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17Calculus Derivatives - Using The First Derivative To Analyze and Graph Functions

Single Variable Calculus
Multi-Variable Calculus

This page gets you started on the two main topics we use the first derivative for to understand what is going on with the graph of a function, increasing and decreasing intervals and maximums and minimums. First, we present a quick introduction on what we mean by increasing and decreasing functions and intervals.

Topics You Need To Understand For This Page

derivatives graphing

Increasing and Decreasing Intervals

In this section, we discuss what is meant by increasing and decreasing by looking at various functions. You need to know this when using critical numbers to determine increasing and decreasing intervals.

plot 1 - an increasing function

A function that is increasing is one that has a positive slope. Plot 1 is an example of an increasing function since it's slope is \(m=2\), which is positive, i.e. \( m > 0\). To keep this straight in my head, I think of myself as walking on the curve (in this case a straight line) from left to right (increasing x). If I am going up, the slope is positive and the function is increasing.

plot 2 - a decreasing function

A decreasing function is one that has a negative slope. In plot 2, the slope is \(m=-2\) and, since the slope is negative, \( m < 0\), this is a decreasing function. Using the same idea as above, I keep this straight by thinking about walking on the curve. If I am going downhill, the slope is negative and, therefore, is a decreasing function.

Some functions are neither increasing nor decreasing. In this case, we say the function is constant, the graph will have a slope of zero and be a horizontal line.

plot 3

Okay, so now that we have the idea of the behavior of entire functions, we are going to work with functions that are increasing in part of the graph and decreasing in others. The idea is break up the function into intervals (sections of the graph based on the x-value) so that only one thing is going on at a time. Let's look at the graph in plot 3.

Notice that there are two things going on. First, the graph is increasing from \(x=-2\) to \(x=-1\). Second, the graph is decreasing from \(x=-1\) to \(x=3\).
We would describe this graph as:
- increasing on the interval \( -2 < x < -1 \), and
- decreasing on the interval \( -1 < x < 3 \)

Notice that we specified open intervals, i.e. we did not include the endpoints. This is usually the way it is done. Also, we do not say anything for \( x < -2 \) and \( x > 3\) because those x-values are not in the domain. This is shown by the obviously filled circles at the points \((-2,0)\) and \((3,-1)\).

Okay, so now that you have a feel for what increasing and decreasing functions look like, let's discuss critical points. We use critical points to know where a function changes from increasing to decreasing or vis-versa.

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