Discovering the Derivative

Discovering the Derivative with a Graphing Calculator

This is an outline of how to introduce the idea that the slope of the line tangent to a graph can be found, or at least approximated, by finding the slope of a line through two very close points in the graph.  It is a set of graphing calculator activities that will use graphs and numbers to lead to the symbolic form of the derivative.

You may work through the activities with your class (which is what I would do) or you could write and distribute them and let your class do them a laboratory exercise. Before starting students should know how to use their calculator to graph, to trace to points on the graph, and how to save and recall the coordinates from the graph to variables on the home screen using the graphing calculator’s store feature.

I suggest you work through these three times (or more) using different functions. I will work with $y={{x}^{2}}$. A good second example is $y={{x}^{3}}$, and a third example to use is $y=\sin \left( x \right)$. Use simple functions, because you will want the students to see the answers without too much trouble.The procedure is the same for all.

Part 1:

1. Begin by asking students to enter$y={{x}^{2}}$ in their calculator asY1 and graph it in a standard, square window. (Do not use the decimal window as “nice” decimals are not necessary or helpful.)

Figure 1

2. Next, have them trace over to different points on the graph (some should go left and others to the right); they should all end up at different points. Then have them zoom-in 6 or 8 times until the graph looks linear. (This is local linearity – functions that are differentiable are locally linear.)
3. Then push TRACE to be sure the cursor is on the graph. The coordinates of the point are on the bottom of the screen. Go to the HOME screen and save the two values as a and b. Think of this first point as (a, b).
4. Return to the graph screen and push TRACE. This should return the cursor to the first point. (If not, close is okay.) Then click to the right or to left once, or twice at most, to move to a nearby point on the graph. Return to the home screen and save the new values to c and d for the second point (c, d).
5. On the home screen use a, b, c, and d to write the slope of the line through the two points. See figure 1. (Go around the room as they are doing this and make sure students are getting this – their slope should be approximately twice a or c.)
6. Return to the equation screen and enter the equation of the line through the two points asY2. (See figure 2). Graph this equation with the parabola.

Figure 2

7. Have the students record their values of a, b, c, d and m on paper (three decimal places will be enough) and also write a description of what they see on the graph and why they think this is so. (This is in case they lose the numbers on their calculator when they do another graph and also because you will need them later in the next part of the exploration.)

Repeat the same steps separately with the other two functions and record the results in the same way. Write their numbers and observations. Discuss the observations with the class.

• Of course, the lines should look tangent to the graphs, but since they contain two points of the graph, they cannot actually be tangent.
• Discuss how a line can be tangent to a graph. How is this different from a tangent to a circle?
• Ask what could be done to make their line even closer to being tangent. (Use points closer together.)

Part 2:

Now you have homework to do. Collect the student’s data and combine it into a list with columns for a, c, and m. The points do not have to be in order. Leave any “wrong” points for discussion; if there are none, you might want to make one up and include it. Do this for each of the three sets of data. Make a copy for each student. Enter the numbers for a and m as lists in your emulator and make a dot-plot of the points (a, m) = (a, slope at x = a).

1. Return the lists of points to the students and ask them to study the list and see if they can see any obvious relationship between the numbers on each line. Answers for y = x2 should be the m is approximately twice either a or c; or maybe some will see that m is approximately a + c. Answers for y = x3 will be less obvious (three times the square of a). Answers for y = sin(x) will not be obvious at all.
2. Using the emulator, separately for each of the three sets of data, make a dot-plot of the numbers (use a square window). Ask the student to discuss what they see. See if they can find an equation of the graph of the dot-plots. Now the equation of the data set for sin(x) should be obvious. Plot their guesses on top of the points and see how close they come.

Part 3:

Now guide the class through the symbolic explanation of what they did. Ask them to explain and write in symbols specifically what they did. The idea here is for you, the teacher, to ask lots of leading questions until the class decides on the best answer.

1. Call the first point (a, f(a)). Let h = “a little bit.” then the second is the point (a plus a little bit, f(a plus a little bit)) or (a + h, f(a + h)). Recall that h may have been negative for some students so the second point may actually be to the left of the first. Then help them come up with

$\displaystyle m\approx \frac{f\left( a+h \right)-f\left( a \right)}{\left( a+h \right)-a}=\frac{f\left( a+h \right)-f\left( a \right)}{h}$

Or they may prefer

$\displaystyle m\approx \frac{f\left( a \right)-f\left( c \right)}{a-c}$

1. Ask how this could be made “less approximate” and more actually equal. (Answer: smaller and smaller value of h.) Ask them to find the value of h = ac for their points. How small are they? How can you make them really small? (Find a limit.)
2. Notice that h cannot be zero in these expressions. Keep hinting until someone comes up with the idea of finding

$\displaystyle \underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( a+h \right)-f\left( a \right)}{h}$  or  $\displaystyle \underset{a\to c}{\mathop{\lim }}\,\frac{f\left( a \right)-f\left( c \right)}{a-c}$

1. Next have the students calculate the limits below by actually doing the algebra. (They will not be able to handle the sin(x) at this point so save that for later.)

$\displaystyle \underset{h\to 0}{\mathop{\lim }}\,\frac{{{\left( x+h \right)}^{2}}-{{\left( x \right)}^{2}}}{h}\text{ and }\underset{h\to 0}{\mathop{\lim }}\,\frac{{{\left( x+h \right)}^{3}}-{{\left( x \right)}^{3}}}{h}$

1. Compare the answers here with the earlier work and guesses, and discuss.

… And now you are ready to define the limits as the ’(a) derivative of f at x = a.

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Graphing Calculators

I was asked to pass the following information along to you. I decided to do so because you may want to know about one of the newest CAS calculator models and because their follow-up offer for attending the summer institute is so generous. If your school and students need help getting calculators and/or you want to keep up with the latest trends, you may be interested in looking into this.

The calculator is the new Hewlett-Packard HP Prime. It is a CAS calculator with really good graphing features. HP is offering the HP Prime AP Summer Institute Program, a 3-Day Institute this summer in either Statistics or Calculus with expert teacher trainers to introduce their new Mathematics Solution, the HP Prime Wireless Classroom. Following the Summer Institute, teachers who attended will receive a donated HP Prime Wireless Classroom Kit for their school with 30 HP Prime Graphing Calculators and the HP Prime Wireless Kit (a $5,000 value!). The institute is an opportunity to improve your knowledge of teaching mathematics with a technology that makes learning intuitive for students and receive the technology to keep for classroom use. For more information click the link https://h30602.www3.hp.com/assets/hpmath/web1.html Some Graphing Calculator History Graphing calculators first came on the market around 1989. In the early 1990s after it was announced that graphing calculators would be required on the AP Calculus exams starting in 1995, there were a series of workshops following the AP calculus reading, then at Clemson University. They were called the Technology Intensive Calculus for Advanced Placement conference or TICAP. Half the readers were invited to stay after the reading for the conference. The next year the other half were invited, and others the third year. Casio, Texas Instruments, Hewlett-Packard, and Sharpe all contributed and provided their calculators to the participants. The Texas Instrument calculators (then the TI-81 and TI-82) emerged the most popular and have since been the most popular in the United States. TI to their credit makes a good product and provided, and still provides, lots of help for teachers in the form of print material, programs for the calculators, workshops, and meetings. They have developed ways to connect the classroom’s calculators to the teacher’s computer. Other manufacturers have done the same, but not on TI’s scale. TI has made improvements in their calculators and other manufacturers have made newer and improved machines as well. While similar in functionality, I think the Casio PRIZM to be a bit better than the latest TI-84 model; it is also a bit less expensive. TI’s ‘Nspire line is an excellent CAS machine. Casio also has several CAS calculators and HP has now come out with the new HP Prime model (mentioned above). There are others. TI has a whopping 92% market share with Casio far behind at 7%. While their machines are excellent, Casio and HP are playing catch-up and have a long way to go. If you are just starting out or have limited funds for your class, you might consider a different brand. I often think the main problem is that the buttons are all in the wrong place! That is, the keyboards are different than the TIs we all learned on. They are different for you, but students who have never learned the old way will have no trouble with the keyboards. You won’t either – just sit down with the guidebook for a couple of hours and you’ll become an expert (or let the kids help you!). You can also use the manufacturer’s online instructions or go to a summer institute such as HP’s mentioned above. Some Graphing Calculator Opinion Now comes the real heresy. Graphing calculators don’t graph all that well. Their screens are small and often crowded. Tablets such as an iPad, or computers do a much better job. (For example, TI-Nspire’s operating system is available as an iPad app that is easy to use and much easier to see (okay maybe I’m getting old and my eyes are not what they once were). Still the many other graphing and mathematics related apps available are fabulous. Graphing, statistics and geometry apps abound and will only increase in number and functionality. This is the future. The reason graphing calculators are still here is because the Educational Testing Service, for good reason, will not let students use any device with a QWERTY keyboard on their exams including the Advanced Placement exams. The primary reason is that they are afraid that students will copy the secure questions using the QWERTY keyboards on iPads and computers. For some reason, students apparently cannot figure out how to do this with the alphabetic keyboards on graphing calculators. (Or as Dan Kennedy once quipped, there is nothing wrong with the old method of writing them on your cuff.) Other more important reasons tablets are not allowed include being able to photograph the questions, get information and help through the internet during the exams, or communicating with others in or out of class with tweets and instant messages. These are real problems that need to be considered, but I cannot believe a work-around is not possible. It must be possible to make an app that will allow only the use of approved apps during exams. After all they have done that for graphing calculators. If technology helps students learn mathematics – and I believe it does – then students should have the best available technology. End of sermon. Take a moment or more to consider the new and improved calculators. Update – iPad’s “Educational Standardized Testing” Option I wrote the paragraphs above a few days ago. This morning the new operating system 8.1.3 for iPad became available. They have a new feature for “educational standardized testing.” You can turn it on under Settings > Accessibility > Guided Access. Once turned on you open any app, triple-click the home button, and the controls for that one app appear on the screen. The individual settings are slightly different for each app. You may turn off the keyboard, turn off the touch screen, or disable the dictionary. On apps with their own buttons you can turn off any or all of the buttons by circling them. A time limit may also be set. It appears for now that each iPad must have these features adjusted individually. Unfortunately, to change or turn the restricted features on again all a student needs is his or her passcode or fingerprint. In addition, there should be a way to turn off all the other apps where students may quite legitimately have notes or homework saved. (Most of my students last year in one-to-one classrooms took most of their notes and did their homework on their iPads.) This is a good start, but it has a long way to go before it can be used in group settings. Stay tuned for updates. Calculators First some history and then an opinion I remember buying my first electronic calculator in the late 1960s. It did addition, subtraction, multiplication, and division, and could remember one number. It displayed 8 digits and had a special button that displayed the next eight digits. I remember using those next eight digits never. To buy it I had to drive 40 minutes and spend$70 – expensive even today.

The square root of 743 computed using the algorithm discussed in the post. The third iteration (fourth answer) is correct to 10 digits.

With it I learned an iterative algorithm for finding square roots: guess the root, divide the guess into the number, average the quotient and the guess, repeat using the average as the new guess.  You could do it all without writing anything down. (See the illustration on a modern calculator – accurate to 8 decimals in only 3 iterations (fourth answer), but then I could find the next 8 with the special button.)

Since then, I’ve had lots of calculators of all sorts.

Graphing calculators hit the general market around 1989 or 1990. This was the same time as the “reform calculus” movement. The College Board announced that the AP calculus exams would require graphing calculators in 1995 – five years to get the country ready.

The College Board held intensive training immediately following the reading. These were the TICAP conferences (Technology Intensive Calculus for Advanced Placement). Half the readers were invited for the first year and the other half for the second, then more for the third year.

Casio, Hewlett-Packard, Texas Instruments all gave participants calculators to use take home. Sharpe lent them calculators (and we haven’t heard of Sharpe since). Sample lessons were taught using Hewlett-Packard CAS calculators and then the same lesson was taught using TI-81s. The HP computer algebra system calculators, with far more features but using the far more complicated reverse Polish notation entry system, lost in the completion to the simpler to use, but less sophisticated TI-81s.

The teachers were not all happy. A friend of mine, due to retire in 2-3 years gave up his AP calculus classes early so he would not need to learn the calculators. Others embraced technology. The AP program forced the graphing calculator into high schools where they were used to improve learning and instruction. Yet even today not all high schools have embraced technology.

The calculator makers, especially Texas Instruments, provided print materials, software, workshops and conferences that helped teachers learn how to use graphing calculators in their classes at all levels.

Technology, as a way to teach, learn, and most importantly, do mathematics, caught on big time. And that was and is a good thing.

I think graphing calculators are very quickly becoming obsolete and should be phased out.

Technology has bypassed graphing calculators. Tablet computers, PCs, Macs, iPads, and the like, even smart phones, can do everything graphing calculators can do. They are more versatile. The larger screens are easier to see and can show more information without crowding.

The initial investment may be more than for a graphing calculator, but once purchased the apps are relatively cheap. There are many free apps that not only do computations and graphing, but CAS operations as well. Interactive geometry and statistics apps are also available.

These, along with online textbooks and internet access, put everything students need to learn math literally at their fingertips. Graphs and other results can be easily copied and printed, or pasted into note-taking apps.

One disadvantage is the initial cost for the hardware (but of course many students already have the hardware). The other disadvantage is the ability to communicate and find help both in the room and around the world during tests. Photographing the questions for later use by others is another concern.  I think (hope) it is just a matter of time before this problem can be overcome perhaps with an app that allows access only to the apps the teachers allow for tests.

Technology, like time, marches on.