zuai-logo

Graphs of Functions & Their Derivatives

Sarah Miller

Sarah Miller

5 min read

Listen to this study note

Study Guide Overview

This study guide covers increasing and decreasing functions using the first derivative, f′(x)f'(x). It explains how to find these intervals by solving inequalities (f′(x)≥0f'(x) \geq 0 for increasing, f′(x)≤0f'(x) \leq 0 for decreasing) and interpreting the graph of f(x)f(x) and f′(x)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

  1. Introduction
  2. Finding Where a Function is Increasing or Decreasing
  3. Graphical Interpretation
  4. Worked Example
  5. Practice Questions
  6. Glossary
  7. 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)f'(x), describes the rate of change of f(x)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)≥0f'(a) \geq 0, then ff is increasing at x=ax = a.
  • If f′(a)≤0f'(a) \leq 0, then ff is decreasing at x=ax = a.
  • If f′(a)=0f'(a) = 0, the point...
Previous Topic - Critical PointsNext Topic - First Derivative Test for Local Extrema

Question 1 of 12

Let's start with an easy one! 🎉 If f′(x)>0f'(x) > 0 on the interval (a,b)(a, b), what does this tell us about the function f(x)f(x) on that interval?

f(x) is constant

f(x) is decreasing

f(x) is increasing

f(x) has a local maximum