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

Copyright © by Houghton Mifflin Company, All rights reserved.Copyright © by Houghton Mifflin Company, All rights reserved.

Calculus Concepts 2/eCalculus Concepts 2/eLaTorre, Kenelly, Fetta, Harris, and CarpenterLaTorre, Kenelly, Fetta, Harris, and Carpenter

Chapter 9Chapter 9Ingredients of Multivariable Change:Ingredients of Multivariable Change:

Models, Graphs, RatesModels, Graphs, Rates


Chapter 9 Key ConceptsChapter 9 Key Concepts• Multivariable FunctionsMultivariable Functions

• Cross-Sectional ModelsCross-Sectional Models

• Contour GraphsContour Graphs

• Partial Rates of ChangePartial Rates of Change

• Slopes of Contour CurvesSlopes of Contour Curves


Multivariable FunctionsMultivariable Functions• A function with two or more input variables A function with two or more input variables

and one output variableand one output variable– Example:Example:

• Volume of box = length x width x height Volume of box = length x width x height • V(l, w, h) = lwhV(l, w, h) = lwh

– ExampleExample• Area of a rectangle = length x widthArea of a rectangle = length x width• A(l,w) = lwA(l,w) = lw


Multivariable Functions: ExampleMultivariable Functions: ExampleThe accumulated value, A, of an investment of P The accumulated value, A, of an investment of P dollars at interest rate 100r% compounded n times a dollars at interest rate 100r% compounded n times a year after t years is a multivariable function given byyear after t years is a multivariable function given by

nt

n

r1P)t,n,r,P(A

nt

n

r1P)t,n,r,P(A

Write the equation of the multivariable function that Write the equation of the multivariable function that gives the future value of a $P investment compounded gives the future value of a $P investment compounded daily at an annual rate of 6% after t years.daily at an annual rate of 6% after t years.

t365

365

06.01P)t,P(A

t365

365

06.01P)t,P(A


Multivariable Functions: Exercise 9.1 #7Multivariable Functions: Exercise 9.1 #7Let P(c, s) be the profit in dollars from the sale of a 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 of fabric when c dollars is the production cost per yard and s dollars is the sales price per yard. Interpret yard and s dollars is the sales price per yard. Interpret the following:the following:a. P(1.2, s)a. P(1.2, s)c. P(1.2, 4.5) = 3.0c. P(1.2, 4.5) = 3.0

a. P(1.2, s) = the profit when the production cost is 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.$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 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 when the production cost is $1.20 per yard and the sales price is $4.50 per yard.sales price is $4.50 per yard.


Cross-Sectional ModelsCross-Sectional Models• Partially model data by holding one input Partially model data by holding one input

variable constant and finding an equation in variable constant and finding an equation in terms of the other input variableterms of the other input variable

• The cross-section of multivariable function The cross-section of multivariable function with two input variables is the curve that with two input variables is the curve that results when a three-dimensional graph is results when a three-dimensional graph is intersected with a planeintersected with a plane


Cross-Sectional Models: ExampleCross-Sectional Models: ExampleBoth figures show the elevation of a piece of land in Both figures show the elevation of a piece of land in relation the the western and southern fences. The relation the the western and southern fences. The cross-section of the elevation function E(e,n) when cross-section of the elevation function E(e,n) when e = 0.8 miles is a single variable function E(0.8,n). e = 0.8 miles is a single variable function E(0.8,n).


Cross-Sectional Models: ExampleCross-Sectional Models: ExampleThe tables show the The tables show the elevation, E, of the land 0.8 elevation, E, of the land 0.8 miles east of the western miles east of the western fence as a function of the fence as a function of the distance from the southern distance from the southern fence, n. fence, n.

En

799.5

799.7

799.9

800.0

800.1

800.1

800.1

800.0

799.9

799.7

799.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

En

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 Using the table data, we can model the elevation in feet above sea level on the elevation in feet above sea level on the farmland measured 0.8 miles east of the farmland measured 0.8 miles east of the western fence aswestern fence as

E(0.8, n) = -2.5nE(0.8, n) = -2.5n22 + 2.497n + 799.490 + 2.497n + 799.490feet above sea level n miles north of the feet above sea level n miles north of the southern fence.southern fence.


Cross-Sectional Models: ExampleCross-Sectional Models: ExampleThe graph of E(0.8, n) = -2.5nThe graph of E(0.8, n) = -2.5n22 + 2.497n + 799.490 + 2.497n + 799.490is the parabola shown below. It models the elevation is the parabola shown below. It models the elevation of the land 0.8 miles east of the western fence.of the land 0.8 miles east of the western fence.


Cross-Sectional Model: Exercise 9.1 #15Cross-Sectional Model: Exercise 9.1 #15The table shows the average yearly consumption of The table shows the average yearly consumption of peaches per person based on the price of peaches and peaches per person based on the price of peaches and the yearly income of the person’s family. Find a cross the yearly income of the person’s family. Find a cross sectional model for a yearly income of $40,000.sectional model for a yearly income of $40,000.

1

2

3

4

5

6

YearlyIncome($10K) 0

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

8.0

8.3

4.7

6.1

6.9

7.4

7.9

8.3

4.6

6.0

6.8

7.4

7.8

8.2

0.10 0.20 0.30 0.40 0.50

Price per pound above $1.50


Cross-Sectional Model: Exercise 9.1 #15Cross-Sectional Model: Exercise 9.1 #15

The function C(p, i) models the per The function C(p, i) models the per capita consumption of peaches where capita consumption of peaches where the price of peaches is $1.50 + p per the price of peaches is $1.50 + p per pound and the person lives in a family pound and the person lives in a family with per capita income $10,000i. The with per capita income $10,000i. The table shows the value of C when i = 40.table shows the value of C when i = 40.

Cp

7.8

7.7

7.6

7.5

7.4

7.4

0

0.10

0.20

0.30

0.40

0.50

A cross-sectional model for a family earning $40,000 A cross-sectional model for a family earning $40,000 annually is C(p, 40) = 0.893pannually is C(p, 40) = 0.893p22 -1.304p + 7.811 -1.304p + 7.811 pounds per person per year.pounds per person per year.


Contour GraphsContour Graphs• Used to represent graphs of multivariable Used to represent graphs of multivariable

functionsfunctions

• Equal and adjacent table values linked with Equal and adjacent table values linked with smooth curvesmooth curve

• Also called topographical mapAlso called topographical map


Contour Graphs: ExampleContour Graphs: ExampleThe blue plane represents the 800-foot elevation level. The blue plane represents the 800-foot elevation level. In the gray region against the western fence and in the In the gray region against the western fence and in the egg-shaped region, the elevation is above 800-feet.egg-shaped region, the elevation is above 800-feet.


Contour Graphs: ExampleContour Graphs: ExampleThe contour lines on the table are drawn at the 796, The contour lines on the table are drawn at the 796, 797, 798, 799, 800, 801, and 802-foot levels.797, 798, 799, 800, 801, and 802-foot levels.


Contour Graphs: Exercise 9.2 #1Contour Graphs: Exercise 9.2 #1The table shows the apparent temperature for a The table shows the apparent temperature for a given temperature and relative humidity. Draw given temperature and relative humidity. Draw contour curves for apparent temperatures of 90°F, contour curves for apparent temperatures of 90°F, 105°F, and 130°F.105°F, and 130°F.


Contour Graphs: Exercise 9.2 #1Contour Graphs: Exercise 9.2 #1


Partial Rates of ChangePartial Rates of Change• The partial derivative of a multivariable The partial derivative of a multivariable

function G(x,y) represents the rate of change function G(x,y) represents the rate of change of a cross-sectional function.of a cross-sectional function.

x

G

x

G

is the partial derivative of G with respect to xis the partial derivative of G with respect to x

y

G

y

G

is the partial derivative of G with respect to yis the partial derivative of G with respect to y

GGxx is the partial derivative of G with respect to xis the partial derivative of G with respect to x

GGyy is the partial derivative of G with respect to xis the partial derivative of G with respect to x


Partial Rates of Change: ExamplePartial Rates of Change: Example

x

G

x

G

To find To find treat x as a variable and y as a constant treat x as a variable and y as a constant

y

G

y

G

To find To find treat y as a variable and x as a constant treat y as a variable and x as a constant

322 yxy24.7x53.2)y,x(G 322 yxy24.7x53.2)y,x(G

2y24.7x06.5x

G

2y24.7x06.5

x

G

22 y3xy48.14y3y2)x24.7(y

G

22 y3xy48.14y3y2)x24.7(

y

G


Partial Rates of ChangePartial Rates of Change• The second partial derivative of a The second partial derivative of a

multivariable function G(x,y) represents the multivariable function G(x,y) represents the rate of change of a partial derivative functionrate of change of a partial derivative function

GGxyxy is the partial derivative of G with respect to x is the partial derivative of G with respect to x then ythen y

GGyxyx is the partial derivative of G with respect to y is the partial derivative of G with respect to y then xthen x

GGxxxx is the partial derivative of G with respect to x is the partial derivative of G with respect to x then xthen x

GGyyyy is the partial derivative of G with respect to y is the partial derivative of G with respect to y then ythen y


Partial Rates of Change: ExamplePartial Rates of Change: Example322 yxy24.7x53.2)y,x(G 322 yxy24.7x53.2)y,x(G

2x y24.7x06.5G 2x y24.7x06.5G

2y y3xy48.14G 2y y3xy48.14G

06.5Gxx 06.5Gxx

y48.14Gxy y48.14Gxy

y48.14Gyx y48.14Gyx

y6x48.14Gyy y6x48.14Gyy


Partial Rate of Change: Exercise 9.3 #11Partial Rate of Change: Exercise 9.3 #11

96.14s75.3)tln(s)s,t(M 96.14s75.3)tln(s)s,t(M Find MFind Mtt, M, Mss, and M, and Mss||t=3t=3 given given

t

s

t

1s)tln(0Mt

t

s

t

1s)tln(0Mt

75.3)tln(

75.30s)tln(1Ms

75.3)tln(

75.30s)tln(1Ms

849.475.3)3ln(M3ts

849.475.3)3ln(M

3ts


Slopes of Contour CurvesSlopes of Contour Curves• When a function of two variables z = f(x,y) is When a function of two variables z = f(x,y) is

held constant at a value c, the slope at any held constant at a value c, the slope at any point on the contour curve f(x,y) = point on the contour curve f(x,y) = cc (that is, (that is, the slope of the line tangent to the the slope of the line tangent to the cc contour contour curve) is given bycurve) is given by

y

x

f

f

yf

xf

dx

dy

y

x

f

f

yf

xf

dx

dy


Slopes of Contour CurvesSlopes of Contour Curves• In order to compensate for a small change In order to compensate for a small change x x

in x to keep f(x,y) constant at in x to keep f(x,y) constant at cc, y must , y must change by approximatelychange by approximately

xΔf

fyΔ,isThat

xΔdx

dyyΔ

y

x

xΔf

fyΔ,isThat

xΔdx

dyyΔ

y

x


Slopes of Contour Curves: ExampleSlopes of Contour Curves: Example Your body-mass index is a measurement of how Your body-mass index is a measurement of how thin you are compared to your height. A person’s thin you are compared to your height. A person’s body mass index is given bybody mass index is given by

sintpoh00064516.0

w4536.0)w,h(B

2 sintpo

h00064516.0

w4536.0)w,h(B

2

where h is your height in inches and w is your where h is your height in inches and w is your weight in pounds. weight in pounds.

dh

dwdh

dwFindFind for a 67-inch, 129-pound teenager. for a 67-inch, 129-pound teenager.


Slopes of Contour Curves: ExampleSlopes of Contour Curves: Example

sintpoh00064516.0

w4536.0)w,h(B

2 sintpo

h00064516.0

w4536.0)w,h(B

2

inchpersintpoh00064516.0

)w4536.0)(2(B

3h

inchpersintpo

h00064516.0

)w4536.0)(2(B

3h

poundpersintpoh00064516.0

4536.0B

2w poundpersintpoh00064516.0

4536.0B

2w

At w = 129 and h = 67, BAt w = 129 and h = 67, Bhh 0.60298 point per inch 0.60298 point per inch

and Band Bww 0.15662 point per pound. 0.15662 point per pound.

inchperpounds3.850.156620.60298

BB

dhdw

w

h inchperpounds3.850.156620.60298

BB

dhdw

w

h


Slopes of Contour Curves: Exercise 9.4 #5Slopes of Contour Curves: Exercise 9.4 #5

.dy

dxFind

yx15)y,x(f 32

.dy

dxFind

yx15)y,x(f 32

330xyf x 330xyf x

x3

y2

yx45

xy30

f

f

dy

dx22

3

y

x

x3

y2

yx45

xy30

f

f

dy

dx22

3

y

x

22y yx45f 22y yx45f

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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