Calculus III - Functions of Several Variables

Show General Notice Show Mobile Notice Show All Notes Hide All Notes

General Notice

I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.

Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.

Paul
February 18, 2026

Mobile Notice

You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 12.5 : Functions of Several Variables

Back to Problem List

6. Identify and sketch the level curves (or contours) for the following function.

\[4z + 2{y^2} - x = 0\]

Show All Steps Hide All Steps

Start Solution

We know that level curves or contours are given by setting \(z = k\). Doing this in our equation gives,

\[4k + 2{y^2} - x = 0\] Show Step 2

A quick rewrite of the equation from the previous step gives us,

\[x = 2{y^2} + 4k\]

So, the level curves for this function will be parabolas opening to the right and starting at 4\(k\).

Note as well that there will be no restrictions on the values of \(k\) that we can use, as there sometimes are.

Show Step 3

Below is a sketch of some level curves for some values of \(k\) for this function.