How do you sketch level curves of multivariable functions?

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In this video we're talking about how to sketch the level curves of a multivariable function.

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0:48 // What are level curves?
4:03 // How to sketch level curves?
5:02 // How to find equations of level curves?
7:52 // How to rewrite the function to graph the multivariable function?
12:15 // How many level curves should you draw?
13:30 // What are the level curves for a plane?
16:24 // Summary for how to sketch level curves

Whenever you're dealing with a multivariable function, the graph of that function will be a three-dimensional figure in space. If you take a perfectly horizontal sheet or plane that's parallel to the xy-plane, and you use that to slice through your three-dimensional figure, then what you get at the intersection of the figure and the plane is a two-dimensional curve.

What we want to be able to do is slice through the figure at all different heights in order to get what we call the "level curves" of a function. Then we want to be able to transfer all those two-dimensional curves into the two-dimensional plane, sketching those in the xy-plane.

This will give us the sketch of level curves of the function. In this video we're going to talk about how to find the level curves both graphically (by looking at a picture of the three-dimensional figure) and algebraically, by replacing z in the multivariable function with a constant c, and then substituting different values for c in order to get equations that are in terms of x and y only and can therefore be graphed in the xy-plane.

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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)

Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”

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WOW! You can not beat that handwriting! So c crisp and clear.
Thank you for sharing.

olefrehr

Dear Krista, I have faithfully watched yours and Christine Breiner’s (of MIT Courseware) Calculus videos. Truthfully, I am beyond amazed at both of your levels of skill and brilliance, and the ability to teach and do so in such a pleasant, down to earth, caring way. Thank you both for all you do and please continue with more videos. I am nowhere near your levels, but I do learn from you, and it is a very enjoyable learning experience each time. Blessings, George.

georgepolasky

Awesome collection of math videos. Very helpful. Thanks a lot. 😃

XX-sfeh

Best video I’ve found on this literally!!!! Thank you

chidinmaadogu

best explanation of level curves ever.

BmCC

Been trying to search for this for 2 weeks. Finally understood. I remembered studying this in engineering some 13-14 years ago but I couldn't remember why the oval shape comes in and why it's slanted this way instead of that way.
Danke!

ValiChandiwala

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I hope you're happy about what you do.

mousaalsaeed

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sistersfusion

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atika.

Great video. Great drawings. Great Explanations. SO simple. Thank you

CSGdewxp

Thank you, old guy going back to school needs all the help he can !

SidviciousWisconsin

This is so helpful! Love your clear explanation and voice!!!thank you

parkjessica

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jahansaid

Added to my education playlist. I only save the best on youtube. Thank you...and my future self who will also probably use this to relearn thanks you as well

StaticSleet