Why Applications?

Posted on January 2, 2024 by Lin McMullin

I guess the obvious answer is so you will have something to use your new knowledge of the definite integral on. This unit is a collection of the first applications. There are many more.

The whole idea is based on the fact that a definite integral is a sum. Thus, they can be used to add things. Complicated shapes can be divided into simple shapes, like rectangles, and their areas can be added.

Here are the first uses.

AVERAGE VALUE OF A FUNCTION ON AN INTERVAL

You know that to average two numbers you add them and divide the total by two. To average a larger set of numbers you add them and divide the sum by the number of numbers. Now you will learn how to average the infinite number of y-values of a function over an interval. To do it you use a definite integral.

HINT: Average value of a function, average rate of change of a function, and the mean value theorem all sound kind of the same. Furthermore, the formulas all look kind of the same. Don’t mix them up. Use their full name, not just “average,” and use the right one.

There is an interesting result found some years ago by a tenth-grade student who took AB Calculus in eighth grade and BC in ninth. He proved that Most Triangles Are Obtuse! using the average value integral.

LINEAR MOTION

This application extends your previous work on objects moving on a line. By adding, with a definite integral, you will be able to determine how far the object moves in a given time. Knowing that and where the object starts, you will be able to find its current position. BC students will also learn how to find the length of a curve in the plane.

THE AREA BETWEEN TWO CURVES

This application extends what you know about finding the area between a curve and the x-axis to finding the area between two curves.

VOLUME OF A SOLID FIGURE WITH A REGULAR CROSS-SECTION

Some solid objects, when sliced in the right way, have regular cross-section. That means, the cut surface is a square, a rectangle, a right triangle, a semi-circle and so on. Think of cutting a piece of butter from a stick of butter – each piece is a square with a little thickness, like a very small pizza box. By using an integral to add their volumes you can find the volume of the original figure. .

VOLUME OF A SOLID OF REVOLUTION – DISK METHOD

A region in the plane is revolved aroud a line resulting in a solid figure. Finding its volume is just a special case of a solid with regular cross section where the cross section is a circle. Each little piece is a disk. By adding their volumes with an integral you can calculate the volume.

VOLUME OF A SOLID OF REVOLUTION – WASHER METHOD

Some solids of revolution have holes through them. These are formed by revolving a plane figure (usually the region between two curves) around an edge or a line outside the figure. The line is often the x– or y-axis. There is a formula for doing this, but understanding what you are doing – doing two disk problems – will help you with doing this. (See Subtract the Hole from the Whole.)

One last hint: You will want to know what the solid figure looks like. In addition to the illustrations in your textbook, there are various places online where you can find pictures of the figures. You can build models of them. There is an excellent iPad app called A Little Calculus that does a good job of illustrating and animating solid figures (and lots of other calculus stuff).

Course and Exam Description Unit 8

Why Linear Motion?

Posted on October 10, 2023 by Lin McMullin

Now that you know how to compute derivatives it is time to use them. The next few topics and my next few posts will discuss some of the applications of derivatives, and some of the things you can use them for.

The first is linear motion or motion along a straight line.

Derivatives give the rate of change of something that is changing. Linear motion problems concern the change in the position of something moving in a straight line. It may be someone riding a bike, driving a car, swimming, or walking or just a “particle” moving on a number line.

The function gives the position of whatever is moving as a function of time. This position is the distance from a known point often the origin. The time is the time the object is at the point. The units are distance units like feet, meters, or miles.

The derivative position is velocity, the rate of change of position with respect to time. Velocity is a vector; it has both magnitude and direction. While the derivative appears to be just a number its sign counts: a positive velocity indicates movement to the right or up, and negative to the left or down. Units are things like miles per hour, meters per second, etc.

The absolute value of velocity is the speed which has the same units as velocity but no direction.

The second derivative of position is acceleration. This is the rate of change of velocity. Acceleration is also a vector whose sign indicates how the velocity is changing (increasing or decreasing). The units are feet per minute per minute or meters per second per second. Units are often given as meters per second squared (m/s²) which is correct but meters per second per second helps you understand that the velocity in meters per second is changing so much per second.

Of these four, velocity may be the most useful. You will learn how to use the velocity (the first derivative) and its graph to determine how the particle moves over intervals of time: when it is moving left or right, when it stops, when it changes direction, whether it is speeding up or slowing down, how far the object moves, and so on. You can also find the position from the velocity if you also know the starting position.

You will work with equations and with graphs without equations. Reading the graphs of velocity and acceleration is an important skill to learn.

The reasoning used in linear motion problems is the same as in other applications. What you do is the same; what it means depends on the context.

Not to scare anyone, but linear motion problems appear as one of the six free-response questions on the AP Calculus exams almost every year as well as in several the multiple-choice questions.

So, let’s get moving!

Course and Exam Description Unit 4 Sections 4.1 and 4.2

Linear Motion (Type 2)

Posted on March 15, 2022 by Lin McMullin

AP Questions Type 2: Linear Motion

We continue the discussion of the various type questions on the AP Calculus Exams with linear motion questions.

“A particle (or car, person, or bicycle) moves on a number line ….”

These questions may give the position equation, the velocity equation (most often), or the acceleration equation of something that is moving on the x– or y-axis as a function of time, along with an initial condition. The questions ask for information about the motion of the particle: its direction, when it changes direction, its maximum position in one direction (farthest left or right), its speed, etc.

The particle may be a “particle,” a person, car, a rocket, etc. Particles don’t really move in this way, so the equation or graph should be considered a model. The question is a versatile way to test a variety of calculus concepts since the position, velocity, or acceleration may be given as an equation, a graph, or a table; be sure to use examples of all three forms during the review.

Many of the concepts related to motion problems are the same as those related to function and graph analysis (Type 3). Stress the similarities and show students how the same concepts go by different names. For example, finding when a particle is “farthest right” is the same as finding when a function reaches its “absolute maximum value.” See my post for Motion Problems: Same Thing, Different Context for a list of these corresponding terms. There is usually one free-response question and three or more multiple-choice questions on this topic.

The position, s(t), is a function of time. The relationships are:

The velocity is the derivative of the position $\displaystyle {s}'\left( t \right)=v\left( t \right)$ . Velocity has direction (indicated by its sign) and magnitude. Technically, velocity is a vector; the term “vector” will not appear on the AB exam.
Speed is the absolute value of velocity; it is a number, not a vector. See my post for Speed.
Acceleration is the derivative of velocity and the second derivative of position, $\displaystyle {{s}'}'\left( t \right)={v}'\left( t \right)=a\left( t \right)$ It, too, has direction and magnitude and is a vector.
Velocity is the antiderivative of acceleration.
Position is the antiderivative of velocity.

What students should be able to do:

Understand and use the relationships above.
Distinguish between position at some time and the total distance traveled during the time period.
The total distance traveled is the definite integral of the speed (absolute value of velocity) $\displaystyle \int_{a}^{b}{{\left| {v\left( t \right)} \right|dt}}$ .
Be sure your students understand the term displacement; it is the net distance traveled or distance between the initial position and the final position. Displacement is the definite integral of the velocity (rate of change): $\displaystyle \int_{a}^{b}{{v\left( t \right)dt}}$
The final position is the initial position plus the displacement (definite integral of the rate of change from x= a to x = t): $\displaystyle s\left( t \right)=s\left( a \right)+\int_{a}^{t}{{v\left( x \right)dx}}$ Notice that this is an accumulation function equation (Type 1).
Initial value differential equation problems: given the velocity or acceleration with initial condition(s) find the position or velocity. These are easily handled with the accumulation equation in the bullet above but may also be handled as an initial value problem.
Find the speed at a given time. Speed is the absolute value of velocity.
Find average speed, velocity, or acceleration
Determine if the speed is increasing or decreasing.
- When the velocity and acceleration have the same sign, the speed increases. When they have different signs, the speed decreases.
- If the velocity graph is moving away from (towards) the t-axis the speed is increasing (decreasing). See the post on Speed.
- There is also a worksheet on speed here
- The analytic approach to speed: A Note on Speed
Use a difference quotient to approximate the derivative (velocity or acceleration) from a table. Be sure the work shows a quotient.
Riemann sum approximations.
Units of measure.
Interpret meaning of a derivative or a definite integral in context of the problem

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

This may be an AB or BC question. The BC topic of motion in a plane, (Type 8: parametric equations and vectors) will be discussed in a later post.

The Linear Motion problem may cover topics primarily from primarily from Unit 4, and also from Unit 3, Unit 5, Unit 6, and Unit 8 (for BC) of the CED.

Free-response examples:

2017 AB 5, Equation stem
2009 AB1/BC1, Graph stem:
2019 AB2 Table stem
2021 AB 2 Equation stem
2022 AB6 Equation stem – velocity, acceleration, position, max/min
2023 AB 2 Equation stem – velocity (given), change of direction, acceleration, speeding up or slowing down, position, total distance.

Multiple-choice examples from non-secure exams:

2012 AB 6, 16, 28, 79, 83, 89
2012 BC 2, 89

Updated: March 15, and May 11, 2022, June 4, 2023

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 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

Linear Motion (Type 2)

Posted on March 6, 2020 by Lin McMullin

AP Questions Type 2: Linear Motion

We continue the discussion of the various type questions on the AP Calculus Exams with linear motion questions.

“A particle (or car, person, or bicycle) moves on a number line ….”

These questions may give the position equation, the velocity equation (most often), or the acceleration equation of something that is moving on the x– or y-axis as a function of time, along with an initial condition. The questions ask for information about motion of the particle: its direction, when it changes direction, its maximum position in one direction (farthest left or right), its speed, etc.

The particle may be a “particle,” a person, car, a rocket, etc. Particles don’t really move in this way, so the equation or graph should be considered to be a model. The question is a versatile way to test a variety of calculus concepts since the position, velocity, or acceleration may be given as an equation, a graph, or a table; be sure to use examples of all three forms during the review.

Many of the concepts related to motion problems are the same as those related to function and graph analysis (Type 3). Stress the similarities and show students how the same concepts go by different names. For example, finding when a particle is “farthest right” is the same as finding the when a function reaches its “absolute maximum value.” See my post for Motion Problems: Same Thing, Different Context for a list of these corresponding terms. There is usually one free-response question and three or more multiple-choice questions on this topic.

The position, s(t), is a function of time. The relationships are:

The velocity is the derivative of the position, ${s}'\left( t \right)=v\left( t \right)$ . Velocity is has direction (indicated by its sign) and magnitude. Technically, velocity is a vector; the term “vector” will not appear on the AB exam.
Speed is the absolute value of velocity; it is a number, not a vector. See my post for Speed.
Acceleration is the derivative of velocity and the second derivative of position, $\displaystyle a\left( t \right)={v}'\left( t \right)={{s}''}\left( t \right)$ . It, too, has direction and magnitude and is a vector.
Velocity is the antiderivative of the acceleration.
Position is the antiderivative of velocity.

What students should be able to do:

Understand and use the relationships above.
Distinguish between position at some time and the total distance traveled during the time period.
The total distance traveled is the definite integral of the speed (absolute value of velocity) $\displaystyle \int_{a}^{b}{\left| v\left( t \right) \right|}\,dt$ .
Be sure your students understand the term displacement; it is the net distance traveled or distance between the initial position and the final position. Displacement, is the definite integral of the velocity (rate of change): $\displaystyle \int_{a}^{b}{v\left( t \right)}\,dt$ .
The final position is the initial position plus the displacement (definite integral of the rate of change from x= a to x = t): $\displaystyle s\left( t \right)=s\left( a \right)+\int_{a}^{t}{v\left( x \right)}\,dx$ Notice that this is an accumulation function equation (Type 1).
Initial value differential equation problems: given the velocity or acceleration with initial condition(s) find the position or velocity. These are easily handled with the accumulation equation in the bullet above, but may also be handled as an initial value problem.
Find the speed at a given time. The speed is the absolute value of the velocity.
Find average speed, velocity, or acceleration
Determine if the speed is increasing or decreasing.
- If at some time, the velocity and acceleration have the same sign then the speed is increasing.If they have different signs the speed is decreasing.
- If the velocity graph is moving away from (towards) the t-axis the speed is increasing (decreasing). See the post on Speed.
- There is also a worksheet on speed here
- The analytic approach to speed: A Note on Speed
Use a difference quotient to approximate the derivative (velocity or acceleration) from a table. Be sure the work shows a quotient.
Riemann sum approximations.
Units of measure.
Interpret meaning of a derivative or a definite integral in context of the problem

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

This may be an AB or BC question. The BC topic of motion in a plane, (Type 8: parametric equations and vectors) will be discussed in a later post.

The Linear Motion problem may cover topics primarily from primarily from Unit 4, and also from Unit 3, Unit 5, Unit 6, and Unit 8 (for BC) of the 2019 CED.

Free-response examples:

Equation stem 2017 AB 5,
Graph stem: 2009 AB1/BC1,
Table stem 2019 AB2

Multiple-choice examples from non-secure exams:

2012 AB 6, 16, 28, 79, 83, 89
2012 BC 2, 89

Teaching Calculus

The pleasure lies not in discovering the truth, but in searching for it.

Tag Archives: Linear Motion

Why Applications?

Why Linear Motion?

Linear Motion (Type 2)

AP Questions Type 2: Linear Motion

Adapting 2021 AB 2

2021 AB 2

Linear Motion (Type 2)

AP Questions Type 2: Linear Motion

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AP Questions Type 2: Linear Motion

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Topics 4.1 – 4.6

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2021 AB 2

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Topics 4.1 – 4.6

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AP Questions Type 2: Linear Motion

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