14.1 Videos Guide

14.1a

• Introduction to multivariable functions

  • Domain

  • Graphs

Exercises:

• The temperature-humidity index $I$ (or humidex, for short) is the perceived air temperature when the actual temperature is T and the relative humidity is $h$, so we can write $I=f(T,\ h)$. The following table of values of $I$ is an excerpt from a table compiled by the National Oceanic & Atmospheric Administration. Using the table, find $f(95,\ 70)$. What is the meaning of $f(80,\ h)$?
  

  Actual temp	Relative humidity ($h$)
  ($T^{\circ}$F)	20	30	40	50	60	70
  80	77	78	79	81	82	83
  85	82	84	86	88	90	93
  90	87	90	93	96	100	106
  95	93	96	101	107	114	124
  100	99	104	110	120	132	144

• Let $F(x,\ y)=1+\sqrt{4-y^{2}}$.
  (a) Evaluate $F(3,\ 1)$.

(b) Find and sketch the domain of $F$.

(c) Find the range of $F$.

14.1b

• Let $g(x,\ y,\ z)=x^{3}y^{2}z\sqrt{10-x-y-z}$.
  Find and describe the domain of $g$.

• Find and sketch the domain of the function.

  • $f(x,\ y)=\sqrt[4]{x-3y}$

  • $g(x,\ y)=\frac{\ln(2-x)}{1-x^{2}-y^{2}}$

14.1c

• Find and sketch the domain of the function.
  $f(x,\ y,\ z)=\ln\left(16-4x^{2}-4y^{2}-z^{2}\right)$

• Sketch the graph of the function.
  $f(x,\ y)=2-x^{2}-y^{2}$

14.1d

• Level curves and contour maps

Exercises:

• Make a rough sketch of a contour map for the function whose graph is shown.

14.1e

• A contour map of a function is shown. Use it to make a rough sketch of the graph of $g$.

• Draw a contour map of the function $f(x,\ y)=xy$ showing several level curves.