– WELCOME TO A VIDEO

ON LEVEL CURVES. THE GOALS OF THIS VIDEO

ARE TO GRAPH THE LEVEL CURVES OF A GIVEN FUNCTION AS WELL AS RECOGNIZE A FUNCTION

GRAPHED USING LEVEL CURVES. LEVEL CURVES, SOMETIMES CALLED

CONTOUR LINES, ARE ANOTHER WAY TO REPRESENT

A FUNCTION Z=F OF XY. BY LETTING Z EQUAL

SOME CONSTANT, WE OBTAIN A SINGLE LEVEL CURVE. REMEMBER, YOU CAN THINK OF Z

AS THE HEIGHT OF THE SURFACE. SO WHAT WE’RE DOING IS TAKING

SLICES OF THE SURFACE AT DIFFERENT HEIGHTS, AND THEN GRAPHING THOSE

ON THE XY PLANE. IF WE DO THIS

FOR SEVERAL VALUES OF C, WE HAVE A SET OF LEVEL CURVES TO

REPRESENT A GIVEN FUNCTION. NOW, THERE ARE QUITE A FEW REAL-

LIFE EXAMPLES OF LEVEL CURVES, AND WE’RE GOING TO TAKE

A QUICK LOOK AT THESE BEFORE WE TAKE A LOOK

AT OUR OWN EXAMPLE. A PRESSURE MAP CAN BE

REPRESENTED USING LEVEL CURVES, AND THE LEVEL CURVES ARE CALLED

ISOBARS. A TEMPERATURE MAP,

WHICH WE OFTEN SEE ON THE NEWS, REPRESENTS A SET OF LEVEL CURVES

CALLED ISOTHERMS, AND THEN AN ALTITUDE MAP

USES LEVEL CURVES TO REPRESENT THE HEIGHT

ABOVE SEA LEVEL. HERE’S A PRESSURE MAP. THIS MAY BE DIFFICULT FOR YOU

TO READ. HOWEVER, EACH OF THESE CURVES

REPRESENTS WHERE THE PRESSURE IS THE SAME. HOWEVER, EACH OF THESE CURVES

REPRESENTS WHERE THE PRESSURE

WOULD BE THE SAME, AND IN THIS REGION HERE,

WE CAN SEE THAT AS WE MOVE TOWARD THE CENTER

OF THIS REGION, THE PRESSURE GETS HIGHER. AND OVER IN THIS REGION,

AS WE APPROACH THE CENTER, THE PRESSURE GETS LOWER. HERE’S A TEMPERATURE MAP

THAT WE OFTEN SEE ON THE NEWS, AND AGAIN, EVERY CURVE

REPRESENTS A REGION WHERE THE TEMPERATURE

WOULD BE THE SAME. SO ON THIS MAP WE CAN SEE, AS WE APPROACH THE CENTER

OF THIS REGION HERE, THE TEMPERATURE IS DROPPING, AND AS WE MOVE TOWARD

THE OUTER REGION, THE TEMPERATURE IS RISING. AND THEN LASTLY, THIS ONE’S

REALLY HARD TO READ, BUT YOU CAN SEE DOWN THE MIDDLE

HERE IT LOOKS LIKE A RIVER, AND THEN THERE ARE

A BUNCH OF CURVES TO THE LEFT AND RIGHT

OF THE RIVER. AND EACH CURVE HERE REPRESENTS

A DIFFERENT ALTITUDE. SO YOU CAN KIND OF ENVISION

HAVING THIS RIVER IN THE MIDDLE AND MAYBE SOME HILLS

OR MOUNTAINS ON EACH SIDE. SO AS WE APPROACH THE RIVER, THE ALTITUDE OR THE HEIGHT ABOVE

SEA LEVEL WOULD BE DROPPING. NOW, FOR OUR PURPOSE, IF WE TAKE A LOOK AT THE GRAPH

OF Z=X SQUARED + Y SQUARED, WE WOULD HAVE THIS SURFACE

GENERATED HERE AS A PARABOLOID. AND THE IDEA BEHIND LEVEL CURVES

IS THAT WE SET THE VALUE OF Z

TO DIFFERENT VALUES, AND IF WE DO THAT,

YOU CAN NOTICE WE’RE GOING TO HAVE A CIRCLE WHERE THE RADIUS IS GOING TO BE

DIFFERENT BASED UPON THE VALUE OF Z. SO LOOKING AT THIS GRAPH HERE, YOU CAN SEE THE CIRCLES THAT WOULD BE CREATED

WITH DIFFERENT VALUES OF Z. AND THEN TO SHOW THE GRAPH

OF THE LEVEL CURVES, WE’LL LOOK DOWN ON THIS LOOKING

ONLY AT THE XY PLANE. LET’S TAKE A LOOK AT THIS

USING MAPLE. AGAIN, HERE IS THE GRAPH

OF OUR PARABOLOID, AND THEN WE SET Z

EQUAL TO A VARIETY OF VALUES, WHICH WE USUALLY CALL C

FOR CONSTANT, AND IT WOULD CREATE THESE

BLACK CIRCLES. AND THEN IF WE ROTATE THIS AND LOOK DOWN UPON IT

ON THE XY PLANE, THIS WOULD PRODUCE THE GRAPH

OF OUR LEVEL CURVES OR CONTOUR LINES. SO THE GRAPH OF THE LEVEL CURVES

WOULD LOOK LIKE THIS, AND AGAIN, YOU CAN SEE

THE CIRCLES THAT WERE CREATED BY USING

DIFFERENT VALUES OF Z. LET’S TAKE A LOOK

AT ANOTHER ONE OF THESE. HERE WE HAVE Z

=X SQUARED – Y SQUARED. HERE’S THE GRAPH OF THE SURFACE. IF WE SET Z EQUAL TO A VARIETY

OF CONSTANT VALUES, IT WOULD CREATE

SEVERAL HYPERBOLAS, AS WE SEE HERE IN BLACK, AND YOU CAN THINK OF THIS

AS SLICING THE SURFACE AT DIFFERENT HEIGHTS

BASED UPON THE VALUE OF Z. AND IF WE LOOK DOWN

ON THE XY PLANE, IT WOULD CREATE THIS GRAPH HERE, WHICH IS WHAT WE CALL

OUR CONTOUR LINES OR OUR LEVEL CURVES. LET’S TAKE A LOOK AT IT HERE. AGAIN, HERE IS OUR SURFACE, AND THEN IF WE SET Z EQUAL TO

A VARIETY OF CONSTANTS, IT WOULD CREATE THE HYPERBOLAS

THAT WE SEE HERE IN BLACK. AGAIN, WHAT WE’RE DOING IS

REALLY JUST SLICING THE SURFACE AT DIFFERENT HEIGHTS, AND THEN IF WE LOOK DOWN

ON THE XY PLANE, AS WE SEE HERE, WE WOULD CREATE THE GRAPH

OF THE CONTOUR LINES OR THE LEVEL CURVES. SO HERE ARE THE LEVEL CURVES

IN 3D, AND THEN WE GRAPH THEM

ON THE XY PLANE. IT WOULD LOOK LIKE THIS. LET’S GO AHEAD AND TRY TO CREATE

A CONTOUR MAP OR SET UP LEVEL CURVES

FOR THE GIVEN FUNCTION F OF XY=

X SQUARED + 2Y SQUARED. AND WE’RE GIVEN THAT WE WANT C

EQUAL TO ZERO, TWO, FOUR, SIX AND EIGHT, AND IT’S USUALLY IMPORTANT

TO PICK VALUES OF C WHERE THE DIFFERENCE

IS CONSISTENT. NOTICE HERE WE’RE INCREASING C

BY TWO EACH TIME. SO WE’RE GOING TO HAVE TO CREATE ONE, TWO, THREE, FOUR, FIVE

DIFFERENT LEVEL CURVES TO CREATE THIS CONTOUR MAP. FIRST, WE’RE GOING TO HAVE

C=0. REMEMBER, AND THESE VALUES OF C

REPRESENT THE VARIABLES THAT WE’LL PLACE Z WITH. SO IF THIS IS OUR FUNCTION, IT’S REALLY Z

=X SQUARED + 2Y SQUARED. SO WHEN C=0, WE’RE GOING TO

HAVE 0=X SQUARED + 2Y SQUARED, AND THIS ONE’S NOT TOO EXCITING, BECAUSE THE ONLY SOLUTION

TO THIS IS WHEN X=0 AND Y=0. SO REALLY, IT’S JUST A POINT

AT THE ORIGIN. NOW, WE’LL LET C=2. WHEN C=2, WE’LL HAVE

2=X SQUARED + 2Y SQUARED. THIS IS ACTUALLY AN ELLIPSE. LET’S GO AHEAD

AND SET IT EQUAL TO ONE SO WE CAN DETERMINE

OUR MAJOR AND MINOR AXES. SO WE’D HAVE 1=X SQUARED/2

+ Y SQUARED. I’M GOING TO PUT THIS OVER ONE. SO THIS IS AN ELLIPSE

CENTERED AT ZERO, AND THE MAJOR AXIS

IS HORIZONTAL. WE HAVE “A” SQUARED=2

AND B SQUARED=1. SO “A” WOULD BE THE SQUARED OF

2, WHICH IS APPROXIMATELY 1.4, AND B WOULD BE EQUAL TO ONE. SO WE’RE GOING TO MOVE

TO THE LEFT AND RIGHT OF THE ORIGIN 1.4 UNITS AND ABOVE AND BELOW THE ORIGIN

ONE UNIT TO CREATE THIS ELLIPSE. SO 1.4 IS APPROXIMATELY HERE,

-1.4 IS APPROXIMATELY HERE, AND WE GO UP ONE AND DOWN ONE. SO NOW, WE CAN CREATE THE LEVEL

CURVE WHEN C=2. ALL RIGHT,

NOW, WE’LL JUST KEEP DOING THIS FOR THESE ADDITIONAL VALUES

OF C. SO NEXT, WE HAVE C=4. SO WE’RE GOING TO HAVE 4

=X SQUARED + 2Y SQUARED. AGAIN, IT’S AN ELLIPSE.

WE’LL DIVIDE EVERYTHING BY FOUR. SO WE’RE GOING TO HAVE 1

=X WOULD BE Y SQUARED/2. SO AGAIN, WE HAVE A HORIZONTAL

MAJOR AXIS, BUT NOW “A”=2, AND B WOULD BE EQUAL

TO THE SQUARE ROOT OF 2, AGAIN,

WHICH IS APPROXIMATELY 1.4. LET’S GO AHEAD

AND GRAPH THIS ELLIPSE. SO NOW, WE’LL GO RIGHT

AND LEFT TWO UNITS, UP AND DOWN 1.4 UNITS, AND NOW, WE’LL SKETCH

THE SECOND LEVEL CURVE. AND NOW,

C IS GOING TO EQUAL SIX, SO WE’LL HAVE

6=X SQUARED + 2Y SQUARED. DIVIDE BY SIX, SO WE’LL HAVE 1

=X SQUARED/6 + Y SQUARED/3. SO NOW, WE HAVE “A”

=THE SQUARE ROOT OF 6, WHICH IS APPROXIMATELY 2.4, AND B WOULD BE THE SQUARE ROOT

OF 3, WHICH IS APPROXIMATELY 1.7. SO NOW, WE’LL MOVE LEFT AND

RIGHT APPROXIMATELY 2.4 UNITS AND UP AND DOWN APPROXIMATELY

1.7 UNITS AND SKETCH OUR ELLIPSE, AND WE HAVE ONE MORE. NOW, WE NEED TO CONSIDER

WHEN C=8. LET’S SEE IF WE CAN SQUEEZE IT

IN DOWN HERE AT THE BOTTOM. IF C=8,

THAT MEANS Z WILL BE 8. SO WE’LL HAVE

8=X SQUARED + 2Y SQUARED. DIVIDE BY EIGHT. IT’S GOING TO GIVE US 1=X

SQUARED/8 + Y SQUARED/4. SO “A” IS THE SQUARE ROOT OF 8,

WHICH IS APPROXIMATELY 2.8. B=SQUARE ROOT OF 4 OR B=2. SO TO CREATE

THE LAST LEVEL CURVE, WE’LL GO LEFT AND RIGHT 2.8

UNITS AND UP AND DOWN 2 UNITS, AND HERE’S OUR LAST LEVEL

CURVE. LET’S GO AHEAD AND TAKE A LOOK

AT THIS USING SOME SOFTWARE. THIS WOULD BE THE GRAPH

OF THE ORIGINAL SURFACE. YOU CAN SEE

IT’S AN ELLIPTICAL PARABOLOID. HERE WE SEE THE LEVEL CURVES

IN 3D THAT WE’VE CREATED WITH THE DIFFERENT VALUES OF C. IF WE ROTATE THIS

TO LOOK ONLY AT THE XY PLANE, WE CAN SEE THE LEVEL CURVES

THAT WE’VE CREATED. I HOPE YOU FOUND THIS VIDEO

HELPFUL.

Great video. Thanks. 🙂

Great! Thanks so much for posting this!

this 10 min vid just taught me what i learned in lectures for a month

You = Win

I love youuuu, this is the only video that i found that actually explains how you find the x and y intercepts! Thank you

How did you come up with the a and b values?

@goddetective The standard form of an ellipse: x/a^2 + y/b^2 = 1 when centered at the origin.

good explanation!

Thank you very much!

MY DUDE!you are beast!!!

awesome. is there a part 2 with more examples?

4:05

Thank you sooo much ! This is just what I needed. 3D models really make it easier to visualize and understand !

absolutely fantastic. thank you.

very nicely explained. keep up the good work !

Great explanation! Thanks a lot.

awesome demonstration

thanks

THANK YOU! 🙂 btw, i was stressed about my math final and the quote at the end of this video completely helped me 🙂

Very good video 🙂

thank you very much. This is helpful to me when reading the "convex optimization".

man you're good at those scetches. i dunno how yo udo it on the computer!

VERY HELPFUL! thanks a lot!

That was helpful. Thank You….

Your videos are my new favorite thing. Sending them to everyone that I know. Thank you so much. I hope to join you in online education in the future!

That is great! Thank you

truly enjoyed your video! thank you!

Thanks for the video, I don't know whether this goes without saying but is it common knowledge to let x=0 to find y when z=c and vise versa for x?

Excellent! I am using the the textbook Calculus for Scientists and Engineers, and the authors' explanation of drawing level curves with an ellipse was fantastically terrible. Thank you for contributing!

Love the arthur ashe quote!

Thank you!!!!!!!!!!!!

you made my day!

This is very useful:) thank you so much

😀 I found this video very helpful 😀 thank you so much sir.

You are genius)Thank you so much)*

You are soo much helpfull, thank you really really much!! 🙂

Thanks a lot!

James is an intelligent individual- THANK YOU!

Thank you sir

yes very helpful 🙂 Thx

You are a life saver… Thank you so much for your videos

@Mathispower4u May I ask, what type of software were you using at 4:10? Thank you

thank you very much dude . very helpful

Thank you so much. How did you know it was an ellipse so quickly, and can you always find the a & b values by set the equation equal to 1 and square rooting the denominator?

thx

when creating the 3D graph from the level curve (the example problem you did of the elliptical parabola) how do you know which way the function extends (up or down)? because you're only given 2 variables… not 3.

@john Barre if I may try to answer, you are given two variables and finding the third. Z depends on X and Y so there are three variables, technically. The up or down depends on the function. Since they are x^2 and y^2 there can never be a negative Z because any number squared will be positive. so this will always go in positive z axis direction. if you are talking about the x^2+2y^2=z equation.

This video helped me understand so much about level curves in a single video, thank you good sir, hope you are doing well.

Thank you sir

nice quote

WTF is a and b??

Dafaq is a and b?

Merci beaucoup!

My professor spent the whole lecture period trying to teach us this and you taught me it in like 5 minutes! thank you soooo much!

Thank you.

This is the first video that explained this in detail.

However, I am confused as to how the (a) and (b)'s you were getting out of the equations.

Very helpful. Thank you

I found it very helpful, thank you for explaining 😀

thanku

thanku

good

good

Thanks this was really helpful!

thank you

Thank you sir

Thanks mate, very helpful indeed!

Thank you! All of your videos help me a lot!

The level curve is convex so what does it basically tells us?

showing it with a graph is really helpful.! ty

thnx

great effort, really appreciated for this video and especially for the quotation at the end

Tq..

Actually very very helpful.Thank u

The video is so so helpful. Thank you so much!!

can someone tell me what the software is used in the display? that one he rotated.

that made a lot of sense, thankyou 🙂

Very nice explanation Sir. It is helpful for the Graduate students to understand the concept of the Level curves.