Book cover for Precalculus with Limits

Precalculus with Limits

Ron Larson

ISBN #9781439049099

2nd Edition

7,319 Questions

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724,371 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers the definition, evaluation, and graphing of exponential functions with various bases, including the natural exponential function with base e. It emphasizes the use of the One-to-One Property in solving equations and explains how transformations affect the graph of an exponential function. Importantly, exponential functions are not only a subject of theoretical interest but are also applied in real-life scenarios such as compound interest in finance and modeling decay in radioactive materials.

Learning Objectives

1

Recognize and evaluate exponential functions with various bases (including base a and base e) and understand their graphs.

2

Apply the One-to-One Property to solve exponential equations.

3

Analyze transformations of exponential functions (shifts, reflections) and describe their effects on key features such as intercepts and horizontal asymptotes.

4

Model real-life phenomena such as compound interest, continuous compounding, and radioactive decay using exponential functions.

5

Understand the relationship between exponential functions and their inverses, the logarithmic functions.

Key Concepts

CONCEPT

DEFINITION

Exponential Function

A function of the form f(x) = a^x, where a > 0 and a ≠ 1. Its graph is continuous, either always increasing (if a > 1) or always decreasing (if 0 < a < 1), with a horizontal asymptote at y = 0.

Natural Exponential Function

An exponential function with the base e, written as f(x) = e^x, where e (approximately 2.71828) is an irrational constant. This function is commonly used in continuous growth or decay models.

One-to-One Property

A property of functions that states if f(a) = f(b) then a = b. For exponential functions, if a^x1 = a^x2, then x1 = x2, which is useful for solving exponential equations.

Transformation of Graphs

Operations such as horizontal shifts, vertical shifts, and reflections that alter the graph of a function without changing its basic shape. For exponential functions, these transformations affect the intercepts and the location of the horizontal asymptote.

Continuous Compounding

A method of calculating interest where the number of compounding periods increases without bound, leading to the formula A = Pe^(rt), where P is the principal, r is the annual interest rate (in decimal), and t is the time in years.

Example Problems

Example 1

Fill in the blanks. Polynomial and rational functions are examples of ________ functions.

Example 2

Fill in the blanks. Exponential and logarithmic functions are examples of nonalgebraic functions, also called ________ functions.

Example 3

Fill in the blanks. You can use the ________ Property to solve simple exponential equations.

Example 4

Fill in the blanks. The exponential function given by $ f(x) = e^x $ is called the ________ ________ function,and the base $ e $ is called the ________ base.

Example 5

Fill in the blanks. To find the amount $ A $ in an account after $ t $ years with principal $ P $ and an annual interest rate $ r $ compounded $ n $ times per year, you can use the formula ________.
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Step-by-Step Explanations

QUESTION

Evaluate f(x) = 2^x for x = 3.5.

STEP-BY-STEP ANSWER:

Step 1: Write the function with the given x value: f(3.5) = 2^(3.5).
Step 2: Recognize that 2^(3.5) means 2 raised to the 3.5 power, which can also be written as 2^(7/2).
Step 3: Use a calculator to compute 2^(3.5) or 2^(7/2).
Step 4: The calculator will yield approximately 11.314.
Final Answer: f(3.5) ≈ 11.314.

Evaluating an Exponential Function

QUESTION

Solve for x: 2^(x + 3) = 2^5.

STEP-BY-STEP ANSWER:

Step 1: Recognize that both sides have the same base (2).
Step 2: Set the exponents equal: x + 3 = 5.
Step 3: Solve for x: x = 5 - 3.
Final Answer: x = 2.

Solving an Exponential Equation Using the One-to-One Property

QUESTION

Determine the balance after 5 years for a principal amount P = 12000 at an annual interest rate r = 9% compounded continuously.

STEP-BY-STEP ANSWER:

Step 1: Write the formula for continuous compounding: A = Pe^(rt).
Step 2: Substitute P = 12000, r = 0.09, and t = 5 into the formula.
Step 3: Compute the exponent: 0.09 × 5 = 0.45.
Step 4: Calculate A = 12000 × e^(0.45) using a calculator.
Step 5: e^(0.45) is approximately 1.5683.
Final Answer: A ≈ 12000 × 1.5683 ≈ 18819.75.

Modeling Compound Interest with Continuous Compounding

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Common Mistakes

  • Failing to use parentheses when entering fractional exponents into a calculator, leading to incorrect evaluations.
  • Mixing up the base in exponential functions (e.g., confusing an exponential function with its decay counterpart by misplacing the negative sign in the exponent).
  • Overlooking the requirement to express percentage rates as decimals in compound interest formulas.
  • Assuming all functions that appear similar (e.g., a^x versus a^(-x)) behave the same, without considering the impact on graph orientation and behavior.