1.2 Rate of Change
Contents
Objectives:[edit]
- Students will be able to describe on what intervals a function is increasing/decreasing
- Students will be able to find the average rate of change of a function on a given interval
- Definitions
- increasing, decreasing, average rate of change
Lesson Guide[edit]
Many Webwork problems require students to enter solutions in interval notation. Although writing an interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\leq x \leq b} is acceptable, to make Webwork smoother we teach interval notation. Ideally, you should make a point of writing all of your (as the instructor) answers in both notations to help students learn to smoothly go between notations. It is important to note that finding the average rate of change is the same as finding the slope of a line. Although we will discuss finding the slope of a line later in the chapter, many students will be familiar with the concept, and it will hopefully help their understanding of the topic.
Warmup[edit]
Have students do Problems 1 and 2.
Review interval notation[edit]
[10 minutes] Introduce interval notation through several examples highlighting the meaning in words, inequalities, and interval notation. Be sure to illustrate intervals graphically on a number line.
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Instructor Example |
Suggested Example |
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This is a sentence that is in place to generate space in the table |
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In groups have students draw a number line on the board and have a representative from a couple of tables come to the board and draw the interval on the number-line.
[5 minutes] Review plotting points from a table on the coordinate plane. A good example would be the graph of the first problem on the worksheet since from here you can also highlight increasing/decreasing.
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Instructor Example |
Suggested Example | ||||||||||
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This is a sentence that is in place to generate space in the table |
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Describe on what intervals a function is increasing/decreasing[edit]
[10 minutes] Ask students on which interval the function from Problem 2 is increasing or decreasing. Accept all answers that are correct even if they are not the largest such intervals. If no students provide a "small" interval as a solution then you should give a "small" interval as a potential solution. Ask students to try and decide which answer is best and use this to motivate the idea that when we ask on what intervals a function is increasing/decreasing we really mean what are the largest intervals on which the function is increasing or decreasing. You should assume that the function is graphed completely. i.e. don't add extra arrows to the problem. If the graphs end at "not-nice" points, just have students estimate.
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = f(x)} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\leq x \leq b} . We say that
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is increasing on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} if the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} increases as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} increases on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} .
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is decreasing on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} if the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} decreases as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} increases on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} .
If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is always increasing (respectively, decreasing) we say it is increasing (respectively, decreasing).
[10 minutes] Have students attempt Problems 3 and 4.
Once a few of the groups have reasonable answers for Problems 3 and 4, ask them how their answer would change if you added the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-1,12)} to Table 1. Use this discussion to help the students realize that the table is an incomplete picture of the function and while we can hypothesize, we actually do not know what happens in-between data points.
Students have difficulty identifying whether functions are increasing or decreasing without looking at a graph, so you may ask them to give a sketch of the points for each part of the first problem. Also, students often don't understand that we are asking for the input values on which the output increases or decreases.
If a group gets ahead you can have them discuss Problem 5. Problem 5 should be thought of as an optional problem for discussion by groups who have finished everything else.
Find the average rate of change of a function on a given interval[edit]
[10 minutes] We can do more than just tell if a function is increasing or decreasing. We can also measure how fast a function is increasing or decreasing using the average rate of change. Draw only the axes, function and line segment from (0,0) to (2,4) then do the example below for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\leq x\leq 2.}
The average rate of change of a function should be introduced through an example. Start by drawing the secant line on the quadratic of Problem 4 between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1,0)} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3,2)} . Then, on the white board, erase the quadratic leaving just the line (or use the document camera and stop projecting the quadratic.). Ask students to compute the slope of this line. Use this to motivate the definition of average rate of change of a function.
We define the average rate of change of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t) \text{ on } [a,b]} by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dfrac{\text{rise}}{\text{run}} = \dfrac{\text{ change in } f(t)}{\text{change in } t} = \dfrac{\Delta f(t)}{\Delta t } = \dfrac{f(b) - f(a) }{b-a}}
Do several examples of computing the average rate of change of a function represented by a table, ordered pairs, graph, or formula. Be sure to use different function names in these examples. At this stage, students aren't used to the flexibility of function notation, and students have a hard time generalizing the average rate of change formula to functions other then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)} .
Be sure to emphasize the units of the average rate of change if the function has specific units given.
Examples:[edit]
[10 minutes] Have students do Problems 6-9.
[5 minutes] Assign Problem 10 to a couple of advanced groups and inform them that they will be asked to present their solutions to the entire class in 5 minutes. Assign Problems 11 and 12 to other groups and inform them that they will present their answers at the board in a few minutes. Problem 10 is more of a challenge and Problems 11 and 12 are a little easier.
[10 minutes] Have people present their solutions to Problems 10-12. Close class by recommending that students complete the Focus Problem as homework. (if the class is moving quickly through the material these problems can be done in class)
The CRA[edit]
[10 Minutes] Take 10 minutes to introduce the CRA. Students should have signed up to take the CRA in the learning commons earlier in the week.
[30 minutes] Give the CRA.
Comments[edit]
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.
I introduced rate of change using this example: "You are driving from Lincoln to Omaha, and the distance you've travelled is given by this graph. [Graph has a small slope marked "Lincoln" for the first quarter, bigger sloped marked "highway" for the middle half, and smaller slope marked "Omaha" for the last quarter]. What is your average speed over the entire trip? What is your average speed on the highway?" I like starting with this problem because speed is an intuitive connection to rate of change, but we don't see it in the workbook until Problem 9. -Juliana
When I first introduced interval notation in class, I asked them what they thought (1,2] meant, and I was told that it is a point. I should have been more specific in my question, or otherwise brought up the notation differently, but my point (haha) is to make others aware that some students will see interval notation and think that they are looking at a point. Just something of which to be mindful. -Kelsey
Also, do not take for granted that students know the meaning of the phrase, "the value of f". I asked students what they thought this meant and I heard crickets for a long time until somebody guessed that it meant "x", the input value. Students may not be familiar with or remember some of the jargon in the definitions, so be careful. - Kelsey
Note: When introducing functions in the previous lesson, a table is used to represent a function. We have to assume that what the table we are handed is complete for the problem (we make an analogous assumption for the graph in 1.2 problem 2). Note that if we take an equation and create a table using some input values, then the table does not give us a complete picture of the function. So in the portion of the lesson plan where it mentions adding in the point (-1, 12) in to the table in section 1.2 problem 3(a), we should modify the language a bit to be more consistent. -Elizabeth
- I'd be cautious here -- there is no reason to not assume the table is NOT 'complete'. The table is itself a function, we don't need the table to represent some underlying graph or equation for it to be a function.
