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Copyright © by Houghton Mifflin Company, All rights reserved. Calculus Concepts 2/e LaTorre, Kenelly, Fetta, Harris, and Carpenter Chapter 9 Ingredients.

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Presentation on theme: "Copyright © by Houghton Mifflin Company, All rights reserved. Calculus Concepts 2/e LaTorre, Kenelly, Fetta, Harris, and Carpenter Chapter 9 Ingredients."— Presentation transcript:

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

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

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

4 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.

5 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.

6 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 variablePartially 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 planeThe cross-section of multivariable function with two input variables is the curve that results when a three-dimensional graph is intersected with a plane

7 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).

8 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 E n 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) = -2.5n n feet above sea level n miles north of the southern fence.

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

10 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 persons family. Find a cross sectional model for a yearly income of $40, Yearly Income ($10K) Price per pound above $1.50

11 Copyright © by Houghton Mifflin Company, All rights reserved. Cross-Sectional Model: Exercise 9.1 #15 The function C(p, i) models the per capita consumption of peaches where the price of peaches is $ 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. C p A cross-sectional model for a family earning $40,000 annually is C(p, 40) = 0.893p p pounds per person per year.

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

13 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.

14 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.

15 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.

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

17 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.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 is the partial derivative of G with respect to y GxGxGxGx is the partial derivative of G with respect to x GyGyGyGy

18 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

19 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 functionThe second partial derivative of a multivariable function G(x,y) represents the rate of change of a partial derivative function G xy is the partial derivative of G with respect to x then y G yx is the partial derivative of G with respect to y then x G xx is the partial derivative of G with respect to x then x G yy is the partial derivative of G with respect to y then y

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

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

22 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 byWhen 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

23 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 approximatelyIn order to compensate for a small change x in x to keep f(x,y) constant at c, y must change by approximately

24 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 persons 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.

25 Copyright © by Houghton Mifflin Company, All rights reserved. Slopes of Contour Curves: Example At w = 129 and h = 67, B h point per inch and B w point per pound.

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


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