Book cover for Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart

ISBN #9781285741550

8th Edition

6,422 Questions

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2,819,387 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces functions as a fundamental concept where one quantity depends on another. It emphasizes the various representations of functions, including graphs, tables, verbal and algebraic expressions. Key topics include determining the domain and range, understanding piecewise functions, and the significance of even and odd symmetries. Moreover, it lays the groundwork for mathematical modeling by showing how functions can represent and predict real-world phenomena.

Learning Objectives

1

Describe the concept of a function and how one quantity depends on another.

2

Identify and explain the different representations of functions: verbal, numerical, graphical, and algebraic.

3

Determine domain and range for a given function and use the vertical line test to validate function graphs.

4

Analyze piecewise defined functions and interpret even and odd functions along with their symmetries.

5

Understand the process of mathematical modeling and the application of functions to real-world problems.

Key Concepts

CONCEPT

DEFINITION

Function

A rule that assigns to each element x in a set D exactly one element, denoted f(x), in a set E.

Domain

The set D of all possible input values (x-values) for which the function is defined.

Range

The set of all possible output values, f(x), obtained by varying x throughout the domain.

Independent Variable

A symbol representing the input of a function, typically denoted by x.

Dependent Variable

A symbol representing the output of a function, typically denoted by f(x) or y, which depends on x.

Vertical Line Test

A test to determine if a graph represents a function; if no vertical line intersects the graph more than once, then it is a function.

Piecewise Defined Function

A function defined by different expressions in different parts of its domain.

Even Function

A function f that satisfies f(-x) = f(x) for every x in its domain, with a graph symmetric about the y-axis.

Odd Function

A function f that satisfies f(-x) = -f(x) for every x in its domain, with a graph symmetric about the origin.

Difference Quotient

An expression of the form [f(a+h) - f(a)]/h, representing the average rate of change of the function over an interval.

Mathematical Modeling

The process of formulating a real-world problem into a mathematical framework to analyze, interpret, and predict behavior.

Example Problems

Example 1

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

Example 2

If $ f(x) = \frac{x^2-x}{x-1} $ and $ g(x) = x $ is it true that $ f = g $?

Example 3

The graph of a function $f$ is given. (a) State the value of $f(1)$ (b) Estimate the value of $f(-1)$ (c) For what values of $x$ is $f(x)=1 ?$ (d) Estimate the value of $x$ such that $f(x)=0$ (e) State the domain and range of $f$. (f) On what interval is $f$ increasing?

Example 4

The graph of a function $ f $ and $ g $ are given. (a) State the values of $ f(-4) $ and $ g(3) $. (b) For what values of x is $ f(x) = g(x) $? (c) Estimate the solution of the equation $ f(x) = -1 $? (d) On what interval is $ f $ decreasing? (e) State the domain and range of $ f $. (f) State the domain and range of $ g $.

Example 5

Figure 1 was recorded by an instrument operated by the California Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use it to estimate the range of the vertical ground acceleration function at USC during the Northridge earthquake.
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Step-by-Step Explanations

QUESTION

How do you determine the domain of f(x) = √(x - 2)?

STEP-BY-STEP ANSWER:

Step 1: Recognize that the square root function is only defined for non-negative numbers.
Step 2: Set the radicand (expression inside the square root) to be greater than or equal to zero: x - 2 ≥ 0.
Step 3: Solve the inequality to get x ≥ 2.
Final Answer: The domain of the function is [2, ∞).

Determining the Domain of a Function

QUESTION

How can you determine if a given graph represents a function?

STEP-BY-STEP ANSWER:

Step 1: Draw or imagine vertical lines across the graph.
Step 2: Check whether any vertical line intersects the graph at more than one point.
Step 3: If no vertical line cuts the graph more than once, then the graph represents a function.
Final Answer: Use the vertical line test to verify the function property.

Applying the Vertical Line Test

QUESTION

Given f(x) = { 1/2 x if x < 2; x^2 if x ≥ 2 }, how do you find f(2) and f(1)?

STEP-BY-STEP ANSWER:

Step 1: Check which piece of the function applies to the given x-value.
Step 2: For x = 2, since 2 ≥ 2, use the rule f(x)= x^2. Thus, f(2) = 2^2 = 4.
Step 3: For x = 1, since 1 < 2, use the rule f(x) = (1/2) x. Thus, f(1) = 1/2.
Final Answer: f(2) = 4 and f(1) = 0.5.

Evaluating a Piecewise Defined Function

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Common Mistakes

  • Assuming any equation represents a function without applying the vertical line test.
  • Misidentifying the domain by overlooking restrictions such as square roots or division by zero.
  • Confusing the independent and dependent variables, especially in verbal descriptions of functions.
  • Overlooking the piecewise nature of some functions and incorrectly applying one formula over the entire domain.
  • Misinterpreting symmetrical graphs and erroneously labeling functions as even or odd without proper verification.