- 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.