3.2 Functions of Several Variables

Functions of Several Variables

Introduction

Functions of Several Variables

Representing Functions of Two Variables

Traces

Suppose you invest money in an account that pays 5% interest compounded continuously. If you have an initial investment of $P$ dollars in the account, then $A$, the amount of money in the account after $t$ years is given by

The variables $P$ and $t$ are independent of each other, so using functional notation we write

Find the amount of money in the account after 7 years if you originally invest 1000 dollars.

Work through this exercise and explain your reasoning step by step.

$f(x,y) = x^2+y^2$

$f(x,y) = \sqrt{x^2+y^2}$

$\displaystyle Q(x,y) = \frac{x+y}{x^2-y^2}$

$\displaystyle s(x,y) = \frac{1}{\sqrt{1-xy^2}}$

Identify the $y = 0.6$ trace for the distance function $f$ defined by $f(x,y) = \frac{x^2 \sin(2y)}{g}$ by highlighting or circling the appropriate cells in Table. Write a sentence to describe the behavior of the function along this trace.

Identify the $x = 150$ trace for the distance function by highlighting or circling the appropriate cells in Table. Write a sentence to describe the behavior of the function along this trace.

In the next several parts, we will be looking at using traces to help us draw an accurate plot of the surface given by $z=g(x,y)=yx^2$. As a first step, find the equation for the $y=1$ trace of $z=g(x,y)=yx^2$ and draw a graph of the $y=1$ trace on the corresponding face in .

Find the equation for each of the following traces and draw a plot of the trace on the corresponding face in . -$y=-1$-$x=1$-$x=-1$

Draw a few traces (at least three more) that correspond to values in the middle of the plot to fill in a plot of the surface given by $z=x^2y$.