Book cover for Applied Calculus for the MLSS A Brief Approach

Applied Calculus for the MLSS A Brief Approach

Soo T. Tan

ISBN #9781285464640

10th Edition

3,245 Questions

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46,959 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Differentiation is a powerful tool in calculus that simplifies finding the derivative of a function, enabling the study of instantaneous rates of change. This process is essential in a variety of real-world applications ranging from environmental conservation to economic marginal analysis. By understanding and applying differentiation rules, one can effectively model and analyze dynamic systems, predict trends, and make informed decisions based on small incremental changes.

Learning Objectives

1

Explain the concept of differentiation and how it simplifies finding derivatives.

2

Apply differentiation rules to analyze the rate of change in various real-world contexts.

3

Utilize differentiation for marginal analysis in economics and other fields.

4

Understand the concept of differentials and their use in approximating changes in functions.

Key Concepts

CONCEPT

DEFINITION

Differentiation

A mathematical process used to find the derivative of a function, which represents the rate at which one quantity changes with respect to another.

Derivative

A measure of how a function changes as its input changes, often interpreted as the instantaneous rate of change or slope of the tangent line.

Marginal Analysis

An analysis technique that uses derivatives to study the rate of change in economic variables, such as cost, revenue, or consumption, with respect to small changes in input.

Differentials

Small changes in the variables of a function; they are used to approximate the change in a function's output resulting from a small change in input.

Example Problems

Example 1

Find the derivative of the function $f$ by using the rules of differentiation. $$f(x)=-3$$

Example 2

Find the derivative of the function $f$ by using the rules of differentiation. $$f(x)=365$$

Example 3

Find the derivative of the function $f$ by using the rules of differentiation. $$f(x)=x^{5}$$

Example 4

Find the derivative of the function $f$ by using the rules of differentiation. $$f(x)=x^{7}$$

Example 5

Find the derivative of the function $f$ by using the rules of differentiation. $$f(x)=x^{3.1}$$
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Step-by-Step Explanations

QUESTION

How do you differentiate the function f(x) = x³ using the power rule?

STEP-BY-STEP ANSWER:

Step 1: Identify the function and the power involved. For f(x) = x³, the power is 3.
Step 2: Recall the power rule which states: d/dx(x^n) = n*x^(n-1).
Step 3: Apply the power rule by multiplying the exponent (3) by x raised to the power of (3-1).
Step 4: Write the resulting derivative: f'(x) = 3x².
Final Answer: The derivative of f(x) = x³ is f'(x) = 3x².

Using Differentiation Rules

QUESTION

How can differentiation be used to find the rate of change of DVD sales over time?

STEP-BY-STEP ANSWER:

Step 1: Define the sales function S(t) where t represents time in years since the DVD release.
Step 2: Identify the differentiation rules applicable to the form of S(t) (e.g., power rule, product rule).
Step 3: Differentiate S(t) with respect to time t to obtain S'(t), which represents the rate of sales change.
Step 4: Interpret S'(t): a positive value indicates increasing sales, while a negative value indicates decreasing sales.
Final Answer: Differentiation provides S'(t), the instantaneous rate at which DVD sales are changing over the first 10 years.

Analyzing Rate of Change in Real-World Applications

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Common Mistakes

  • Confusing the derivative with the original function, leading to misinterpretation of rate of change.
  • Forgetting to apply the correct differentiation rule (such as the power rule, product rule, or chain rule) based on the function's form.
  • Misunderstanding the concept of differentials and assuming they provide exact changes rather than approximations.
  • Overlooking the significance of the derivative's sign, which indicates whether the function is increasing or decreasing.
  • Failing to connect the mathematical concept of differentiation to its practical applications in real-world scenarios.