# Rate & Accumulation (Type 1)

### The Free-response Questions

There are ten general categories of AP Calculus free-response questions. As I’ve done in part years, I will consider each individually over the next few weeks posting on Tuesdays and Fridays. These will be updates of last year’s posts. A list of all the posts is at the end of this post.

NOTE: The type number I’ve assigned to each type DO NOT correspond to the 2019 CED Unit numbers. Many AP Exam questions have parts from different Units. The CED Unit numbers will be referenced in each post.

### AP  Questions Type 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 an 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

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

• Write an integral expression that gives the amount at a general time. BE CAREFUL, the dt must be included at the correct place. Think of the integral sign and the dt as parentheses around the integrand.
• Find the average value of a function
• 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.

The Rate – Accumulation question may cover topics primarily from Unit 4, Unit 5, Unit 6 and Unit 8 of the 2019 CED.

Typical free-response examples:

Typical multiple-choice examples from non-secure exams:

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

Schedule of future posts for reviewing for the 2019 Exams

NOTE: The type number I’ve assigned to each type DO NOT correspond to the 2019 CED Unit numbers. Many AP Exam questions have parts from different Units. The CED Unit numbers will be referenced in each post.

Tuesday February 25 – AP Exam Review 2020 https://wp.me/p2zQso-286
Friday, February 28 – Reviewing Resources 2020 https://wp.me/p2zQso-28f
Tuesday March 3, 2020: Rate and accumulation questions (Type 1) https://wp.me/p2zQso-265
Friday March 6, 2020: Linear motion problems (Type 2) https://wp.me/p2zQso-269
Tuesday March 10, 2020: Graph analysis problems (Type 3) https://wp.me/p2zQso-26e
Friday March 13, 2020: Area and volume problems (Type 4) https://wp.me/p2zQso-26m
Tuesday March 17, 2020: Table and Riemann sum questions (Type 5) https://wp.me/p2zQso-27K
Friday March 20, 2020: Differential equation questions (Type 6) https://wp.me/p2zQso-27U
Tuesday March 24, 2020: Other questions (Type 7) https://wp.me/p2zQso-28q
Friday March 27, 2020: Sequences and Series questions (Type 10) BC Topic https://wp.me/p2zQso-298

Updated January 31, 2019

This site uses Akismet to reduce spam. Learn how your comment data is processed.