# Type 1 Questions: Rate and Accumulation

The Free-response Questions

There are ten general categories of AP Calculus free-response questions, listed below. These are usually the subject of individual free-response questions. Keep in mind that two or more types may be included in the same free-response question and are also the topics of shorter multiple-choice questions. There are links to all the types here.

• Type 1 questions – Rate and accumulation questions
• Type 2 questions – Linear motion problems
• Type 3 questions – Graph analysis problems
• Type 4 questions – Area and volume problems
• Type 5 questions – Table and Riemann sum questions
• Type 6 questions – Differential equation questions
• Type 7 questions – miscellaneous
• Type 8 questions – Parametric and vector questions (BC topic)
• Type 9 questions – Polar equations
• Type 10 questions – Sequences and Series

AP Type Questions 1: Rate and Accumulation

These questions are often in context with a lot of words describing a situation in which some things are changing. There are usually two rates acting in opposite ways (sometimes called in-out question). Students are asked about the change that the rates produce over some time interval either separately or together.

The rates are often fairly complicated functions. If they are on the calculator allowed section, students should store the functions in the equation editor of their calculator and use their calculator to do any graphing,  integration, or differentiation that may be necessary.

The main idea is that over the time interval [a, b] the integral of a rate of change is the net amount of change

$\displaystyle \int_{a}^{b}{{f}'\left( t \right)dt}=f\left( b \right)-f\left( a \right)$

If the question asks for an amount, look around for a rate to integrate.

The final (accumulated) amount is the initial amount plus the accumulated change:

$\displaystyle f\left( x \right)=f\left( {{x}_{0}} \right)+\int_{{{x}_{0}}}^{x}{{f}'\left( t \right)}\,dt$,

where ${{x}_{0}}$ is the initial time, and  $f\left( {{x}_{0}} \right)$ is the initial amount. Since this is one of the main interpretations of the definite integral the concept may come up in a variety of situations.

What students should be able to do:

• Be ready to read and apply; often these problems contain a lot of writing which needs to be carefully read.
• Recognize that rate = derivative.
• Recognize a rate from the units given without the words “rate” or “derivative.”
• Find the change in an amount by integrating the rate. The integral of a rate of change gives the amount of change (FTC):

$\displaystyle \int_{a}^{b}{{f}'\left( t \right)dt}=f\left( b \right)-f\left( a \right)$.

• Find the final amount by adding the initial amount to the amount found by integrating the rate. If $x={{x}_{0}}$ is the initial time, and $f\left( {{x}_{0}} \right)$  is the initial amount, then final accumulated amount is

$\displaystyle f\left( x \right)=f\left( {{x}_{0}} \right)+\int_{{{x}_{0}}}^{x}{{f}'\left( t \right)}\,dt$,

• Understand the question. It is often not necessary to as much computation as it seems at first.
• Use FTC to differentiate a function defined by an integral.
• Explain the meaning of a derivative or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) how numerical argument applies in context.
• Explain the meaning of a definite integral or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) how the limits of integration apply in context.
• Store functions in their calculator recall them to do computations on their calculator.
• If the rates are given in a table, be ready to approximate an integral using a Riemann sum or by trapezoids.
• Do a max/min or increasing/decreasing analysis.

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.

Typical free-response examples:

Typical multiple-choice examples from non-secure exams:

• 2012 AB 8, 81, 89
• 2012 BC 8 (same as AB 8)

Updated January 31, 2019, March 12, 2021

# Rate and Accumulation Questions (Type 1)

The Free-response Questions

The free-response questions fall into 10 general categories or types. The multiple-choice questions fall largely into the same categories plus some straight-forward questions asking students to find limits, derivatives, and integrals. Often two or more type are combined into one question. The types are the following.

1. Rate and Accumulation
2. Linear motion
3. Graph Analysis
4. Area / Volume
5. Table and Riemann sum
6. Differential Equation (and slope fields)
7. Others (implicit differentiation, related rates, theorems, et. al.)
8. Parametric Equations (BC only)
9. Polar Equations (BC only)
10. Sequences and Series (BC only)

My numbering has changed over the years. This numbering follows this index where each type is referenced to free-response and multiple-choice questions of the same type.

I will discuss each type individually over the next few weeks starting today with Type 1.

AP Type Questions 1: Rate and Accumulation

These questions are often in context with a lot of words describing a situation in which some things are changing. There are usually two rates acting in opposite ways. Students are asked about the change that the rates produce over some time interval either separately or together.

The rates are often fairly complicated functions. If they are on the calculator allowed section, students should store the functions in the equation editor of their calculator and use their calculator to do any integration or differentiation that may be necessary.

The main idea is that integral of a rate of change over the time interval [a, b] is the net amount of change

$\displaystyle \int_{a}^{b}{{f}'\left( t \right)dt}=f\left( b \right)-f\left( a \right)$

If the question asks for an amount, look around for a rate to integrate.

The final (accumulated) amount is the initial amount plus the accumulated change:

$\displaystyle f\left( x \right)=f\left( {{x}_{0}} \right)+\int_{{{x}_{0}}}^{x}{{f}'\left( t \right)}\,dt$,

where ${{x}_{0}}$ is the initial time, and  $f\left( {{x}_{0}} \right)$ is the initial amount. Since this is one of the main interpretations of the definite integral the concept may come up in a variety of situations.

What students should be able to do:

• Be ready to read and apply; often these problems contain a lot of writing which needs to be carefully read.
• Recognize that rate = derivative.
• Recognize a rate from the units given without the words “rate” or “derivative.”
• Find the change in an amount by integrating the rate. The integral of a rate of change gives the amount of change (FTC):

$\displaystyle \int_{a}^{b}{{f}'\left( t \right)dt}=f\left( b \right)-f\left( a \right)$.

• Find the final amount by adding the initial amount to the amount found by integrating the rate. If $x={{x}_{0}}$ is the initial time, and $f\left( {{x}_{0}} \right)$  is the initial amount, then final accumulated amount is

$\displaystyle f\left( x \right)=f\left( {{x}_{0}} \right)+\int_{{{x}_{0}}}^{x}{{f}'\left( t \right)}\,dt$,

• Understand the question. It is often not necessary to as much computation as it seems at first.
• Use FTC to differentiate a function defined by an integral.
• Explain the meaning of a derivative or its value in terms of the context of the problem.
• Explain the meaning of a definite integral or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) how the limits of integration apply in context.
• Store functions in their calculator recall them to do computations on their calculator.
• If the rates are given in a table, be ready to approximate an integral using a Riemann sum or by trapezoids.
• Do a max/min or increasing/decreasing analysis.

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.

For some previous posts on this subject see January 2123, 2013. This post is revised from the post of March 1, 2013

Next Posts:

Tuesday March 7: Type 2 Linear Motion

Friday March 10: Type 3: Graph Analysis

# The Rate/Accumulation Question

AP Type Questions 1

These questions are often in context with a lot of words describing a situation in which some things are changing. There are usually two rates given acting in opposite ways. Students are asked about the change that the rates produce over some time interval either separately or together.

The rates are often fairly complicated functions. If they are on the calculator allowed section, students should store them in the equation editor of their calculator and use their calculator to do any integration or differentiation that may be necessary.

The integral of a rate of change is the net amount of change

$\displaystyle \int_{a}^{b}{{f}'\left( t \right)dt}=f\left( b \right)-f\left( a \right)$

over the time interval [a, b]. If the question asked for an amount, look around for a rate to integrate.

The final accumulated amount is the initial amount plus the accumulated change:

$\displaystyle f\left( x \right)=f\left( {{x}_{0}} \right)+\int_{{{x}_{0}}}^{x}{{f}'\left( t \right)}\,dt$,

where ${{x}_{0}}$ is the initial time, and  $f\left( {{x}_{0}} \right)$ is the initial amount. Since this is one of the main interpretations of the definite integral the concept may come up in a variety of situations.

What students should be able to do:

• Be ready to read and apply; often these problems contain a lot of writing which needs to be carefully read.
• Recognize that rate = derivative.
• Recognize a rate from the units given without the words “rate” or “derivative.”
• Find the change in an amount by integrating the rate. The integral of a rate of change gives the amount of change (FTC):

$\displaystyle \int_{a}^{b}{{f}'\left( t \right)dt}=f\left( b \right)-f\left( a \right)$.

• Find the final amount by adding the initial amount to the amount found by integrating the rate. If $x={{x}_{0}}$ is the initial time, and $f\left( {{x}_{0}} \right)$  is the initial amount, then final accumulated amount is

$\displaystyle f\left( x \right)=f\left( {{x}_{0}} \right)+\int_{{{x}_{0}}}^{x}{{f}'\left( t \right)}\,dt$,

• Understand the question. It is often not necessary to as much computation as it seems at first.
• Use FTC to differentiate a function defined by an integral.
• Explain the meaning of a derivative or its value in terms of the context of the problem.
• Explain the meaning of a definite integral or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) how the limits of integration apply in context.
• Store functions in their calculator recall them to do computations on their calculator.
• If the rates are given in a table, be ready to approximate an integral using a Riemann sum or by trapezoids.
• Do a max/min or increasing/decreasing analysis.

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.

For some previous posts on this subject see January 21, 23, 2013