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

This chapter lays the foundation for understanding how functions, limits, and derivatives interconnect to describe and analyze change. A function defines a direct relationship between inputs and outputs, the concept of limits helps examine function behavior near critical points, and the derivative, derived from the limit of the difference quotient, quantifies instantaneous change. These mathematical concepts are essential for tackling a range of real-world problems from science to social economics.

Learning Objectives

1

Explain and model the relationship between two variables using functions.

2

Understand the concept of limits and how they describe the behavior of functions near specific points.

3

Define and compute the derivative as the instantaneous rate of change using the limit of the difference quotient.

4

Apply the concepts of functions, limits, and derivatives to solve real-world problems in fields such as economics and physics.

Key Concepts

CONCEPT

DEFINITION

Function

A rule that assigns each element in a set of inputs (domain) a unique element in a set of outputs (range).

Domain

The set of all possible input values for which a function is defined.

Range

The set of all possible output values produced by a function.

Limit

A value that a function or sequence approaches as the input or index approaches some value.

Derivative

A measure of the rate at which a function is changing at any given point, computed as the limit of the difference quotient.

Difference Quotient

An expression used to calculate the derivative of a function, defined as (f(x+h) - f(x))/h, as h approaches zero.

Tangent Line

A line that touches a curve at a single point and has the same slope as the curve at that point.

Secant Line

A line that intersects a curve at two or more points, used to approximate the slope of the curve between those points.

Example Problems

Example 1

Let $f$ be the function defined by $f(x)=5 x+6 .$ Find $f(3), f(-3), f(a), f(-a),$ and $f(a+3).$

Example 2

Let $f$ be the function defined by $f(x)=5 x+6 .$ Find $f(3), f(-3), f(a), f(-a),$ and $f(a+3).$

Example 3

Let $g$ be the function defined by $g(x)=3 x^{2}-6 x-3 .$ Find $g(0), g(-1), g(a), g(-a),$ and $g(x+1).$

Example 4

Let $h$ be the function defined by $h(x)=x^{3}-x^{2}+x+1 .$ Find $h(-5), h(0), h(a),$ and $h(-a).$

Example 5

Let $f$ be the function defined by $f(x)=2 x+5 .$ Find $f(a+h), f(-a), f\left(a^{2}\right), f(a-2 h),$ and $f(2 a-h).$
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Step-by-Step Explanations

QUESTION

Using the limit definition, find the derivative of the function f(x) = x².

STEP-BY-STEP ANSWER:

Step 1: Write the difference quotient: (f(x+h) - f(x)) / h, which for f(x)=x² becomes ((x+h)² - x²) / h.
Step 2: Expand (x+h)² to get x² + 2xh + h², so the numerator becomes x² + 2xh + h² - x².
Step 3: Simplify the numerator to obtain 2xh + h².
Step 4: Factor h out of the numerator: h(2x + h).
Step 5: Cancel the common factor h in the numerator and denominator, yielding 2x + h.
Step 6: Take the limit as h approaches 0. Since h becomes 0, the expression simplifies to 2x.
Final Answer: The derivative f'(x) = 2x.

Derivative

QUESTION

Explain how a function uniquely maps each input to one output using a real-world example.

STEP-BY-STEP ANSWER:

Step 1: Identify the independent variable, such as time, and the dependent variable, such as profit.
Step 2: Define a function, for example, f(t) = profit at time t, which indicates that each value of t corresponds to a specific profit.
Step 3: Emphasize that for every input value t, the function rule f provides a unique output (profit).
Step 4: Use the machine analogy: input an element (time) into the machine (function) and receive a unique output (profit).
Final Answer: A function uniquely matches each input to one output, ensuring no ambiguity in the relationship, such as mapping each time value to its respective profit.

Function Mapping

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Common Mistakes

  • Confusing the concept of a function with a vague input-output relationship, without ensuring the uniqueness of the mapping.
  • Misinterpreting the limit process, particularly when the value of h approaches zero in the difference quotient.
  • Mixing up secant lines (average rate of change) with tangent lines (instantaneous rate of change).
  • Overlooking the importance of checking the domain of a function when modeling real-world phenomena.