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

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers the foundational properties of functions, emphasizing the differences between linear and nonlinear functions. It explains how to determine and graph the domain and range, evaluate functions by substituting specific inputs, and apply the vertical line test to ascertain if a graph represents a function. Through real-world examples such as the Dow Jones Industrial Average and energy consumption projections, the section demonstrates the practical relevance of these concepts. Additionally, it introduces even and odd functions to explore symmetry in graphs.

Learning Objectives

1

Differentiate between linear and nonlinear functions and recognize their graphs.

2

Define and determine the domain and range of both linear and nonlinear functions.

3

Apply the vertical line test to decide whether a graph represents a function.

4

Evaluate functions by substituting given input values and interpreting output values.

5

Analyze real-world applications (e.g., energy consumption projections) using function concepts.

Key Concepts

CONCEPT

DEFINITION

Function

A rule that assigns to each element of one set exactly one element from another set.

Independent Variable

The input variable (often x or t) whose value determines the output of the function.

Dependent Variable

The output variable (often y or f(x)) that depends on the input value.

Domain

The set of all possible values of the independent variable that can be used in the function.

Range

The set of all possible output values of the function as the independent variable varies over its domain.

Linear Function

A function of the form y = ax + b which graphs as a straight line.

Nonlinear Function

A function whose graph is not a straight line; often modeled by quadratic, step, or other types of curves.

Vertical Line Test

A method to determine if a graph represents a function: if any vertical line intersects the graph more than once, the graph is not a function.

Even Function

A function for which f(-x) = f(x), showing symmetry about the y-axis.

Odd Function

A function for which f(-x) = -f(x), showing symmetry about the origin.

Example Problems

Example 1

Which of the following rules define $y$ as a function of $x ?$

Example 2

Which of the following rules define $y$ as a function of $x ?$

Example 3

$$\begin{array}{|c|c|}\hline x & {y} \\ \hline 25 & {5} \\ \hline 9 & {3} \\ \hline 1 & {1} \\ \hline 0 & {0} \\ \hline 1 & {-1} \\ \hline 9 & {-3} \\ \hline 25 & {-5} \\ \hline\end{array}$$

Example 4

$$\begin{array}{|c|c|}\hline x & {y} \\ \hline 9 & {3} \\ \hline 4 & {2} \\ \hline 1 & {1} \\ \hline 0 & {0} \\ \hline 1 & {-1} \\ \hline 4 & {-2} \\ \hline 9 & {-3} \\ \hline\end{array}$$

Example 5

Which of the following rules define $y$ as a function of $x ?$ $y=x^{5}+1$
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Step-by-Step Explanations

QUESTION

Given g(x) = -x² + 4x - 5, find g(3).

STEP-BY-STEP ANSWER:

Step 1: Substitute x = 3 into the function: g(3) = -3² + 4(3) - 5.
Step 2: Compute the square: -9 + 12 - 5.
Step 3: Combine the values: (-9 + 12) = 3, then 3 - 5 = -2.
Final Answer: g(3) = -2.

Evaluating a Function

QUESTION

Find the domain and estimate the range for a function modeled by a graph, such as the Dow Jones industrial average over one year.

STEP-BY-STEP ANSWER:

Step 1: Identify the independent variable (time in days, t) and the dependent variable (y, the index value).
Step 2: From the graph, determine the minimum and maximum t values (e.g., t = 0 to t = 365).
Step 3: Examine the graph’s lowest and highest y-values to estimate the range (for example, y from 13,100 to 16,600).
Final Answer: Domain: [0, 365]; Range: Approximately [13,100, 16,600].

Determining Domain and Range

QUESTION

How are the energy consumptions of the United States, China, and India changing over time?

STEP-BY-STEP ANSWER:

Step 1: Identify that the independent variable is time (year) and the dependent variable is energy consumption measured in quadrillion Btu.
Step 2: Observe from the graph whether the functions for the US, China, and India are linear or nonlinear.
Step 3: For each country, estimate values from the graph (e.g., US consumption in 2015, China’s consumption when reaching 150 quadrillion Btu, etc.).
Step 4: Use the vertical line test on each country’s graph to confirm they define valid functions.
Final Answer: The analysis involves determining the type of function (linear vs nonlinear), identifying the domain (year range) and range (consumption levels), and estimating specific values from the graph for each country.

Analyzing Energy Consumption Functions

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Common Mistakes

  • Confusing the independent variable with the dependent variable.
  • Misidentifying the domain by ignoring restrictions like division by zero or negative radicands in even roots.
  • Assuming that a function must have a unique output for every input incorrectly when multiple inputs yield the same output.
  • Mistaking symmetry requirements for even/odd functions, leading to incorrect classifications.
  • Forgetting to apply the vertical line test when determining if a graph represents a function.