Graphs of Functions & Their Derivatives

Sarah Miller

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Study Guide Overview

This study guide covers increasing and decreasing functions using the first derivative, $f'(x)$ . It explains how to find these intervals by solving inequalities ( $f'(x) \geq 0$ for increasing, $f'(x) \leq 0$ for decreasing) and interpreting the graph of $f(x)$ and $f'(x)$ . The guide includes a worked example, practice questions, a glossary of key terms like critical points, and key takeaways summarizing the first derivative test and graphical analysis techniques.

#Increasing & Decreasing Functions

#Table of Contents

Introduction
Finding Where a Function is Increasing or Decreasing
- Using the First Derivative
- Determining Intervals of Increase and Decrease
Graphical Interpretation
Worked Example
Practice Questions
Glossary
Summary and Key Takeaways

#Introduction

Understanding where a function is increasing or decreasing is a fundamental concept in calculus. This involves analyzing the first derivative of the function to determine the behavior of the function at various points.

#Finding Where a Function is Increasing or Decreasing

#Using the First Derivative

The first derivative of a function, $f'(x)$ , describes the rate of change of $f(x)$ .

If the rate of change is positive, the function is increasing.
If the rate of change is negative, the function is decreasing.

This allows us to determine whether a function is increasing or decreasing at a specific point:

If $f'(a) \geq 0$ , then $f$ is increasing at $x = a$ .
If $f'(a) \leq 0$ , then $f$ is decreasing at $x = a$ .
If $f'(a) = 0$ , the point could be a local minimum, maximum, or a point of inflection.

#Determining Intervals of Increase and Decrease

To find intervals where a function is increasing or decreasing, follow these steps:

Increasing Intervals: Solve the inequality $f'(x) \geq 0$ .
Decreasing Intervals: Solve the inequality $f'(x) \leq 0$ .

Exam Tip

The definitions for where a function is increasing or decreasing include the endpoints. However, exam guidelines often permit the point to be awarded even if the endpoints are not included. For example, " $f(x)$ is increasing for $1 < x < 5$ ."

#Graphical Interpretation

Sketching a graph of both $f(x)$ and $f'(x)$ can help identify where a function is increasing or decreasing:

On the graph of $f(x)$ :
- An upward slope from left to right indicates the function is increasing.
- A downward slope from left to right indicates the function is decreasing.
On the graph of $f'(x)$ :
- The portion of the graph above the $x$ -axis indicates the function is increasing.
- The portion of the graph below the $x$ -axis indicates the function is decreasing.

Key Concept

Between the critical points at $a$ and $b$ , if the cubic is decreasing, the graph of the derivative will be below the $x$ -axis between $a$ and $b$ .

#Worked Example

Find the interval(s) on which the graph of $f(x) = \frac{1}{4}x^4 + \frac{1}{3}x^3 - 3x^2 + 4$ is decreasing.

**Solution:**

The function is decreasing where $f'(x) \leq 0$ .
Find $f'(x)$ : $f'(x) = x^3 + x^2 - 6x$
Solve the inequality $f'(x) \leq 0$ : $x^3 + x^2 - 6x \leq 0$ $x(x^2 + x - 6) \leq 0$ $x(x + 3)(x - 2) \leq 0$
The easiest way to solve a cubic inequality is to graph it. Using a calculator or plotting it manually, we find:
- The function is decreasing for $x \leq -3$ and $0 \leq x \leq 2$ .
The intervals where $f(x)$ is decreasing are: $(-\infty, -3] \cup [0, 2]$

#Practice Questions

Practice Question

Determine the intervals where the function $f(x) = x^3 - 3x^2 + 2x$ is increasing and decreasing.

Practice Question

Find the intervals of increase and decrease for the function $f(x) = \sin(x)$ on the interval $[0, 2\pi]$ .

Practice Question

Given the function $f(x) = e^x - x$ , find the critical points and determine whether the function is increasing or decreasing at these points.

#Glossary

First Derivative: The rate of change of a function, denoted as $f'(x)$ .
Critical Point: A point on the graph where $f'(x) = 0$ or $f'(x)$ is undefined.
Increasing Function: A function where $f'(x) > 0$ over an interval.
Decreasing Function: A function where $f'(x) < 0$ over an interval.

#Summary and Key Takeaways

#Summary

The first derivative, $f'(x)$ , helps determine where a function is increasing or decreasing.
Solve $f'(x) \geq 0$ for increasing intervals and $f'(x) \leq 0$ for decreasing intervals.
Graphical interpretation aids in visualizing these intervals.

#Key Takeaways

First Derivative Test: Use $f'(x)$ to find where the function is increasing or decreasing.
Critical Points: Important for identifying local maxima, minima, and points of inflection.
Graphical Analysis: Helps to clearly see the behavior of the function.

Exam Tip

Always double-check your intervals and endpoints, especially in exam scenarios.

#Additional Guidelines

Exam Strategy: Always clearly show your work when solving inequalities and graphing functions.
Real-World Applications: Understanding increases and decreases in functions is crucial in fields like economics, engineering, and the natural sciences.
Difficulty Rating: This topic is considered intermediate. It builds on basic derivative concepts and introduces more complex analysis.

These notes are designed to help you thoroughly understand and apply the concepts of increasing and decreasing functions. Practice regularly and consult these notes to reinforce your learning.