Book cover for Calculus and Its Applications

Calculus and Its Applications

Larry J. Goldstein, David C. Lay, David I. Schneider

ISBN #9780134437774

14th Edition

3,033 Questions

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63,186 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter extends the fundamentals of calculus to functions of several variables, addressing how to analyze and optimize systems influenced by multiple factors. Key techniques include computing partial derivatives, determining extrema through both unconstrained and constrained optimization methods like Lagrange multipliers, and applying double integrals for area and volume computations. Understanding these concepts is critical for solving complex problems in fields ranging from economics to engineering.

Learning Objectives

1

Explain the concept of functions of several variables and illustrate real-life examples where multiple variables affect outcomes.

2

Compute and interpret partial derivatives to analyze how functions change with respect to different variables.

3

Identify, classify, and solve optimization problems involving maxima and minima in functions of several variables.

4

Apply the method of Lagrange multipliers for constrained optimization problems.

5

Utilize the least squares method and double integrals as advanced tools in multivariable calculus applications.

Key Concepts

CONCEPT

DEFINITION

Functions of Several Variables

Mathematical functions that depend on two or more independent variables, used to model complex real-world phenomena.

Partial Derivative

The derivative of a multivariable function with respect to one variable while keeping other variables constant.

Maxima and Minima

The highest (maximum) or lowest (minimum) values a function attains, important for optimization problems.

Lagrange Multipliers

A strategy for finding the local extrema of a function subject to equality constraints by introducing auxiliary variables.

Least Squares Method

A statistical technique used to find the best-fitting curve by minimizing the sum of squares of the offsets (errors) of points from the curve.

Double Integrals

Integrals used to compute areas or volumes, integrating functions of two variables over a two-dimensional region.

Example Problems

Example 1

Let $f(x, y)=x^{2}-3 x y-y^{2} .$ Compute $f(5,0), f(5,-2),$ and $f(a, b)$.

Example 2

Let $g(x, y)=\sqrt{x^{2}+2 y^{2}} .$ Compute $g(1,1), g(0,-1),$ and $g(a, b)$.

Example 3

Let $g(x, y, z)=x /(y-z) .$ Compute $g(2,3,4)$ and $g(7,46,44)$.

Example 4

Let $f(x, y, z)=x^{2} e^{\sqrt{y}+z^{2}} .$ Compute $f(1,-1,1)$ and $f(2,3,-4)$.

Example 5

Let $f(x, y)=x y .$ Show that $f(2+h, 3)-f(2,3)=3 h$.
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Step-by-Step Explanations

QUESTION

How do you compute the partial derivative of the function f(x,y) = x^2y + 3xy^2 with respect to x?

STEP-BY-STEP ANSWER:

Step 1: Identify the function f(x,y) = x^2y + 3xy^2.
Step 2: Treat y as a constant and differentiate with respect to x.
Step 3: Differentiate x^2y with respect to x. Since y is constant, the derivative is 2xy.
Step 4: Differentiate 3xy^2 with respect to x. With y^2 as constant, the derivative is 3y^2.
Step 5: Combine the results. The partial derivative with respect to x is f_x(x,y) = 2xy + 3y^2.
Final Answer: f_x(x,y) = 2xy + 3y^2.

Partial Derivative

QUESTION

How do you apply Lagrange multipliers to maximize a function f(x,y) subject to a constraint g(x,y)=c?

STEP-BY-STEP ANSWER:

Step 1: Write down the function f(x,y) that must be optimized and the constraint g(x,y) = c.
Step 2: Set up the equations by equating the gradient of f with 位 times the gradient of g: 鈭噁 = 位鈭噂.
Step 3: Solve the set of equations: f_x = 位g_x, f_y = 位g_y, along with g(x,y) = c.
Step 4: Solve for x, y, and 位 simultaneously.
Step 5: Verify which solution gives the maximum (or minimum) based on the context of the problem.
Final Answer: The optimal values of x and y are obtained from the simultaneous solution of the equations and satisfy the constraint.

Lagrange Multipliers

QUESTION

How do you evaluate a double integral of f(x,y) over a rectangular area [a, b]脳[c, d]?

STEP-BY-STEP ANSWER:

Step 1: Write the double integral as 鈭玔a to b] 鈭玔c to d] f(x,y) dy dx (or reverse the order).
Step 2: Integrate f(x,y) with respect to y while treating x as a constant.
Step 3: Evaluate the inner integral from y = c to y = d to obtain an expression in x.
Step 4: Integrate the resulting expression with respect to x over the interval [a, b].
Step 5: Arrive at the numerical value, which is the double integral of the function over the given region.
Final Answer: The evaluated double integral over [a, b]脳[c, d].

Double Integrals

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Common Mistakes

  • Confusing partial derivatives with total derivatives, especially in how other variables are treated as constants.
  • Neglecting the constraint when using Lagrange multipliers, which leads to incorrect solutions in optimization problems.
  • Overlooking the proper order of integration or limits in double integrals, resulting in computational errors.
  • Misapplying the least squares method by not correctly setting up the error minimization problem.