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## Lecture 4 MA 261

by: Viktoryia Zhuleva

108

4

4

# Lecture 4 MA 261 MA 26100

Viktoryia Zhuleva
Purdue
GPA 3.0

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Lecture 4 notes with examples (plus graphs)
COURSE
Multivariable Calculus
PROF.
Patricia E Bauman
TYPE
Class Notes
PAGES
4
WORDS
CONCEPTS
Math, Calculus, AP Calculus BC, Purdue
KARMA
Free

## Popular in Applied Mathematics

This 4 page Class Notes was uploaded by Viktoryia Zhuleva on Sunday January 24, 2016. The Class Notes belongs to MA 26100 at Purdue University taught by Patricia E Bauman in Fall 2015. Since its upload, it has received 108 views. For similar materials see Multivariable Calculus in Applied Mathematics at Purdue University.

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Date Created: 01/24/16
Lecture 4: Recall: a trace is the intersection of the surface with one of the coordinate planes x=k, y=k and z=k. Traces will be used to graph quadratic surfaces in 3-dimensional space. How to graph in 3-D. Step 1: Identify the general traces x=k g=k, z=k and select one family that is easy to draw. Step 2: Compute the k=0 traces and graphs. Sketch each of the graphs in 3D. Step 3: Fill in the "outline" with graphs and traces from Step 1. 2 2 Example: Use traces to sketch graph z=x +4y 1) Create general traces x=k: z=k +4y (parabola). 2 2 y=k: z=x +4k (parabola) z=k: k=x +4y (ellipse) So the name of the shape is elliptic paraboloid. 2) Make traces --> set k=0 for each case. x=0: z=4y 2 2 y=0: z=x z=0: x +4y =0 (here, at the origin it is just going to be a dot, because x=0 and y=0; later on, when it grows up along z-axis, it will take a shape of an ellipce) 3) Graph all the traces in 3D. (Make sure to consider points of intersections of separate traces on one 3D graph) 2 2 Example: Graph z=2y -x 1) Set up k-constants for each two-dimensional space: 2 2 a. x=k  z=2y -k - parabola b. y=k  z=2k -x – parabola c. z=k  k=2y -x – hyperbola (recall general formulas for hyperbola and ellipse here to stop confusing these two shapes) 2) Create k=0 traces a. z=y 2 b. z=-x 2 c. 2y -x =0  2y =x  ±√2y=x 3) Sketch the traces: Standard Graphs of Quadratic Surfaces Name Function Picture Ellipsoid ????2 ????2 ????2 2 + 2+ 2 = 1 ???? ???? ???? Elliptic Paraboloid ????2 ????2 ???? 2+ 2= ???? ???? ???? Hyperbolic Paraboloid ????2 ????2 ???? 2− 2= ???? ???? ???? Elliptic Double Cone ????2 ???? 2 ????2 + = ????2 ????2 ????2 2 2 2 Hyperboloid of One Sheet ???? + ???? − ???? = 1 ????2 ????2 ????2 2 2 2 Hyperboloid of Two Sheets − ???? − ???? + ???? = 1 ????2 ????2 ????2

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