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

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63,186 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section delves into the fundamental nature of exponential functions and the natural logarithm, emphasizing their properties and interrelationships. Students learn how exponential growth and decay models apply to financial investments, biological populations, and nuclear decay, and gain skills in differentiating these functions. Key takeaways include understanding the differentiation techniques for exponential and logarithmic functions and applying logarithmic properties to simplify and solve equations.

Learning Objectives

1

Understand and describe the properties and behavior of exponential functions.

2

Explain exponential growth and decay and their applications in real-world scenarios.

3

Differentiate exponential functions and compute the derivative of the natural logarithm function ln(x).

4

Apply the properties of the natural logarithm to solve exponential and logarithmic equations.

Key Concepts

CONCEPT

DEFINITION

Exponential Function

A function in which the variable appears in the exponent, often written as f(x) = a^x where a is a positive constant. They model situations where the rate of change is proportional to the current amount.

Exponential Growth

A process in which a quantity increases at a rate proportional to its current value, such as investments growing at a constant percentage or populations increasing rapidly.

Exponential Decay

A process in which a quantity decreases at a rate proportional to its current value, typical in radioactive decay where the substance decreases over time.

Natural Logarithm Function

The inverse function of the natural exponential function, typically denoted as ln(x), which is defined for x > 0.

Differentiation of Exponential Functions

The process of finding the derivative of an exponential function, which often involves applying the chain rule. For example, the derivative of e^(kx) is k*e^(kx).

Derivative of ln(x)

The derivative of the natural logarithm function ln(x) is 1/x, applicable for x > 0.

Properties of the Natural Logarithm Function

Characteristics such as ln(ab) = ln(a) + ln(b) and ln(a^b) = b*ln(a), which are used to simplify and solve logarithmic equations.

Example Problems

Example 1

Write each expression in the form $2^{k x}$ or $3^{k x}$, for a suitable constant $k$. $$4^{x},(\sqrt{3})^{x},\left(\frac{1}{9}\right)^{x}$$

Example 2

Write each expression in the form $2^{k x}$ or $3^{k x}$, for a suitable constant $k$. $$27^{x},(\sqrt[3]{2})^{x},\left(\frac{1}{8}\right)^{x}$$

Example 3

Write each expression in the form $2^{k x}$ or $3^{k x}$, for a suitable constant $k$. $$8^{2 x / 3}, 9^{3 x / 2}, 16^{-3 x / 4}$$

Example 4

Write each expression in the form $2^{k x}$ or $3^{k x}$, for a suitable constant $k$. $$9^{-x / 2}, 8^{4 x / 3}, 27^{-2 x / 3}$$

Example 5

Write each expression in the form $2^{k x}$ or $3^{k x}$, for a suitable constant $k$. $$\left(\frac{1}{4}\right)^{2 x},\left(\frac{1}{8}\right)^{-3 x},\left(\frac{1}{81}\right)^{x / 2}$$
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Step-by-Step Explanations

QUESTION

Find the derivative of f(x) = e^(2x).

STEP-BY-STEP ANSWER:

Step 1: Recognize that f(x) = e^(2x) is an exponential function with an exponent that is a function of x.
Step 2: Apply the chain rule. The derivative of e^(u) with respect to x is e^(u) multiplied by the derivative of u with respect to x.
Step 3: Here, u = 2x, so compute du/dx = 2.
Step 4: Multiply: f'(x) = e^(2x) * 2.
Final Answer: f'(x) = 2e^(2x).

Differentiation of Exponential Function

QUESTION

Differentiate f(x) = ln(x).

STEP-BY-STEP ANSWER:

Step 1: Recognize that ln(x) is the natural logarithm function defined for x > 0.
Step 2: Use the known derivative result for the natural logarithm: d/dx [ln(x)] = 1/x.
Final Answer: f'(x) = 1/x.

Derivative of ln(x)

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Common Mistakes

  • Confusing exponential growth with linear growth, leading to incorrect modeling of real-world scenarios.
  • Misapplying the chain rule when differentiating exponential functions with composite exponents.
  • Forgetting the domain restrictions of the natural logarithm function (x must be > 0).
  • Overlooking the inverse relationship between the natural logarithm and the exponential function.