1. Calculus Concepts 2/e LaTorre, Kenelly, Fetta, Harris, and Carpenter
Chapter 9
Ingredients of Multivariable Change:
Models, Graphs, Rates
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2. 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
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3. 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
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4. 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.
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5. 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.
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6. 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
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7. 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).
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8. 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) = -2.5n n
feet above sea level n miles north of the southern fence.
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9. 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.
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10. 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
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11. Cross-Sectional Model: Exercise 9.1 #15p C
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.
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) = 0.893p p pounds per person per year.
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12. 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
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13. 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.
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14. Contour Graphs: Example
The contour lines on the table are drawn at the 796, 797, 798, 799, 800, 801, and 802-foot levels.
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15. 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.
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16. Contour Graphs: Exercise 9.2 #1
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17. 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
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18. 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
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19. 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
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20. Partial Rates of Change: Example
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21. Partial Rate of Change: Exercise 9.3 #11
Find Mt, Ms, and Ms|t=3 given
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22. 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
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23. 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
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24. 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.
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25. Slopes of Contour Curves: Example
At w = 129 and h = 67, Bh  point per inch and Bw  point per pound.
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26. Slopes of Contour Curves: Exercise 9.4 #5
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