# Category Archives: Calculus

## Introduction to Curve Sketching, Part 2

While students were working on this antiderivative from a Desmos Activity called Sketchy Derivatives, I heard several students ask how they know where to place the y-intercept. We didn’t have time to answer that question during the first lesson of the unit, but we started with it during the second lesson. Should the cubic start decreasing or increasing? How do you know?

[Yes, I do know that I can anonymize the names. However, by the time I thought of that, these two responses were no longer adjacent. It would be nice to be able to drag the responses to different locations in case you want to compare/contrast several specific responses at the same time. ☺]

Then we looked at these. They all have the same basic shape. Is one more right than the other?

Students began to think about all of the curves that have y’=2x, and so while they know something about the importance of the constant, one won’t be more right than the other until we learn about area under the curve.

I had the Second Derivative Grapher from Calculus Nspired queued during the previous class, but I decided it would be better for students to “do” instead of “discuss” for the last few minutes of class. We looked at it next to make the relationship between the second derivative and concavity of the original function more clear.  Students put all of this information together to begin to analyze a function given the graph of its derivative. The results of our formative assessments seem to indicate that students have a better understanding of the relationship between f-f’-f” than they have in the past. I’ll know for sure later today, though, after their summative assessment. And so the journey continues …

## Introduction to Curve Sketching, Part 1

Learning Intentions:

Level 4:

I can use the graph of the derivative to sketch a graph of the original function.

Level 3:

I can use the graph of the original function to deduce information about the first and second derivatives.

I can use the graph of the derivative to deduce information about the second derivative and the original function.

I can use the graph of the second derivative to deduce information about the first derivative and the original function.

Level 2:

I can determine when a function is concave up or concave down and where it has points of inflection.

Level 1:

I can determine when a function is increasing or decreasing and where it has maxima and minima.

We were on the first day of a new unit. I included two questions on the opener to ensure students know what we mean by increasing/decreasing and concave up/concave down intervals. As expected students were familiar with increasing/decreasing and not so familiar with concave up/concave down.

Based on the results, we discussed what it means to be concave up and concave down. Someone asked how we would be able to tell for sure where the graph changes concavity, which we get to learn during the unit.

We started the lesson with a few Quick Polls for students to determine which graph was the derivative, given the graphs of a function and its derivative. The polls were based on Graphical Derivatives from Calculus Nspired. I sent the poll, asked students to answer individually, stopped the poll, asked students to explain their thinking to a partner. If needed, I sent the poll again to see whether they wanted to change their response after talking with their partner. I had 6 polls prepared. I sent 3.

I listened while students shared their thinking. I selected three conversations for the whole class.

1. A student who knew which was which based on the power rule, which she learned during the last unit.
2. A student who knew that the slope of the tangent line at the minimum of the parabola should be zero, which is the value of the line at z=0.
3. A student who noticed that the line (derivative) was negative (below the x-axis) when the parabola was decreasing and positive (above the x-axis) when the parabola was increasing.

Again, as I listened to the pairs talking, I selected a few students to share their thinking with the whole class.

1. The first student who shared used the maximum and minima to determine which had to be the derivative, since the derivative is zero at those x-values.
2. The second student thought about what the slope of the tangent line would be at certain x-values and whether the y-values of the other function complied.
3. A third student volunteered a fourth student to discuss her thinking: she noted that the graph of the function (b) changed concavity at the max/min of the derivative (b). After students talked, I sent the poll again to see if anyone was convinced otherwise. Two students briefly discussed how they used increasing/decreasing and concavity to determine the derivative. Next we began to solidify what increasing/decreasing and concave up/concave down intervals look like using Derivative Analysis from Calculus Nspired.

I asked students to notice and note.

Where is the function increasing? Where is it decreasing?
What is the relationship between the slope of the tangent line and where the function is increasing and decreasing?  Where is the function concave up? Where is it concave down? What does the tangent line have to do with where the function is concave up and concave down? Can you look at a graph and estimate intervals of concavity? I was able to see what students were noting on paper and hear what they were noting in our conversation, but I didn’t send any polls during this part of the lesson.

Next we looked at Derivative Grapher from Calculus Nspired. We changed the graph to f(x)=cos(x). We already know the derivative is f’(x)=sin(x). What if we were only given the graph of the derivative? How could we use that graph to determine information about the original function?  I had more for us to discuss as a whole class, but I wanted to know what they had learned before the class ended. I used a Desmos Activity called Sketchy Derivatives to see what students had learned – given a function, sketch its derivative; and given a derivative, sketch an antiderivative. The original activity was from Michael Fenton. I modified it to go back and forth between sketching the derivative and antiderivative instead of doing all derivatives first and all antiderivatives second, and I added a few questions so that students could begin to clarify their thinking using words.

We spent the last minutes of class looking at an overlay of some of their sketches.

Could you figure out exactly where to sketch the horizontal line? Most students have the vertex of the parabola near the right x-coordinate. Should the antiderivative be concave up or concave down? Most students have the derivative crossing the x-axis near the correct location. The bell rang. Another #lessonclose failure. But thankfully, there are do-overs as the journey continues …

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Posted by on October 17, 2016 in Applications of Differentiation, Calculus

## Derivative Rules

Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.

Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

Learning Objective 2.1C: Calculate Derivatives

Essential Knowledge 2.1C2: Specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric.

Essential Knowledge 2.1C3: Sums, differences, products, and quotients of functions can be differentiated using derivative rules.

Mathematical Practice for AP Calculus (MPAC) 1: Reasoning with definitions and theorems

Students can: develop conjectures based on exploration with technology.

How do you provide students the opportunity to develop conjectures?

After determining the derivative of a few quadratic functions using the definition, we use our TI-Nspire Computer Algebra System (CAS) software to explore derivatives. We use the power rule to make conjectures about the product rule. (I think that I saw this suggestion in a Mathematics Teacher magazine in the early 90s, but I can’t find the reference now.) We know what the derivative should be, because we know the derivative of x^5. How could we use f, f ‘, g, and g ‘ to get to what we know is the derivative from the power rule?

Once students made conjectures about the product rule, we formalized the rule.

I asked students to predict the derivative of f(x)=sin(3x). As expected, many thought that it would be f ’(x)=cos(3x). When we looked at the graph of the derivative of f(x), students realized that f ‘(x)=3cos(3x). We used CAS to explore the chain rule (power and composite) in more detail.

Students practiced “Notice and Note”. Several generalized the chain power rule before I asked.

Once students knew the chain rule, we used the chain rule to derive the quotient rule.

And so the journey providing opportunities for students to make sense of rules instead of just telling them rules continues …

Posted by on September 14, 2016 in Calculus, Derivatives

## Seeing the Definition of Derivative

Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.

Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

Mathematical Practice for AP Calculus (MPAC) 2: Connecting Concepts

1. relate the concept of a limit to all aspects of calculus
2. Students can connect concepts to their visual representation with and without technology.

How do you introduce the definition of a derivative?

We start with the visual of a tangent line at a point and a secant line containing the point. We don’t need calculus to determine the slope of the secant line. We do need calculus to determine the slope of the tangent line.

How might we use the slope of the secant line to determine the slope of the tangent line? 1 Comment

Posted by on September 9, 2016 in Calculus, Derivatives

## Seeing the Derivative

Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.

Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

Mathematical Practice for AP Calculus (MPAC) 2: Connecting Concepts

1. Students can connect concepts to their visual representation with and without technology.

How do you introduce the concept of a derivative?

We start with visuals of the derivative (from a Getting Started with Calculus activity called Derivative Trace).       Tangent Line Demonstration from Calculus Nspired has some similar ideas.

We then look at the Derivative Grapher to connect the slope of the tangent line to the graph of the derivative, changing the original function as requested by the students. We begin to develop some common language around derivatives before we formalize what is a derivative, before we formalize the definition of a derivative.

Posted by on September 8, 2016 in Derivatives

## Seeing Limits

Big Idea 1 from the 2016-2017 AP Calculus Curriculum Framework is Limits.

Enduring Understanding 1.1: The concept of a limit can be used to understand the behavior of a function.

Mathematical Practice for AP Calculus (MPAC) 2: Connecting Concepts

Students can connect concepts to their visual representation with and without technology.

Mathematical Practice for AP Calculus (MPAC) 4: Multiple Representations

Students can associate tables, graphs, and symbolic representations of functions.

Students can develop concepts using graphical, symbolical, verbal, or numerical representations with and without technology.

We begin calculus with a discussion of limits. I throw a dart at the dart board and ask what just happened.

You hit the bullseye.

(I really did this year – on the second try – after missing the board completely on the first try.) But how did the dart make it to the board? It had to go half the distance to the board. And then half the distance again. And again. And again. How did the dart make it to the board? The other demonstration is at the recommendation of my grandfather, now 101, who taught high school mathematics for over 50 years. I take a measured piece of yarn (1 meter), cut it in half, and begin to make a pile of the “halves”. How long does the yarn measure in the pile if we put it end to end? It gets closer and closer to measuring 1 meter, but does it ever make it to 1 meter?

How do you provide opportunities for students to “see” limits?

I used to [ineffectively] wave my hands. Now we use technology. Students won’t use this type of visualization while they’re taking a test, but we find it invaluable while they’re learning.       Posted by on August 24, 2016 in Calculus, Limits & Continuity

## Infestation to Extermination

We used a problem from the Calculus Nspired activity Infestation to Extermination recently during our unit on differential equations:

The rate of increase of bugs is proportional to the number of bugs in a certain area. When t=0, there are 2 bugs and they are increasing at a rate of 3 bugs/day.

What does this mean?

I set the mode to individual and watched as students worked.

Many recognized that the rate of change changes.

Several used the initial condition to write a statement about the rate of change. Eventually, we went back to the given information to decipher what it was saying. And then we (anti)derived the model for exponential growth, which of course students recognized using in a previous math course. So what are the constants for this particular model?

I sent a poll to collect their model honestly having no idea that the bell was going to ring in less than two minutes. A few students correctly answered before the end of class. Productive struggle isn’t fast.

I should have paid better attention to the time … I really had no idea it had taken us as long as it did. But students were engaged in “grappling with mathematical ideas and relationships” the entire time. That’s got to be better for their learning than them watching me tell them how to work the problem.

What opportunities are you giving your students to struggle productively? Even if you don’t “cover” as much as you think you should?

And so the #AskDontTell journey continues …

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Posted by on March 8, 2015 in Calculus

## The Fundamental Theorem of Calculus

How do you provide an opportunity for your students to figure out the relationship between differentiation and antidifferentiation?

We have used the Calculus Nspired activities The First Fundamental Theorem of Calculus and The Second Fundamental Theorem of Calculus for several years now to improve our understanding of the relationship between a function and its accumulation function. I actually do print the student handouts for these activities and give students time during class to make sense of the relationship between differentiation and antidifferentiation.

It was time for our whole class discussion.

We defined the accumulation function using a definite integral. What do we know? Students had figured out earlier that the definite integral of f(x) from a to a would be 0 and concluded that F(a)=0. Students recognized that the value of the definite integral of f(x) from a to d would be F(d) and that the value of the definite integral of f(x) from a to c would be F(c).   Suppose we want to calculate the definite integral of f(x) from c to d. They could tell area-wise that was equivalent to finding the definite integral of f(x) from a to d and subtracting the definite integral of f(x) from a to c, which of course gives us F(d)-F(c). And what does F(x) have to do with f(x)?

They could tell from the exploration that F(x) is the antiderivative of f(x).

Really? You mean we don’t have to do the limit-sum-infinite-number-of-rectangles every time? Really. You’ve earned the Fundamental Theorem of Calculus. We talked for a little while about average value using the Calculus Nspired activity MVT for Integrals, and then checked in on their understanding.

As we moved into the second part of the Fundamental Theorem of Calculus, I posed a question to see how they would answer. (Remember that at this point, they’ve been using the FTOC for about 20 minutes.) I was excited about a few students getting it right. Without discussing the correct responses with the whole group (I showed their answers but had Show Correct Answer deselected, I sent another question, which unearthed their misconception and revealed my initially bad question. The students who got the answer correct in the first question had gotten it correct the wrong way, but their mistake wasn’t revealed because sin(-π)=0.

By now we were past the bell, and so we started over the next lesson with the second part of the Fundamental Theorem of Calculus.

It’s always exciting to find both the right questions to ask (the ones that reveal student misconceptions) and the wrong questions to ask (the ones that hide student misconceptions) so that I can continue asking the right ones and discontinue asking the wrong ones. In this lesson, I found at least one of each. And so the journey finding and asking the right questions continues …

Posted by on February 27, 2015 in Calculus

## Area between Curves

Our learning goals for the Applications of Definite Integrals unit in calculus are the following:

I can calculate and use the area between two curves.

I can use the disc and washer methods to calculate and use the volume of a solid.

I can use the shell method to calculate and use the volume of a solid.

I can calculate and use volumes of solids created by known cross sections.

During the lesson focusing on the first goal, we used a scenario from a TImath activity The Area Between to start our conversation.

I rarely send the TNS documents as is to my students or give them a copy of the printed student handout (even though I learn from both in my own planning of how the lesson will play out). This activity gave the following information on the first two pages:

Suppose you are building a concrete pathway. It is to be 1/3 foot deep.

To determine the amount of concrete needed, you will need to:

– calculate area (the integral of the top function minus the bottom function

– calculate volume (area multiplied by depth)

The borders for the pathway can be modeled on the interval -2π ≤ x ≤ 2π by

f(x)=sin(0.5x)+3

g(x)=sin(0.5x)

On the next page, graph the functions. Use the Integral tool to calculate the area under f1 and f2. Then, use the Text and Calculate tools to find the volume of the pathway.

Which takes away any opportunity for students to engage in productive struggle.

Suppose you are building a concrete pathway that is to be 1/3 foot deep. The borders for the pathway can be modeled on the interval -2π ≤ x ≤ 2π by f(x)=sin(0.5x)+3 and g(x)=sin(0.5x).

(I’m fully aware that giving them even this much information takes away from the modeling process … but there is always give and take, and for this lesson, the learning goal wasn’t whether they could determine functions for modeling the sidewalk.)

They decided to graph the functions. And talked about how they could calculate the area between the curves.

They had never used the Integral tool for graphs, much less the Bounded Area tool, so they oohed and aahed gasped in amazement.

Sydney asked: Is that the only way to get the area between the curves?

(I knew that she was looking for and making use of structure, composing and decomposing the sidewalk into regions with equal area).

We made Sydney the Live Presenter, and she used the Integral tool to calculate the area between f(x)=sin(0.5x)+3 and the x-axis from -2π to 2π. So how can we calculate the amount of concrete needed? The integral and bounded area tools are helpful for visualizing what you’re calculating, but you can’t use those tools on the AP Exam.

And so the students decided to calculate the area between the curves and then multiply by 1/3 to get the volume of the pathway.

Because they were able to tell me what to do, I almost didn’t send a Quick Poll to collect a definite integral that would calculate the volume. I wanted to hurry up and get to a card-matching activity similar to Michael Fenton’s that I knew would be helpful, but instead I eased the hurry syndrome and sent the poll. What I saw and heard was well worth the time that it took.

Can you spot the students’ misconception? Several students were multiplying the definite integral by and by 1/3, to represent height times base times depth, instead of recognize that the definite integral represented height times base (area), and not just height. (They knew this … we had summed the areas of an infinite number of rectangles for a certain base to calculate area under the curve. But they obviously didn’t know this like they needed to.)

When we calculated their integral, we didn’t get (1/3)*37.699, as expected.

Next I purposefully choose a region for which the upper and lower boundaries changed.  We had a nice look for and make use of structure discussion about different ways to write a definite integral for calculating the area of the region. Many of you might notice that there is more opportunity to look for and make use of structure for the concrete pathway. I never asked whether you really need calculus to calculate the volume of the pathway. Nevertheless, I feel like I found two good problems/items/tasks to push and probe student thinking. And there’s always next year, as the journey continues …

Posted by on February 25, 2015 in Calculus

## An Infinite Number of Rectangles

We have started our unit on the definite integral for a few years now with Lin McMullin’s The Old Pump.

I love watching students work without yet having developed Riemann Sums. Many use areas of rectangles to approximate the amount of water in the tank, but even then, they don’t all do it the same way. That work leads us to developing the idea of estimating area between a curve and the x-axis using Riemann Sums and the Trapezoidal Rule. And then we are finally ready to determine the exact area between a curve and the x-axis using a Riemann Sum with an infinite number of rectangles.

We practice reason abstractly and quantitatively throughout these lessons. Once we’ve thought about numerical approximations for area between a curve and the x-axis, we spend some time writing a Riemann Sum to represent area and evaluating its limit as the number of rectangles approaches ∞. I want them to be able to go backwards, too. So we start with a limit, and I ask them what definite integral will have the same value.

Which is apparently not as difficult as the groans suggested when I first gave it to them.  But we are always working on our Mathematical Flexibility, and while I was pleased that everyone can get a definite integral, I was disappointed that they all did it the same way. Jill Gough has provided us with a leveled learning progression for Mathematical Flexibility. Can you write another definite integral for which the area can be calculated using the given limit?

It took a while. But students made progress. Some made use of the symmetry of the graph of y=x2 to write a second integral. Some figured out that translating the parabola and the limits of integration one unit to the right would result in a region that has the same area.

Those were the types of answers I was expecting. But I also got answers I wasn’t expecting.

Some of my students were on the path to Level 4 of reason abstractly and quantitatively, beginning to generalize the idea of translating the parabola and the limits of integration c units to the right, resulting in a region that has the same area. They didn’t quite make it, as their limits were shifted to the right c but their parabola shifted to the left c. I was still impressed by their jump to Level 4, finding connections between pathways.

Our TI-Nspire CAS software let us check our results and helped us attend to precision.

And so the journey continues … learning more from my students and our technology every day about mathematical flexibility.