Calculus Concepts 2/e LaTorre, Kenelly, Fetta, Harris, and Carpenter Chapter 9 Ingredients of Multivariable Change: Models, Graphs, Rates Copyright © by Houghton Mifflin Company, All rights reserved.

Copyright © by Houghton Mifflin Company, All rights reserved. Chapter 9 Key Concepts Multivariable Functions Cross-Sectional Models Contour Graphs Partial Rates of Change Slopes of Contour Curves Copyright © by Houghton Mifflin Company, All rights reserved.

Multivariable Functions A function with two or more input variables and one output variable Example: Volume of box = length x width x height V(l, w, h) = lwh Example Area of a rectangle = length x width A(l,w) = lw Copyright © by Houghton Mifflin Company, All rights reserved.

Multivariable Functions: Example The accumulated value, A, of an investment of P dollars at interest rate 100r% compounded n times a year after t years is a multivariable function given by Write the equation of the multivariable function that gives the future value of a $P investment compounded daily at an annual rate of 6% after t years. Copyright © by Houghton Mifflin Company, All rights reserved.

Multivariable Functions: Exercise 9.1 #7 Let P(c, s) be the profit in dollars from the sale of a yard of fabric when c dollars is the production cost per yard and s dollars is the sales price per yard. Interpret the following: a. P(1.2, s) c. P(1.2, 4.5) = 3.0 a. P(1.2, s) = the profit when the production cost is $1.20 per yard and the sales price is s dollars per yard. c. P(1.2, 4.5) = 3.0 means the profit is $3.00 per yard when the production cost is $1.20 per yard and the sales price is $4.50 per yard. Copyright © by Houghton Mifflin Company, All rights reserved.

Cross-Sectional Models Partially model data by holding one input variable constant and finding an equation in terms of the other input variable The cross-section of multivariable function with two input variables is the curve that results when a three-dimensional graph is intersected with a plane Copyright © by Houghton Mifflin Company, All rights reserved.

Cross-Sectional Models: Example Both figures show the elevation of a piece of land in relation the the western and southern fences. The cross-section of the elevation function E(e,n) when e = 0.8 miles is a single variable function E(0.8,n). Copyright © by Houghton Mifflin Company, All rights reserved.

Cross-Sectional Models: Example The tables show the elevation, E, of the land 0.8 miles east of the western fence as a function of the distance from the southern fence, n. E n 799.5 799.7 799.9 800.0 800.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 E n 799.2 798.9 798.5 798.1 797.6 1.1 1.2 1.3 1.4 1.5 Using the table data, we can model the elevation in feet above sea level on the farmland measured 0.8 miles east of the western fence as E(0.8, n) = -2.5n2 + 2.497n + 799.490 feet above sea level n miles north of the southern fence. Copyright © by Houghton Mifflin Company, All rights reserved.

Cross-Sectional Models: Example The graph of E(0.8, n) = -2.5n2 + 2.497n + 799.490 is the parabola shown below. It models the elevation of the land 0.8 miles east of the western fence. Copyright © by Houghton Mifflin Company, All rights reserved.

Cross-Sectional Model: Exercise 9.1 #15 The table shows the average yearly consumption of peaches per person based on the price of peaches and the yearly income of the person’s family. Find a cross sectional model for a yearly income of $40,000. 1 2 3 4 5 6 Yearly Income ($10K) 5.0 6.4 7.2 7.8 8.2 8.6 4.8 6.2 7.0 7.6 8.0 8.4 4.9 6.3 7.1 7.7 8.1 8.5 4.7 6.1 6.9 7.5 8.3 7.4 7.9 4.6 6.0 6.8 0.10 0.20 0.30 0.40 0.50 Price per pound above $1.50 Copyright © by Houghton Mifflin Company, All rights reserved.

Cross-Sectional Model: Exercise 9.1 #15 p C The function C(p, i) models the per capita consumption of peaches where the price of peaches is $1.50 + p per pound and the person lives in a family with per capita income $10,000i. The table shows the value of C when i = 40. 0.10 0.20 0.30 0.40 0.50 7.8 7.7 7.6 7.5 7.4 A cross-sectional model for a family earning $40,000 annually is C(p, 40) = 0.893p2 -1.304p + 7.811 pounds per person per year. Copyright © by Houghton Mifflin Company, All rights reserved.

Copyright © by Houghton Mifflin Company, All rights reserved. Contour Graphs Used to represent graphs of multivariable functions Equal and adjacent table values linked with smooth curve Also called topographical map Copyright © by Houghton Mifflin Company, All rights reserved.

Contour Graphs: Example The blue plane represents the 800-foot elevation level. In the gray region against the western fence and in the egg-shaped region, the elevation is above 800-feet. Copyright © by Houghton Mifflin Company, All rights reserved.

Contour Graphs: Example The contour lines on the table are drawn at the 796, 797, 798, 799, 800, 801, and 802-foot levels. Copyright © by Houghton Mifflin Company, All rights reserved.

Contour Graphs: Exercise 9.2 #1 The table shows the apparent temperature for a given temperature and relative humidity. Draw contour curves for apparent temperatures of 90°F, 105°F, and 130°F. Copyright © by Houghton Mifflin Company, All rights reserved.

Contour Graphs: Exercise 9.2 #1 Copyright © by Houghton Mifflin Company, All rights reserved.

Partial Rates of Change The partial derivative of a multivariable function G(x,y) represents the rate of change of a cross-sectional function. is the partial derivative of G with respect to x Gx is the partial derivative of G with respect to x is the partial derivative of G with respect to y Gy is the partial derivative of G with respect to x Copyright © by Houghton Mifflin Company, All rights reserved.

Partial Rates of Change: Example To find treat x as a variable and y as a constant To find treat y as a variable and x as a constant Copyright © by Houghton Mifflin Company, All rights reserved.

Partial Rates of Change The second partial derivative of a multivariable function G(x,y) represents the rate of change of a partial derivative function Gxx is the partial derivative of G with respect to x then x Gxy is the partial derivative of G with respect to x then y Gyy is the partial derivative of G with respect to y then y Gyx is the partial derivative of G with respect to y then x Copyright © by Houghton Mifflin Company, All rights reserved.

Partial Rates of Change: Example Copyright © by Houghton Mifflin Company, All rights reserved.

Partial Rate of Change: Exercise 9.3 #11 Find Mt, Ms, and Ms|t=3 given Copyright © by Houghton Mifflin Company, All rights reserved.

Slopes of Contour Curves When a function of two variables z = f(x,y) is held constant at a value c, the slope at any point on the contour curve f(x,y) = c (that is, the slope of the line tangent to the c contour curve) is given by Copyright © by Houghton Mifflin Company, All rights reserved.

Slopes of Contour Curves In order to compensate for a small change x in x to keep f(x,y) constant at c, y must change by approximately Copyright © by Houghton Mifflin Company, All rights reserved.

Slopes of Contour Curves: Example Your body-mass index is a measurement of how thin you are compared to your height. A person’s body mass index is given by where h is your height in inches and w is your weight in pounds. Find for a 67-inch, 129-pound teenager. Copyright © by Houghton Mifflin Company, All rights reserved.

Slopes of Contour Curves: Example At w = 129 and h = 67, Bh  -0.60298 point per inch and Bw  0.15662 point per pound. Copyright © by Houghton Mifflin Company, All rights reserved.

Slopes of Contour Curves: Exercise 9.4 #5 Copyright © by Houghton Mifflin Company, All rights reserved.