# Linear Motion (Type 2)

### 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 positions(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 xa 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:

• Equation stem 2017 AB 5,
• Graph stem: 2009 AB1/BC1,
• Table stem 2019 AB2
• Equation stem 2021 AB 2
• Equation stem 2022 AB6 – velocity, acceleration, position, max/min

Multiple-choice examples from non-secure exams:

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

Revised March 15, and May 11, 2022

# Unit 9 – Parametric Equations, Polar Coordinates, and Vector-Valued Functions

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 and Vector Equations

Implicit Differentiation of Parametric Equations

A Vector’s Derivatives

Adapting 2012 BC 2 (A parametric equation question)

Polar Curves

Polar Equations for AP Calculus

Extreme Polar Conditions

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.

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.

1. Find the amount processed and the amount shipped after hour.
2. Is the amount on hand increasing or decreasing at time t = 1? Explain your reasoning.
3. 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.
4. 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.

# Parametric Equations, Polar Coordinates, and Vector-Valued Functions – Unit 9

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

# Linear Motion (Type 2)

### 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 positions(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 xa 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

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

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

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 4.2 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 $\displaystyle \tfrac{\infty }{\infty }$. (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.

Posts on these topics include:

Motion Problems

Motion Problems: Same Thing, Different Context

Speed

A Note on Speed

Related Rates

Related Rate Problems I

Related Rate Problems II

Good Question 9 – Related rates

Linear Approximation

Local Linearity 1

Local Linearity 2

L’Hospital’s Rule

Locally Linear L’Hôpital

L’Hôpital Rules the Graph

Determining the Indeterminate

Determining the Indeterminate 2

Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.

2019 CED – Unit 1: Limits and Continuity

2019 CED – Unit 2: Differentiation: Definition and Fundamental Properties.

2019 CED – Unit 3: Differentiation: Composite , Implicit, and Inverse Functions

2019 CED – Unit 4 Contextual Applications of the Derivative  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 5 Analytical Applications of Differentiation  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 6 Integration and Accumulation of Change

2019 – CED Unit 7 Differential Equations  Consider teaching after Unit 8

2019 – CED Unit 8 Applications of Integration   Consider teaching after Unit 6, before Unit 7

2019 – CED Unit 9 Parametric Equations, Polar Coordinates, and Vector-Values Functions

2019 CED Unit 10 Infinite Sequences and Series