• Determine intervals on which a function is increasing or decreasing.
  • Apply the First Derivative Test to find relative extrema of a function.

Assignment

  • Vocabulary and teal boxes
  • p233 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 57, 58, 59–65 odd,75, 76, 80, 81–86, 90 97, 99, 101, 115–117

Additional Resources

Increasing and Decreasing Functions

The theme of this chapter is curve sketching, or how to determine the features of a curve without technology. In the first section we found out how to find extreme points. After a short detour with the Mean Value Theorem—which is used to prove a lot of the ideas in this section—we’re back to curve sketching with increasing and decreasing functions.

This is a pretty straightforward idea on the surface. If the derivative is positive, the slope is positive, so the function is increasing. Negative means decreasing. However, testing every number is not very productive, so we’ll combine this with the idea of critical points. Critical points were all candidates for local extrema, so that means in between those critical points, the slope will be consistently positive or negative.

Example

Let’s look at the function $f(x)=x^3-\frac{3}{2}x^2$. It’s derivative is $f’(x)=3x^2 - 3x$. There are no places where $f’$ doesn’t exist, so we can just set it equal to zero to find our critical points. Factoring gives us $0=(3x)(x-1)$, so at $x=0$ and $x=1$.

From here, it’s helpful to set up a table or a sign chart to keep track of your intervals. Regardless of how you choose to organize, pick a value in each interval and plug it into your derivative to see if it’s positive or negative. In our case, we have three intervals: $x < 0$, $0 < x < 1$, and $x > 1$.

With the end intervals, pick massive numbers to make your life easier. We only need a sign, so the exact value is unnecessary. In this case, our derivative is an even degree polynomial with a positive coeffecient, so the end intervals are positive, meaning the original function is increasing there.

For the other interval, use either the original or the factored derivative. One might be easier than the other depending on the number you plug in. Factored might be preferred since you just need the sign. Anything between 0 and 1 is positive, so $3x$ is always positive, but $x-1$ will always be negative, so a decreasing interval.

The First Derivative Test

This is pretty straightforward: if consecutive intervals switch from increasing to decreasing, then you have a relative maximum. It’s a minimum when going from decreasing to increasing.

This doesn’t wholly replace what you learned in 3.1. The first derivative test is good for curve sketching, while finding absolute extrema still requires evaluating and comparing critical points.