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Chapter 17 Multivariable Calculus

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**INTRODUCTORY MATHEMATICAL ANALYSIS**

0. Review of Algebra Applications and More Algebra Functions and Graphs Lines, Parabolas, and Systems Exponential and Logarithmic Functions Mathematics of Finance Matrix Algebra Linear Programming Introduction to Probability and Statistics

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**INTRODUCTORY MATHEMATICAL ANALYSIS**

Additional Topics in Probability Limits and Continuity Differentiation Additional Differentiation Topics Curve Sketching Integration Methods and Applications of Integration Continuous Random Variables Multivariable Calculus

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**Chapter 17: Multivariable Calculus**

Chapter Objectives To discuss functions of several variables and to compute function values. To compute partial derivatives. To develop the notions of partial marginal cost, marginal productivity, and competitive and complementary products. To find partial derivatives of a function defined implicitly. To compute higher-order partial derivatives. To find the partial derivatives of a function by using the chain rule.

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**Chapter 17: Multivariable Calculus**

Chapter Objectives To apply the second-derivative test for a function of two variables. To find critical points for a function. To develop the method of least squares. To compute double and triple integrals.

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**Chapter Outline Functions of Several Variables Partial Derivatives**

Chapter 17: Multivariable Calculus Chapter Outline 17.1) Functions of Several Variables Partial Derivatives Applications of Partial Derivatives Implicit Partial Differentiation Higher-Order Partial Derivatives Chain Rule Maxima and Minima for Functions of Two Variables 17.2) 17.3) 17.4) 17.5) 17.6) 17.7)

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**Chapter Outline Lagrange Multipliers Lines of Regression**

Chapter 17: Multivariable Calculus Chapter Outline Lagrange Multipliers Lines of Regression Multiple Integrals 17.8) 17.9) 17.10)

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**17.1 Functions of Several Variables**

Chapter 17: Multivariable Calculus 17.1 Functions of Several Variables Example 1 – Functions of Two Variables A function can involve 2 or more variables, e.g. a is a function of two variables. Because the denominator is zero when y = 2, the domain of f is all (x, y) such that y 2. b. h(x, y) = 4x defines h as a function of x and y. The domain is all ordered pairs of real numbers. c. z2 = x2 + y2 does not define z as a function of x and y.

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**The plane intersects the x-axis when y = 0 and z = 0.**

Chapter 17: Multivariable Calculus 17.1 Functions of Several Variables Example 3 – Graphing a Plane Sketch the plane Solution: The plane intersects the x-axis when y = 0 and z = 0.

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**Sketch the surface z = x2 is a parabola.**

Chapter 17: Multivariable Calculus 17.1 Functions of Several Variables Example 5 – Sketching a Surface Sketch the surface Solution: z = x2 is a parabola.

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**Chapter 17: Multivariable Calculus**

17.2 Partial Derivatives Partial derivative of f with respect to x is given by Partial derivative of f with respect to y is given by

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**Solution: The partial derivatives are**

Chapter 17: Multivariable Calculus 17.2 Partial Derivatives Example 1 – Finding Partial Derivatives If f(x, y) = xy2 + x2y, find fx(x, y) and fy(x, y). Also, find fx(3, 4) and fy(3, 4). Solution: The partial derivatives are Thus, the solutions are

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**By partial differentiating, we get**

Chapter 17: Multivariable Calculus 17.2 Partial Derivatives Example 3 – Partial Derivatives of a Function of Three Variables If find Solution: By partial differentiating, we get

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**By partial differentiating, we get**

Chapter 17: Multivariable Calculus 17.2 Partial Derivatives Example 4 – Finding Partial Derivatives If find Solution: By partial differentiating, we get

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**17.3 Applications of Partial Derivatives**

Chapter 17: Multivariable Calculus 17.3 Applications of Partial Derivatives Example 1 – Marginal Costs Interpretation of rate of change: A company manufactures two types of skis, the Lightning and the Alpine models. Suppose the joint-cost function for producing x pairs of the Lightning model and y pairs of the Alpine model per week is where c is expressed in dollars. Determine the marginal costs ∂c/∂x and ∂c/∂y when x = 100 and y = 50, and interpret the results.

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**Solution: The marginal costs are**

Chapter 17: Multivariable Calculus 17.3 Applications of Partial Derivatives Example 1 – Marginal Costs Solution: The marginal costs are Thus, and

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**Chapter 17: Multivariable Calculus**

17.3 Applications of Partial Derivatives Example 3 – Marginal Productivity A manufacturer of a popular toy has determined that the production function is P = √(lk), where l is the number of labor-hours per week and k is the capital (expressed in hundreds of dollars per week) required for a weekly production of P gross of the toy. (One gross is 144 units.) Determine the marginal productivity functions, and evaluate them when l = 400 and k = 16. Interpret the results.

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**Solution: Since , Thus, Chapter 17: Multivariable Calculus**

17.3 Applications of Partial Derivatives Example 3 – Marginal Productivity Solution: Since , Thus,

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**17.4 Implicit Partial Differentiation**

Chapter 17: Multivariable Calculus 17.4 Implicit Partial Differentiation Example 1 – Implicit Partial Differentiation We will look into how to find partial derivatives of a function defined implicitly. If , evaluate when x = −1, y = 2, and z = 2. Solution: Using partial differentiation, we get

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**17.5 Higher-Order Partial Derivatives**

Chapter 17: Multivariable Calculus 17.5 Higher-Order Partial Derivatives Example 1 – Second-Order Partial Derivatives We obtain second-order partial derivatives of f as Find the four second-order partial derivatives of Solution:

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**Solution: By implicit differentiation,**

Chapter 17: Multivariable Calculus 17.5 Higher-Order Partial Derivatives Example 3 – Second-Order Partial Derivative of an Implicit Function Determine if Solution: By implicit differentiation, Differentiating both sides with respect to x, we obtain Substituting ,

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**Chapter 17: Multivariable Calculus**

17.6 Chain Rule If f, x, and y have continuous partial derivatives, then z is a function of r and s, and

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**The demand functions for the cameras and film are given by**

Chapter 17: Multivariable Calculus 17.6 Chain Rule Example 1 – Rate of Change of Cost For a manufacturer of cameras and film, the total cost c of producing qC cameras and qF units of film is given by The demand functions for the cameras and film are given by where pC is the price per camera and pF is the price per unit of film. Find the rate of change of total cost with respect to the price of the camera when pC = 50 and pF = 2.

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**Solution: By the chain rule,**

Chapter 17: Multivariable Calculus 17.6 Chain Rule Example 1 – Chain Rule Example 3 – Chain Rule Solution: By the chain rule, Thus, a. Determine ∂y/∂r if Solution: By the chain rule,

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**Solution: By the chain rule,**

Chapter 17: Multivariable Calculus 17.6 Chain Rule Example 3 – Chain Rule b. Given that z = exy, x = r − 4s, and y = r − s, find ∂z/∂r in terms of r and s. Solution: By the chain rule, Since x = r − 4s and y = r − s,

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**17.7 Maxima and Minima for Functions**

Chapter 17: Multivariable Calculus 17.7 Maxima and Minima for Functions of Two Variables Relative maximum at the point (a, b) is shown as RULE 1 Find relative maximum or minimum when RULE 2 Second-Derivative Test for Functions of Two Variables Let D be the function defined by If D(a, b) > 0 and fxx(a, b) < 0, relative maximum at (a, b); If D(a, b) > 0 and fxx(a, b) > 0, relative minimum at (a, b); If D(a, b) < 0, then f has a saddle point at (a, b); If D(a, b) = 0, no conclusion.

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**Find the critical points of the following functions. a. Solution: **

Chapter 17: Multivariable Calculus 17.7 Maxima and Minima for Functions of Two Variables Example 1 – Finding Critical Points Find the critical points of the following functions. a. Solution: Since we solve the system and get b. we solve the system and get and

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**we solve the system and get**

Chapter 17: Multivariable Calculus 17.7 Maxima and Minima for Functions of Two Variables Example 1 – Finding Critical Points c. Since we solve the system and get

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**Chapter 17: Multivariable Calculus**

17.7 Maxima and Minima for Functions of Two Variables Example 3 – Applying the Second-Derivative Test Examine f(x,y) = x3 + y3 − xy for relative maxima or minima by using the second derivative test. Solution: We find critical points, which gives (0, 0) and (1/3, 1/3). Now, Thus, D(0, 0) < 0 no relative extremum at (0, 0). D(1/3,1/3)>0 and fxx(1/3,1/3)>0 relative minimum at (1/3,1/3) Value of the function is

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**Examine f(x, y) = x4 + (x − y)4 for relative extrema. **

Chapter 17: Multivariable Calculus 17.7 Maxima and Minima for Functions of Two Variables Example 5 – Finding Relative Extrema Examine f(x, y) = x4 + (x − y)4 for relative extrema. Solution: We find critical points at (0,0) through D(0, 0) = 0 no information. f has a relative (and absolute) minimum at (0, 0).

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**Chapter 17: Multivariable Calculus**

17.7 Maxima and Minima for Functions of Two Variables Example 7 – Profit Maximization A candy company produces two types of candy, A and B, for which the average costs of production are constant at $2 and $3 per pound, respectively. The quantities qA, qB (in pounds) of A and B that can be sold each week are given by the joint-demand functions where pA and pB are the selling prices (in dollars per pound) of A and B, respectively. Determine the selling prices that will maximize the company’s profit P.

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**Since ∂2P/∂p2A< 0, we indeed have a maximum.**

Chapter 17: Multivariable Calculus 17.7 Maxima and Minima for Functions of Two Variables Example 7 – Profit Maximization Solution: The total profit is given by The profits per pound are pA − 2 and pB − 3, The solution is pA = 5.5 and pB = 6. Since ∂2P/∂p2A< 0, we indeed have a maximum.

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**Chapter 17: Multivariable Calculus**

17.8 Lagrange Multipliers Example 1 – Method of Lagrange Multipliers Lagrange multipliers allow us to obtain critical points. The number λ0 is called a Lagrange multiplier. Find the critical points for z = f(x,y) = 3x − y + 6, subject to the constraint x2 + y2 = 4. Solution: Constraint Construct the function Setting , we solve the equations to be

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**How should the output be distributed in order to minimize costs?**

Chapter 17: Multivariable Calculus 17.8 Lagrange Multipliers Example 3 – Minimizing Costs Suppose a firm has an order for 200 units of its product and wishes to distribute its manufacture between two of its plants, plant 1 and plant 2. Let q1 and q2 denote the outputs of plants 1 and 2, respectively, and suppose the total-cost function is given by How should the output be distributed in order to minimize costs?

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**Solution: We minimize c = f(q1, q2), given the constraint **

Chapter 17: Multivariable Calculus 17.8 Lagrange Multipliers Example 3 – Minimizing Costs Solution: We minimize c = f(q1, q2), given the constraint q1 + q2 = 200.

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**Chapter 17: Multivariable Calculus**

17.8 Lagrange Multipliers Example 5 – Least-Cost Input Combination Find critical points for f(x, y, z) = xy+ yz, subject to the constraints x2 + y2 = 8 and yz = 8. Solution: Set We obtain 4 critical points: (2, 2, 4) (2,−2,−4) (−2, 2, 4) (−2,−2,−4)

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**Chapter 17: Multivariable Calculus**

17.9 Lines of Regression

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**Chapter 17: Multivariable Calculus**

17.9 Lines of Regression Example 1 – Determining a Linear-Regression Line By means of the linear-regression line, use the following table to represent the trend for the index of total U.S. government revenue from 1995 to 2000 (1995 = 100).

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**Solution: Perform the arithmetic**

Chapter 17: Multivariable Calculus 17.9 Lines of Regression Example 1 – Determining a Linear-Regression Line Solution: Perform the arithmetic

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**Chapter 17: Multivariable Calculus**

17.10 Multiple Integrals Example 1 – Evaluating a Double Integral Definite integrals of functions of two variables are called (definite) double integrals, which involve integration over a region in the plane. Find Solution:

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**Find Solution: Example 3 – Evaluating a Triple Integral**

Chapter 17: Multivariable Calculus 17.10 Lines of Regression Example 3 – Evaluating a Triple Integral Find Solution:

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