Unit 4 – Contextual Applications of the Derivative

Posted on October 5, 2021 by Lin McMullin

Unit 4 covers rates of change in motion problems and other contexts, related rate problems, linear approximation, and L’Hospital’s Rule. (CED – 2019 p. 82 – 90). These topics account for about 10 – 15% of questions on the AB exam and 6 – 9% of the BC questions.

You may want to consider teaching Unit 5 (Analytical Applications of Differentiation) before Unit 4. Notes on Unit 5 will be posted next Tuesday September 29, 2020

Topics 4.1 – 4.6

Topic 4.1 Interpreting the Meaning of the Derivative in Context Students learn the meaning of the derivative in situations involving rates of change.

Topic 4.2 Linear Motion The connections between position, velocity, speed, and acceleration. This topic may work better after the graphing problems in Unit 5, since many of the ideas are the same. See Motion Problems: Same Thing, Different Context

Topic 4.3 Rates of Change in Contexts Other Than Motion Other applications

Topic 4.4 Introduction to Related Rates Using the Chain Rule

Topic 4.5 Solving Related Rate Problems

Topic 4.6 Approximating Values of a Function Using Local Linearity and Linearization The tangent line approximation

Topic 4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms. Indeterminate Forms of the type $\displaystyle \tfrac{0}{0}$ and $\displaystyle \tfrac{{\pm \infty }}{{\pm \infty }}$ . (Other forms may be included, but only these two are tested on the AP exams.)

Topic 4.1 and 4.3 are included in the other topics, topic 4.2 may take a few days, Topics 4.4 – 4.5 are challenging for many students and may take 4 – 5 classes, 4.6 and 4.7 two classes each. The suggested time is 10 -11 classes for AB and 6 -7 for BC. of 40 – 50-minute class periods, this includes time for testing etc.

This is a re-post and update of the third in a series of posts from last year. It contains links to posts on this blog about the differentiation of composite, implicit, and inverse functions for your reference in planning. Other updated post on the 2019 CED will come throughout the year, hopefully, a few weeks before you get to the topic.

Posts on these topics include:

Motion Problems

Motion Problems: Same Thing, Different Context

Speed

A Note on Speed

Adapting 2021 AB 2

Adapting 2021 AB 4 / BC 4

Related Rates

Adapting 2021 BC 2

Posted on August 3, 2021 by Lin McMullin

Seven of nine. This week we continue our look at the 2021 free-response questions with an eye to ways to adapt and expand the questions. Hopefully, you will find ways to use this and other free-response questions to help your students learn more and be better prepared for the exams.

2021 BC 2

This is a Parametric and Vector Equation (Type 8) question and contains topic from Unit 9 of the current Course and Exam Description. The vector equation of the velocity of a particle moving in the xy-plane is given along with the position of the particle at t = 0. No units were given.

The stem for 2021 BC 2 is next. (Note the $\displaystyle \left\langle \text{ } \right\rangle$ notation for vectors. Any of the usual notations may be used by students, but be sure to show them the others in case the one their book usage is different than the exam’s.)

Part (a): Students were asked to find the speed and acceleration of the particle at t = 1.2. This is a calculator active questions and the students were expected, but not actually required, to use their calculator. With their calculator in parametric mode, students should begin by entering the velocity as xt1(t) and yt1(t).

Discussion and ideas for adapting this question:

There is little I can suggest here other than changing the time.
At the given time and other times, you can ask in what direction is the particle moving and which way the acceleration is pulling the velocity.
Ask student to do this without using their calculator. The answer need not be simplified or expressed as a decimal.

Part (b): Asked the students to find the total distance traveled by the particle over a given the time interval. This must be done on a calculator. Be sure your students know how to enter the expression using the already entered values for xt1(t) and yt1(t). The calculator entry should look like this.

$\displaystyle \int_{0}^{{1.2}}{{\sqrt{{{{{\left( {\text{xt}1(t)} \right)}}^{2}}+{{{\left( {\text{yt1}(t)} \right)}}^{2}}}}}}dt$

Discussion and ideas for adapting this question:

Use different intervals.
Discuss the similarities with the number line distance formula. In linear motion, the distance is simply the integral of the absolute value of the velocity. Since $\displaystyle \int_{a}^{b}{{\left| {v\left( t \right)} \right|}}dt=\int_{a}^{b}{{\sqrt{{{{{\left( {v(t)} \right)}}^{2}}}}}}dt$ , this is the same formula reduced to one dimension.

Part (c): The situation is reduced to a one-dimensional problem: students were asked to find the coordinates of the point at which the particle is farthest left and explain why there is no point farthest to the right.

Discussion and ideas for adapting this question:

Discuss how to do this and how students should present their answer and explanation.
Show that this is the same as an extreme value problem and done the same way (i.e., find where the derivative is zero, and show that this is a minimum (farthest left), etc.).
Discuss how you know there is no maximum and interpret this in the context of the equation.

For further exploration. Try graphing the path of the particle. Discuss how to do that with your class. See what they suggest. Here a few approaches.

The first thought may be to integrate the velocity vector as an initial value problem. Unfortunately, this cannot be done. Neither the x-component nor the y-component can be integrated in terms of Elementary Functions. Even WolframAlpha.com is no help.
Having entered the velocity vector as xt1(t) and yt1(t), as suggested above, enter something like this depending on your calculator’s syntax and then graph in a suitable window. Compare the graph with the previous analysis in part (c)?

$\displaystyle \text{x2t}(t)=-2+\int_{0}^{t}{{\text{x1t}(t)dt}}$

$\displaystyle \text{y2t}(t)=5+\int_{0}^{t}{{\text{y2t}(t)dt}}$

You may also try expressing the components of velocity as a Taylor series centered at some positive number, a, not at zero. Integrate that to get an approximation to graph. Be sure to adjust things so the initial point is on the graph. WolframAlpha will help here. The one problem here is that the y-component is not defined for negative numbers. Therefore, zero cannt be then center and the largest the interval of convergence can be is [0, 2a] (Why?) and may not even by that large. This is an interesting approach mathematically but will not help with most of the graph.

Personal opinion: I do not think much of this question because all the first two parts require is entering the formula in your calculator and computing the answer, and the third part is really an AB level question. Just my opinion.

Next week 2021 BC 5

I would be happy to hear your ideas for other ways to use this question. Please use the reply box below to share your ideas.

Adapting 2021 AB 2

Posted on June 29, 2021 by Lin McMullin

Two of nine. Continuing the series started in my last post, this post looks at the AB Calculus 2021 exam question AB 2. The series looks at each question with the aim of showing ways to use the question in with your class as is or by adapting and expanding it. Like most of the AP Exam questions there is a lot more you can ask from the stem and a lot of other calculus you can discuss.

2021 AB 2

This is a Linear Motion Problem (Type 2) and has topics from Unit 4 of the current Course and Exam Description. Two particles are moving on the x-axis and the questions ask about their motion individually and relative to each other. The velocity and initial position are given for each particle. Parts (a), (c), and (d) are typical; (b) is the core of the problem.

The stem is:

Part (a): Students are asked to find the position of each particle at time t = 1.

Discussion and ideas for adapting this question:

The expected approach is to calculate for each particle the initial position plus the displacement from t = 0 to t = 1. So, for P the computation is $P\left( 1 \right)=5+\int_{0}^{1}{{\sin \left( {{{t}^{{1.5}}}} \right)}}dt$ and similarly for Q(1). This is a calculator allowed question and students should use their calculator to find the answer and not do it by hand.
A different approach is to work it as an initial value differential equation problem. This will work but takes longer than the approach suggested above.
In class, it is worth discussing both methods.
You can adapt this by using a different time.
Another question is to find (only) the displacement if each particle over some time interval. Displacement has been asked in other years.
Ask “Will the particles ever collide? If so when and justify your answer. (Answer: no)

Part (b): Students were asked to determine if the particles are moving apart or towards each other at time t = 1. This is the main question and requires a careful analysis of their motion.

Discussion and ideas for adapting this question:

To determine this, students need to consider the velocity of the particles and their position (from part (a)). P is to the left of Q and moving right. Q is to the right of P and moving left, therefore, the distance between them is decreasing.
You can practice this analysis by using different times.
Ask students to carefully describe the motion of one or both particles: when it is moving left and right, when it changes direction, find the local maximum and minimum positions, etc. Notice that this is really the same as analyzing the shape of a graph. The connection between the two problems will help students understand both better. See: Motion Problems: Same Thing, Different Context

Part (c): A question about speed.

Discussion and ideas for adapting this question:

A typical question. Students should compare the signs of the velocity and acceleration of the particle. If they are the same, the speed is increasing; if different, decreasing.
You may ask this of the other particle.
You may ask this at different times.
See previous posts on speed here and here.

Part (d): Students were required to find the total distanced traveled by Q on the interval [0, π].

Discussion and ideas for adapting this question:

Since speed is the absolute value of the velocity, integrate the absolute value of the velocity. Do this on a calculator.
Adapt this by using a different interval.
Adapt this by using the other particle.
Another (longer) way to approach this question is to find where the particle changes direction by finding where the velocity changes from negative to positive and/or vice versa (i.e., the local extreme values). Then find the distanced traveled on each part of the “trip,” and add or subtract. This will reinforce a lot of the concepts involved in linear motion; that is why it is worth doing. As for the exam, integrating the absolute value is the way to go. However, if this were a non-calculator question, then it would have to be done this way. Find a simpler velocity and try it both ways.
To integrate the absolute value by hand, it is necessary to break the interval into subintervals depending on where the velocity is positive or negative. This is the same as the approach in the bullet immediately above. This, too, is worth showing to reinforce the definition of absolute value.

2021 revised as an in-out question.

There was some unhappiness over the fact that the 2021 AB Calculus exam did not have an in-out questions (Rate and Accumulation Type 1). However, this question does have two rates going in opposite directions. So, just to be ornery, I rewrote it as an in-out questions by changing the context and units while keeping the same velocity functions. The point is that the situation tested can be reframed in other ways. Seeing the same thing in different dress may help students concentrate on the calculus involved. Here it is:

A factory processes cement at the rate of $\displaystyle {{v}_{p}}\left( t \right)=\sin \left( {{{t}^{{1.5}}}} \right)$ tons per hour for $\displaystyle 0\le t\le \pi$ hours. At time t = 0 the amount on hand is P = 5 tons.

The factory ships the cement at a rate given by $\displaystyle {{v}_{Q}}\left( t \right)=\left( {t-1.8} \right){{1.25}^{t}}$ tons per hour for $\displaystyle 0\le t\le \pi$ hours. At time t = 0 the amount shipped is 10 tons.

Find the amount processed and the amount shipped after hour.
Is the amount on hand increasing or decreasing at time t = 1? Explain your reasoning.
At what rate is the rate at which the cement is being shipped changing at t = 1? Is the amount being shipped increasing or decreasing at t = 1? Explain your reasoning.
Find the total amount of cement processed over the time interval $\displaystyle 0\le t\le \pi$ .

Next week 2021 AB 3/ BC 3.

I would be happy to hear your ideas for other ways to use this questions. Please use the reply box below to share your ideas.

Contextual Applications of the Derivative – Unit 4

Posted on September 22, 2020 by Lin McMullin

You may want to consider teaching Unit 5 (Analytical Applications of Differentiation) before Unit 4. Notes on Unit 5 will be posted next Tuesday September 29, 2020

Topics 4.1 – 4.6

Topic 4.1 Interpreting the Meaning of the Derivative in Context Students learn the meaning of the derivative in situations involving rates of change.

Topic 4.3 Rates of Change in Contexts Other Than Motion Other applications

Topic 4.4 Introduction to Related Rates Using the Chain Rule

Topic 4.5 Solving Related Rate Problems

Topic 4.6 Approximating Values of a Function Using Local Linearity and Linearization The tangent line approximation

Topic 4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms. Indeterminate Forms of the type $\displaystyle \tfrac{0}{0}$ and $\displaystyle \tfrac{\infty }{\infty }$ . (Other forms may be included, but only these two are tested on the AP exams.)

Posts on these topics include:

Motion Problems

Motion Problems: Same Thing, Different Context

Speed

A Note on Speed

Related Rates

2019 CED Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Posted on January 7, 2020 by Lin McMullin

Unit 9 includes all the topics listed in the title. These are BC only topics (CED – 2019 p. 163 – 176). These topics account for about 11 – 12% of questions on the BC exam.

Comments on Prerequisites: In BC Calculus the work with parametric, vector, and polar equations is somewhat limited. I always hoped that students had studied these topics in detail in their precalculus classes and had more precalculus knowledge and experience with them than is required for the BC exam. This will help them in calculus, so see that they are included in your precalculus classes.

Topics 9.1 – 9.3 Parametric Equations

Topic 9.1: Defining and Differentiation Parametric Equations. Finding dy/dx in terms of dy/dt and dx/dt

Topic 9.2: Second Derivatives of Parametric Equations. Finding the second derivative. See Implicit Differentiation of Parametric Equations this discusses the second derivative.

Topic 9.3: Finding Arc Lengths of Curves Given by Parametric Equations.

Topics 9.4 – 9.6 Vector-Valued Functions and Motion in the plane

Topic 9.4 : Defining and Differentiating Vector-Valued Functions. Finding the second derivative. See this A Vector’s Derivatives which includes a note on second derivatives.

Topic 9.5: Integrating Vector-Valued Functions

Topic 9.6: Solving Motion Problems Using Parametric and Vector-Valued Functions. Position, Velocity, acceleration, speed, total distance traveled, and displacement extended to motion in the plane.

Topics 9.7 – 9.9 Polar Equation and Area in Polar Form.

Topic 9.7: Defining Polar Coordinate and Differentiation in Polar Form. The derivatives and their meaning.

Topic 9.8: Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve

Topic 9.9: Finding the Area of the Region Bounded by Two Polar Curves. Students should know how to find the intersections of polar curves to use for the limits of integration.

Timing

The suggested time for Unit 9 is about 10 – 11 BC classes of 40 – 50-minutes, this includes time for testing etc.

Previous posts on these topics :

Parametric Equations

Vector Valued Functions

Polar Form

2019 CED Unit 4: Contextual Applications of the Derivative

Posted on October 7, 2019 by Lin McMullin

Unit 4 covers rates of change in motion problems and other contexts, related rate problems, linear approximation and L’Hospital’s Rule. (CED – 2019 p. 82 – 90). These topics account for about 10 – 15% of questions on the AB exam and 6 – 9% of the BC questions.

Topics 4.1 – 4.6

Topic 4.1 Interpreting the Meaning of the Derivative in Context Students learn the meaning of the derivative in situations involving rates of change.

Topic 4.3 Rates of Change in Contexts Other Than Motion Other applications

Topic 4.4 Introduction to Related Rates Using the Chain Rule

Topic 4.5 Solving Related Rate Problems

Topic 4.6 Approximating Values of a Function Using Local Linearity and Linearization The tangent line approximation

Topic 4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms. Indeterminate Forms of the type $\displaystyle \tfrac{0}{0}$ and $\displaystyle \tfrac{\infty }{\infty }$ . (Other forms may be included, but only these two are tested on the AP exams.)

Posts on these topics include:

Motion Problems

Motion Problems: Same Thing, Different Context

Speed

A Note on Speed

Related Rates