Section 12.1: Derivatives and Graphs
Learning Objectives:
- Determine if a function is increasing, decreasing, or constant on a certain interval
- Calculate the critical numbers of a function
- Find the local extrema of a function
- Write out the First Derivative Test
- Apply the First Derivative Test to determine local extrema of a function
When is a function increasing? Decreasing?
Look at figure 12.2. The slope of the tangent line is ______when the car travels uphill (the function is ______), and the slope of the tangent line is ______when the car travels downhill (the function is ______).
Suppose has a derivative at each point in an open interval centered at , then
- If , is increasing
- If , is ______
- If , is constant.
The point is called a critical pointwhen one of these two things occur, either:
Look at figure 12.6. Generally speaking, when a graph has a peak it is called a local ______and when a graph has a valley it is called a local ______.
Copy down the statement in the shaded box towards the bottom of page 682 about local extrema and critical numbers, and then read the caution statement:
Read example 4 a) on page 683. Notice that they first find the critical numbers (and ), and then they compare to the graph of the function to see which number is a max and which number is a min. The idea here is that if we have an idea of what the function looks like, we can use that to help identify what critical numbers are max/mins.
#1) Let’s find the local extrema of .
Step 1: Calculate :
Step 2: Find critical numbers. Where does Are there any points where does not exist?
Step 3: Identify if a critical number is a max or min. Think about the shape of the graph of the function, sketch if needed.
The problem with the above strategy is that it requires us to have a good idea of what the graph of the function looks like. Without that information, the critical numbers alone don’t say much about whether a local max or a local min occurs. Luckily we have the first derivative test.
Write down the First Derivative Test below:
1.
2.
3.
Figure 12.12 on the top of page 684 provides a good graphical idea of what the first derivative test says.
Read Example 5.
#2)We already know from #1 that is the only critical number from . Instead of checking the graph, let’s use the first derivative test to see if is the location of a max or a min.
Step 1: We must choose an interval containing 0 but not containing any other critical numbers. Luckily, since 0 is the only critical number in this example, we can choose any number to the left of zero, and any number to the right. Let ____ and _____ (your choice).
Step 2: Fill in the table by evaluating the derivative:
/ / 0 // 0
So we can see from the table that we go from decreasing (since ) to increasing (since ). This is case 2 in the first derivative test, and we get the same answer as before.
Read Example 7. Now try the following. Compare your final answer to the correct answer found at the end of the section on page 691.
#3)Checkpoint 8.
Step 1: Compute the derivative.
Step 2: Find critical numbers.
Step 3: Apply 1st derivative test to determine any local extrema.