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

Group icon
63,186 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers the fundamentals of functions, including their definitions, graphical representations, and algebraic operations. It emphasizes the importance of being able to graph various types of functions and solve for key characteristics like zeros, especially for quadratic functions using factoring or the quadratic formula. Additionally, the section explores the application of exponent and power functions in real-world scenarios, reinforcing the practical use of mathematical concepts.

Learning Objectives

1

Explain the concept of a function and accurately graph various types of functions.

2

Identify and analyze important functions including linear, quadratic, exponential, and power functions.

3

Demonstrate the algebra of functions through operations such as addition, subtraction, multiplication, and division.

4

Solve for zeros of functions using both factoring and the quadratic formula.

5

Apply function concepts to real-world problems and interpret their graphical representations.

Key Concepts

CONCEPT

DEFINITION

Function

A rule or relation that assigns exactly one output to each input from a specific domain.

Graph of a Function

A visual representation of a function showing the relationship between input values (domain) and output values (range).

Quadratic Function

A function of the form f(x) = ax² + bx + c, whose graph is a parabola.

Quadratic Formula

A formula to find the zeros of a quadratic function: x = (-b ± √(b² - 4ac)) / (2a).

Exponential Function

A function where the variable appears in the exponent, typically in the form f(x) = a * b^x.

Power Function

A function of the form f(x) = k * x^n where n is a constant exponent.

Example Problems

Example 1

Draw the following intervals on the number line. $$[-1,4]$$

Example 2

Draw the following intervals on the number line. $$(4,3 \pi)$$

Example 3

Draw the following intervals on the number line. $$[-2, \sqrt{2})$$

Example 4

Draw the following intervals on the number line. $$\left[1, \frac{3}{2}\right]$$

Example 5

Draw the following intervals on the number line. $$(-\infty, 3)$$
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

How do you find the zeros of a quadratic function f(x) = ax² + bx + c using factoring or the quadratic formula?

STEP-BY-STEP ANSWER:

Step 1: Identify the coefficients a, b, and c in the quadratic equation.
Step 2: Check if the quadratic can be factored easily. If so, factor the equation to find the solutions.
Step 3: If factoring is not straightforward, apply the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
Step 4: Calculate the discriminant (b² - 4ac) to determine the nature of the roots.
Final Answer: The zeros of the function are given by the values of x obtained through factoring or the quadratic formula.

Zeros of a Quadratic Function

QUESTION

What are the step-by-step processes for graphing a function based on its equation?

STEP-BY-STEP ANSWER:

Step 1: Determine the domain and range of the function.
Step 2: Identify key points, such as intercepts, vertex (for quadratic functions), and any symmetry.
Step 3: Analyze specific features like slopes, asymptotes, and intervals of increase or decrease.
Step 4: Plot the determined points on the coordinate plane and sketch the curve.
Final Answer: By following these steps, you can create an accurate graph of the function that reflects its behavior.

Graphing Functions

Scroll left
Scroll right

Common Mistakes

  • Misidentifying the domain and range of a function.
  • Confusing the process of finding zeros with simply graphing intercepts.
  • Incorrectly applying the quadratic formula, especially with sign errors.
  • Forgetting to check the possibility of factoring before jumping directly to the quadratic formula.
  • Overlooking the impact of exponent rules when working with power functions.