Book cover for Calculus with Applications

Calculus with Applications

Margaret L. Lial • Raymond N. Greenwell • Nathan P. Ritchey

ISBN #9781292108971

11th Edition

3,612 Questions

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224,424 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section extends calculus to functions of several variables by introducing tools to evaluate and graph such functions. Key topics include the usage of independent and dependent variables, methods for evaluating functions at specific inputs, and ways to graph surfaces through traces, intercepts, and level curves. Additionally, the section introduces partial derivatives and demonstrates their application in real-world problems, such as optimizing production in economics.

Learning Objectives

1

Explain the concept of functions of several variables and differentiate between independent and dependent variables.

2

Evaluate functions of two or more variables by substituting specific values into the function.

3

Graph functions of two variables using traces, level curves, and intercepts in three-dimensional space.

4

Understand and compute partial derivatives and interpret their significance in real applications such as production functions.

Key Concepts

CONCEPT

DEFINITION

Function of Several Variables

An expression in which the output (dependent variable) is determined by more than one input (independent variables), for example, f(x, y) or f(x, y, z).

Domain

The set of all ordered tuples of real numbers for which the function is defined.

Range

The set of all possible values taken by the function.

Level Curves

Curves in the xy-plane obtained by setting the function equal to a constant (z = k); they help visualize surfaces by representing cross sections.

Partial Derivative

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

Cobb-Douglas Production Function

A special form production function, typically expressed as z = Ax^α y^(1-α), representing the output produced given certain amounts of labor (x) and capital (y).

Example Problems

Example 1

Let $f(x, y)=2 x-4 y+7 .$ Find the following. (a) $f(3,-1)$ (b) $f(-5,1)$ (c) $f(-5,-4)$ (d) $f(0,7)$

Example 2

Let $f(x, y)=6 x-7 y+3 .$ Find the following. (a) $f(4,-1)$ (b) $f(-5,1)$ $(c) \quad f(-5,-3)$ (d) $f(0,7)$

Example 3

Let $f(x, y)=\sqrt{4 y^{2}+5 x^{2}} .$ Find the following. (a) $f(5,-4) \quad$ (b) $f(-5,3)$ (c) $f(-1,-3)$ (d) $f(0,6)$

Example 4

Let $f(x, y)=\sqrt{y^{2}+4 x^{2}} .$ Find the following. (a) $f(1,-3)$ (b) $f(-3,5)$ (c) $f(-1,-2)$ (d) $f(0,8)$

Example 5

Let $f(x, y)=e^{x}+\ln (x+y) .$ Find the following. (a) $f(1,0)$ (b) $f(2,-1)$ (c) $f(0, e)$ (d) $f\left(0, e^{2}\right)$
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Step-by-Step Explanations

QUESTION

Evaluate the function f(x, y) = 4x² + 2xy + 3/y at x = -1 and y = 3.

STEP-BY-STEP ANSWER:

Step 1: Substitute x = -1 and y = 3 into the function.
Step 2: Compute the term 4x² = 4*( (-1)² ) = 4*1 = 4.
Step 3: Compute the term 2xy = 2 * (-1) * 3 = -6.
Step 4: Compute the term 3/y = 3/3 = 1.
Step 5: Add the results: 4 + (-6) + 1 = -1.
Final Answer: f(-1, 3) = -1.

Evaluating a Function of Two Variables

QUESTION

Graph the plane given by the equation 2x + y + z = 6 in the first octant.

STEP-BY-STEP ANSWER:

Step 1: Find the x-intercept by setting y = 0 and z = 0, so 2x = 6; hence x = 3. The point is (3, 0, 0).
Step 2: Find the y-intercept by setting x = 0 and z = 0, so y = 6. The point is (0, 6, 0).
Step 3: Find the z-intercept by setting x = 0 and y = 0, so z = 6. The point is (0, 0, 6).
Step 4: Plot these intercepts and draw the plane through them in the first octant.
Final Answer: The plane is graphically represented by the points (3, 0, 0), (0, 6, 0), and (0, 0, 6).

Graphing a Plane

QUESTION

For the profit function P(x, y) = 40x² - 10xy + 5y² - 80, determine the effect on profit when x (smartphones sold) increases by one unit while y is constant.

STEP-BY-STEP ANSWER:

Step 1: Compute the partial derivative of P with respect to x, denoted as ∂P/∂x.
Step 2: Differentiate term by term: ∂(40x²)/∂x = 80x, ∂(-10xy)/∂x = -10y, ∂(5y² - 80)/∂x = 0.
Step 3: The resulting partial derivative is ∂P/∂x = 80x - 10y.
Step 4: Evaluate at the given values if needed to find the rate of change.
Final Answer: The rate of change of profit with respect to x is 80x - 10y, which quantifies the effect on profit when x changes.

Partial Derivative Example

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Common Mistakes

  • Confusing the domain restrictions when a function includes a denominator (e.g., 3/y; ensure y ? 0).
  • Mixing up the interpretation of independent and dependent variables in multi-variable contexts.
  • Assuming the graph of a function of two variables is a simple curve, rather than a surface in three dimensions.
  • Overlooking the need to consider multiple intercepts (x-, y-, z-intercepts) when graphing planes in three-dimensional space.