Examples 12.1, 12.2	Examples 12.3, 12.4, 12.5	Examples 12.6
Examples 12.7	Examples 12.8	Examples 12.9

If it is not already on your hard drive, you will need to download the free DPGraph Viewer to view some of the pictures linked to on this site.

QuickTime free download.

 

	Notes and links for edition 8
	Just for fun--A Bouncing Ball.
	Here are some terrific class notes for Calculus III from Paul Dawkins at Lamar University.
	Here is a 3-D parametric grapher found by Denis Untilov.
	Traces Animes  A really nice set of 2D and 3D WIMS graphing tools by XIAO Gang which will produce pictures and graphs that can then be saved and printed or used on a web site.
	Three D Graphing Utility and other math resources including an Excel three D grapher.
	A visually appealing 3D Graphing Utility but you are limited in the types of graphs you can create.
	Allegany College of Maryland has a Calculus III course online.  Unit 3 is on Functions of Several Variables.  Click on Unit 3 at the top of the page when you get there.
	Online Calculus III  from the Springfield Technical Community College distance learning project is another place you might find help.
	Hotmath  You can look at solutions to problems in exercise sets from a wide variety of mathematics textbooks including Calculus by Larson, Hostetler, Edwards, sixth , seventh, and eighth editions.  They have chapters 1 - 12 available.  Only a few solutions are still free (solutions to problems 15, 25, 35 in each section are free).
	DPGraph animation of a spout and a ball and a spout and two semi-transparent balls and birth.
	DPGraph animations:  Bubble and Stars and Cylinder Fun (check out z-slice from above) and
	Filling a Paraboloid and another Paraboloid Fill and one more Filling of a Paraboloid
	Examples All
	Lake Tahoe CC Examples
	Maple Worksheet on Partial Derivatives

12.1  Multivariable Functions, Surfaces, and Contours:  You should be familiar with the surfaces described back in section 10.6 including surfaces of revolution, quadric surfaces, and cylinders.  Quadric Surfaces has some nice animations of some of the surfaces described in 10.6.  Here is a DPGraph animation of a surface deforming into a variety of quadric surfaces.  You will need to download the free DPGraph Viewer to view it.  Here are some DPGraph pictures of a cylinder, a surface of revolution, and Quadric surfaces:  Cylinder, Surface of Revolution, Surface of Revolution 2, Ellipsoid, Hyperboloid(both), Elliptic Cone, Paraboloid (both). 

You need to be able to find the domain and range of a function of two variables, z = f(x,y), and also to be able to graph the domain (P846: 17-27).  The domain of the indicated function is pictured at the right.  Click here for a DPGraph of the function.

 

You need to be able to find and graph a level curve for a function of two variables, z = f(x,y), for a given z = c (P847: 51-56).  Click here or on the picture at the right to view a DPGraph of z = ay2 - bx2 and z = c.  The initial values in the DPGraph picture are a = 1, b = 1, and c = 1.  Use the scrollbar to vary c and observe level curves.  You can also use the scrollbar to vary a and b and to look at x-, y-, and z-slices.  A contour plot for the graph of the function is pictured below.

Here is another DPGraph level curve demonstration that I will go over in class.  One application of level curves involves the construction of contour maps.  Given a surface with varying heights along it, level curves (curves along which the height on the surface is the same) can be constructed.  Sometimes we begin with a contour map in which regions of approximately the same height are shaded the same color and construct from this a representation of the surface.  One example would be this shaded relief map of Arizona.  You can find the color scheme for this map here.  You should also be familiar with traces in planes parallel to one of the coordinate planes.  

To the right is a NASA topographic map of Mars (sinusoidal projection--you can describe what this means for EC--Terri Schein did and included in her research is this nice web site on cylindrical map projections).  Click on the thumbnail picture on the right to enlarge.

	Click this link for a Viking 2 panorama of the surface of Mars.
	U.S. Geological Survey Flagstaff Field Center shaded relief maps of states
	Permian Basin Including Texas-New Mexico from the Geological Data Services
	UNISYS Current Temperature Contour Image
	Daily UV Index Contour Map from the U.S. Environmental Protection Agency.
	Maps relating to the environment from Discover
	Relief map of Florida found by Michael Chase
	1998 Florida Wild Fires also from MC

The figure on the left shows the level curves for f(x,y) = 64 - x2 - 2y2.  The level curves correspond to c = 0 (red), c = 8 (green), c = 16, 24, 32, 40, 48, 56, c = 63 (yellow), and c = 63.9 (black).  Click on the figure to see an animation.  Below the level curve drawing is a Maple generated contour plot for the function.  In the animation you will see a level curve in red changing as you move "up" the surface, i.e., as z = c increases.  You will also see an animated point and vector in black moving with the animated red level curve.  The length of this black vector gives the relative speed at which the black point on the animated red level curve is moving along the x-axis.  One can observe that the speed of the black point is increasing as c increases.  One can also observe that the distance between the level curves is increasing as c increases when the change in c is constant (as it is from red to orange).  This indicates that the surface is becoming less "steep".  In fact at the origin we are directly under a relative maximum point where the tangent plane would be horizontal.  Click here to see an animation with c going from 32 to 64 with a step size of 4 and click here to see an animation with c going from 48 to 64 with a step size of 2.  Here is a DPGraph picture of the surface along with the horizontal plane whose equation is z = b.  In the graph each unit on the z-axis represents 4 units so z = 4 means z = 16.  Use the scrollbar to vary b from 0 to 16 (i.e., z from 0 to 64) to see the level curves corresponding to the curve of intersection of the graph of  f(x,y) and the horizontal plane.

12.2  You should know the definition of the limit of a function of two variables and the definition of continuity of a function of two variables (and the extensions to functions of three variables).  Here are exercises on limits and continuity with solutions to some of the problems (no solutions yet for these).  Be able to compute limits by direct substitution and also be able to show that a particular limit does not exist because the limit is different over two different paths (P857: 21-24) in a case where direct substitution yields an indeterminate form (or show the limit does not exist over a particular path).

12.3  Know the definition of partial derivative and be able to compute a "simple" partial derivative by computing the appropriate limit (P865: 29).  Be able to apply the appropriate differentiation formulas to compute partial derivatives.  Here are some exercises on partial derivatives with solutions to some of the problems.  Be able to find the slope at a point on a surface in the x- and y-directions (P865: 33-36).  Be able to demonstrate the equality of mixed partials for a function satisfying the necessary continuity requirements (P866: 69-72).  Here is an example of a function whose mixed partials are NOT equal.  One use of calculus relating to surfaces involves showing that essentially parallel rays of light striking a parabolic surface are directed through the focus (or rays of light emanating from the focus will be parallel after reflecting off the parabolic reflector).  For EXTRA CREDIT use Theorem 9.2, Reflective Property of a Parabola, on page 652 in your text to prove this.  Follow this link to learn more about liquid mirror telescopes.  They use this property of parabolic reflectors.

12.4  Be able to compute a total differential for a given function of more than one variable (P874: 1-10).  Understand the definition of differentiability, that differentiability implies continuity, and that the existence of partial derivatives does not guarantee differentiability (see example 5, P873) or a similar example.

12.5  Know and be able to apply the chain rules for functions of several variables including their use in differentiating implicitly (P882: 39-42).  Be able to apply the chain rule to find a rate of change in a problem similar to P883: 52.  Here are some exercises on the chain rule with solutions to some of the problems.

Summary of main differentiation formulas 12.3-12.5     Powerpoint Presentation

12.6  Be able to find the directional derivative of a given function at a point P in the direction of Q (or in the direction of a vector u) (P893: 1-20).  Here are some of my notes on the directional derivative.  Be able to find the gradient of a given function and the maximum and minimum value of its directional derivative at an indicated point (P893: 31-38).  Here is a link to a nice Gradient Visualization with an interactive Java applet demo under Normal to Level Curves on the left.  A nice exploration involving the gradient and directional derivative is Hiking and Climbing in Yosemite.  Be able to use the directional derivative in an application problem (P895: 73-74, P903: 53-54).  An example would be to find and graph the path followed by a heat-seeking object starting at a given point on a surface or in a region where the temperature field is given by T(x,y) or T(x,y,z).  Try to find the path followed by a heat seeking particle starting at the point (2,4) on a surface whose temperature field is given by T(x,y) = 100 - x² - 2y².  Click here to see the graph of the solution along with an animated vector moving along the solution in the direction of motion and click here to see the animation along with the solution plus the solution to a temperature distribution problem involving T(x,y,z).  In solving this type of problem you will need to solve some ordinary differential equations (ODES) by the method of separation of variables.  SOSMath Separable ODE Tutorial  To have a better sense of what the solution to a first order ODE is take a look at this First Order DE Solution Grapher.  The Vanderbilt DE Toolkit can solve a variety of differential equations.  Here is another example with a nice animation involving a nutrient seeking being (little orange ball). 

Powerpoint Presentation on the Relationship Between the Gradient and Level Curves 

 

12.7  Be able to find an equation for the tangent plane and symmetric equations (or parametric) for the normal line to a given surface at an indicated point (P903: 29-34).  Here are some of my notes on normal lines and tangent planes.  Click here for some nice tangent plane animations.  Here are some exercises on tangent planes and differentials with solutions to some of the problems.  Be able to find the point(s) on a surface where the tangent plane is horizontal (parallel to the xy-coordinate plane) (P903: 51-52).  The picture on the right shows part of the graph of  z = f(x,y) = 9 - x2 - y2   and the plane tangent to the surface at (1,1,7).  Click on the picture to see an animation.  Write the equation of the surface as F(x,y,z) = x2 + y2 + z - 9 = 0 grad F = < 2x , 2y , 1 > grad F(1,1,7) = n = < 2, 2 , 1 > so the equation of the tangent plane will be 2x + 2y + z = d    and substituting in (1,1,7) yields d = 11.        2x + 2y + z = 11

Click here to zoom in on the point of tangency.

 

Pictured at the right is the graph of over the indicated window.     The function is undefined if x = 0 and y = 0 and there is no tangent plane corresponding to x = 0 and y = 0.  Furthermore, zooming in on the point (0,0,0) does not create a "flattening" effect the way it does on any point where there is a tangent plane.  Try changing the window settings on this DPGraph picture of the surface to [-2,2,-2,2,-2,2] and then to [-1,1,-1,1,-1,1] and then to [-0.1,0.1,-0.1,0.1,-0.1,0.1] to see for yourself.  Then change the window to [0.9,1.1,0.9,1.1,0.4,0.6] and see what happens.   Why would changing the window to [0.9,1.1,0.9,1.1,0.9,1.1] not be very rewarding?

12.8  Be able to find any relative minimum points, relative maximum points, and saddle points (here is a manipulable saddle point animation) on the graph of a given function of two variables, z = f(x,y), and identify what kind of point each is (P911: 7-28).  Here is a link to a proof and discussion of the Second Partials Test.  Click here to see a DPGraph drawing of a paraboloid that can be transformed back and forth between an elliptic paraboloid (relative minimum point demo) and a hyperbolic paraboloid (saddle point demo).  Here is an example with a picture of the graph of the surface and also a contour plot.  Be able to find the absolute maximum and absolute minimum values of a given function over a region R, identify the corresponding points on the surface, and include a demonstration of your evaluation along each boundary of R (P912: 53-62).  Click here to see a worked out example involving finding points on a surface where the tangent plane is horizontal, using the Second Partials Test to identify the points as relative maximum points, relative minimum points, or saddle points, and finding the absolute maximum and absolute minimum value of a continuous function over a closed region of the xy-coordinate plane.

12.9  Applied Optimization Problems:  Be able to do optimization problems similar to those done in class and assigned for homework.  Click here to see two examples of optimization problems from Section 12.9 (12.9 #3 and a variation on 12.9 #10).  Be sure to understand how to apply the optimization theory in section 12.9 to least squares regression analysis.  Check out this ExploreMath.com animated demo of linear regression.  Here are some examples of applied optimization problems.  Here is a tutorial on regression analysis that includes a regression utility and here is a polynomial regression utility.

Here is another linear regression demo and a scatter plot applet which also demonstrates linear regression.

Click here to see how you could use polynomial regression and a surface of revolution to model a goblet (and other things).  There is an Extra Credit opportunity at the end of my demonstration.

TAKE-HOME PROBLEMS

These problems may be done in advance at home and must be turned in at the end of the exam on exam day.

1.  For the data points (1,1), (2,3), (3,6), (4,7), and (5,9) find the best fit linear function (y = ax + b) based on a least squares criteria.  Show the system of equations to be solved in finding a and b by setting S_a and S_b each equal to zero.  S(a,b) is the function giving the sum of the squares of the errors.  Make sure you follow the instructions for this problem given in class.

2.  Find the box of largest volume that can be inscribed in the ellipsoid

 

3.  In the problem pictured below, determine the values for x and y that will minimize total construction costs.  The idea is to lay pipe from point P to Point Q.  It costs 3 million dollars per mile to lay the pipe through the blue area, 2 million dollars per mile to lay the pipe through the green area, and 1 million dollars per mile to lay the pipe along the boundary between the green area and the brown area.  Consider the colored regions to be rectangles, x to represent the horizontal distance for the pipe in the blue region and y to be the horizontal distance for the pipe in the green region.  The blue region and the green region are each 1 mile wide and the horizontal distance from P to Q is 5 miles.  You must thoroughly investigate the costs on the boundary of the region over which you would be applying the cost function.  You may assume you would not minimize total cost by laying pipe in the negative x direction, negative y direction, or by laying pipe past point Q and then coming back.  Thus this region (see the bottom picture on the right) would be described by  Click on the top picture at the right to see an animation of some of the possible paths.

EXTRA CREDIT PROBLEM

At time t = 0 one plane is directly above a second plane at an altitude of 1 mile.  The second plane is at an altitude of 1/2 mile.  The first plane is flying along a somewhat elliptical path described by  r(t) = < 4 - 4cos(t) , 3sin(t) , 1 + t/(2pi) >  and the second plane is flying along a hyperbolic path described by  r(t) = < (8/31/2)tan(t/3) , 0 , -1/2 + sec(t/3) >.   These do represent the actual position functions for the airplanes with distance in miles and time in minutes.  In how many minutes after t = 0 will the planes collide?  Approximate the distance between the planes and the rate of change of the distance between the planes 30 seconds before they collide.  Approximate the rate of change of the distance between the planes at the instant just before they collide. Animation with scales Animation without scales Animation without scales and with rotation

Top of the page

          This site contains links to other Internet sites.  These links are not endorsements of any products or services in such sites, and no information  

           in such site has been endorsed or approved by this site.

        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats