Assignment
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
2
Academic year
2022
Knotch
Views
30
Visualizing surfaces in three dimensions. Graphing in three dimensions, an example So, let's say we have a more complicated function, how do we see it? We will examine the graph of f(x, y) = 1 − x2 − y2.First, we set x = 0; when x = 0, y = 1. Then we evaluate f at (0, 1). This yields f(0) =1.Next, we set y = 0; when y = 0, x = −1. Then we evaluate f at (−1, 0). This yieldsf(−1) = −1. Finally, since f(−1) < 0 and f(0) > 0, the surface z = 1 − x2 − y2 is concaveupward in the plane consisting of points (x, y) with both x and y greater than −1 andless than or equal to zero. Then z becomes 1 minus y squared. This is a parabola pointing downward andstopping at 1. So we should draw maybe this downward parabola. It starts at 1, andit cuts the y-axis at 1. When y is 1, that gives us 0. So we might have an idea of what it might look like, or maybe not. Let's get moreslices—the horizontal ones—and see what happens in the xz-plane, which is avertical plane that's coming toward us.So in the xz- plane, which we set y equal to 0 and we get z equals 1 minus xsquared, it's again a parabola coming downward. So we are going to draw aparabola that goes downward and also forward and backward. So we're starting tohave a slightly better idea but we still don't know whether a cross-section of this thingmight be round, square, something else. So if we want more confirmation we mightwant to figure out maybe where this surface intersects the xy- plane?
So we hit the xy-plane when z equals 0. That means 1 minus x squared minus ysquared should be 0. That becomes x squared plus y squared equals 1. That is acircle of radius 1. That's the unit circle.We can now imagine that we slice the unit circle by a vertical plane, producing adownward parabola that is actually the surface of revolution for a given ratio ofdiameter to radius. One way to determine the graph is to guess what it looks like, and then plug thevalues into an equation. Another way is to ask your computer to solve for the pointson the graph.
Visualizing Surfaces in Three Dimensions
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