - 蟺)}{0 - 蟺} \\ = & \frac{- \sin (蟺)}{- 蟺} \\ = & 0 \\ f (\frac{蟺}{2}, \frac{蟺}{3}) & = & \frac{\sin (\frac{蟺}{2} - \frac{蟺}{3})}{(\frac{蟺}{2} - \frac{蟺}{3})} \\ = & \frac{\frac{1}{2}}{\frac{蟺}{6}} \\ = & \frac{3}{蟺} \end{array}$

For domain, denominator can not be equal to 0.

$\begin{array}{rcl} x - y & = & 0 \\ x & = & y \end{array}$

So, domain is $R^{2} - \{(x, y) | x - y = 0\}$

Let $w = x - y$

So, width="93">F(w)=sinww

$\begin{array}{rcl} F^{'} (w) & = & \frac{w \cos w - \sin w}{w} \\ 0 & = & \frac{w \cos w - \sin w}{w} \\ w \cos w - \sin w & = & 0 \\ w & = & \tan w \end{array}$

So, one of the solution is 0.

Function F(w) is not defined for this solution.

Also,

$\lim_{w 鈫� 0} \frac{\sin w}{w} = 1$

So, maximum value of F(w) is 1.

Take w = c

width="58">c=tanc

$\begin{array}{rcl} F (c) & = & \frac{\sin c}{c} \\ = & \cos c \end{array}$

So, minimum value of function is $\cos c$

Using the maximum and minimum value of function, the range of function is written as [c, 1] where c is the non-zero solution of width="66">w=tanw

So, range of the function is R.

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Most popular questions from this chapter

$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Find \frac{鈭� x}{鈭� s} and \frac{鈭� x}{鈭� t} .$

Use Theorem 12.33 to find the indicated derivatives in Exercises 27鈥�30. Express your answers as functions of two variables.

$\frac{鈭� z}{鈭� t} w h e n z = x^{2} y^{3}, x = t \sin s, a n d y = s \cos t$

$Let z = (f {(x, y))}^{n} . Show that \frac{鈭� z}{鈭� x} = n (f {(x, y))}^{n - 1} \frac{鈭� f}{鈭� x} and \frac{鈭� z}{鈭� y} = n (f {(x, y))}^{n - 1} \frac{鈭� f}{鈭� y} .$

$The pressure P of an ideal gas is given by P (n, T, V) = \frac{n R T}{V}, where n is the amount of the gas, R is a constant, T is the temperature of the gas in degrees Kelvin, and V is the volume of the gas . Find \frac{鈭� P}{鈭� n}, \frac{鈭� P}{鈭� V}, and \frac{鈭� P}{鈭� T} . What are the physical interpretations of these partial derivatives ?$

Evaluate the following limits, or explain why the limit does not exist.

$\lim_{(x, y) 鈫� (4, 蟺)} \tan^{- 1} (\frac{y}{x})$

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