super fun book of frq’s - mr. hanson's math classes · 2019. 8. 13. · super fun book of...
TRANSCRIPT
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A.P. Calculus BC
Super Fun
book of FRQ’s
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Table of Contents:
This page (table of contents)…………………………………………………………………………………… page 2
Notes for Derivatives and Integrals you must know………………………………………………… page 3
Released Q’s given each year (2010-2019)……………………………………………………………….. page 4
How to use………………………………………………………………………………………………………………. page 4
AB topic Questions on the BC Exam
Riemann Sum Notes and Comments……………………………………………………………………….. pages 5-8
Riemann Sum Problem Sets (2010-2019)…………………………………………………………………. pages 9-28
Area Under the Curve Notes and Comments………..…………………………………………………. pages 29-32
Area Under the Curve Problem Sets (2010-2019)…………….………………………………………. pages 33-54
Contextual Rates of Change Notes and Comments.…………………………………………………. pages 55-58
Contextual Rates of Change Problem Sets (2010-2019)……………………………………………. pages 59-74
Differential or Revolving Solid FRQ’s—AB topics only Notes and Comments……….... pages 75-82
Differential or Revolving Solid FRQ’s—AB topics only Problem Sets (2010-2019).……. pages 83-94
Mixed AB and BC topic Questions on the BC exam
Random / Revolving / Differential FRQ’s—AB and BC mixed Notes and Comments……… pages 95-96
Random / Revolving / Differential FRQ’s—AB and BC mixed Problem Sets (2010-2019)……………. pages 97-116
BC topic Questions on the BC exam
Parametric Functions Notes and Comments…….………..………………………….……………….. pages 117-120
Parametric Functions Problem Sets (2010-2019)…….………..…………………………………….. pages 121-130
Polar Functions Notes and Comments…….………………...…………………………………….…….. pages 131-134
Polar Functions Problem Sets (2010-2019)…….………………..…………………………….……….. pages 135-144
Taylor Polynomials Notes and Comments…….………..…………………………………………….. (none)
Taylor Polynomials Problem Sets (2010-2019)…….………..………………….………………….. pages 145-158
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Derivatives you must know
Integrals you must know
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Released AP Questions given each year…arranged by FRQ style and by year
How to use this to ensure an awesome AP score in May
The following are all suggestions:
1) Complete EACH problem at least one time throughout the year.
2) Use the scoring documents to score your work (use a DIFFERENT COLOR…this way you will remember when
looking back through the booklet which questions you may have had more difficulty with).
3) Study the scoring documents to get a feel for how points will be awarded on the different parts of the FRQ’s
4) Use other websites to check solutions or work (askmrcalculus.com is one recommended site)
5) Search out video solutions by googling CALC AB FRQ / (the year) / (the question number). There is likely to be
multiple video solutions for each. While I cannot attest to the quality or even the correctness of these websites,
I have yet to see one posted which has blatantly incorrect work demonstrated.
6) For those style questions you have struggled with the most…do additional questions from years prior to the year
2010.
7) Try at least a few of each question style BY YOURSELF. It is important for you to know what you are able to do on
your own and not always in a group setting (obviously the AP test will be done solo).
8) Enjoy the challenge these questions present. More difficulty will bring a higher sense of reward and satisfaction.
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The Riemann Sum (common concepts worked in on the AP test…this is not an all-inclusive list of topics)
1st) Notice the units of the function you are starting with!!
‘Amount Function’ vs ‘Rate of Change function’
How can you tell which type of function you are working with in the Data?
Find the derivative at a value and state the derivative’s meaning (make sense of the unit!!)
a) Do you need to demonstrate work? If so, how?
b) Find H’(6) and state the meaning of this value (includes indicated units of measure)
c) Find R’(7) and state the meaning of this value (includes indicating units of measure)
Finding the Riemann sum (what does it tell you…UNITS?!)
Approximate the value of ∫ 𝑅(𝑡)𝑑𝑡8
0
a) with a right riemann sum
b) with a left riemann sum
c) with a trapezoidal sum
Riemann Sum FRQ
Riemann Sum FRQ
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Approximate the value of ∫ 𝐻(𝑡)𝑑𝑡10
2
a) with a right riemann sum
b) with a left riemann sum
c) with a trapezoidal sum
d) How would the table look different if they asked for a midpoint sum?
Special topic related to the different function (f vs f’) that is often asked
‘Amount Function’ vs ‘Rate of Change function’
1
8∫ 𝐻(𝑡)𝑑𝑡
10
2 ∫ |𝑅(𝑡)|𝑑𝑡
8
0
Riemann Sum FRQ
Riemann Sum FRQ
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‘Amount Function’ vs ‘Rate of Change function’
Is the estimate determined by the riemann sum an over or under estimate?
a) what must be stated in the problem in order for this to be asked?
b) Is the right reimann sum of 1
8∫ 𝐻(𝑡)𝑑𝑡
10
2 an over or under estimate and explain your reasoning
c) Is the left reimann sum of 1
8∫ 𝐻(𝑡)𝑑𝑡
10
2 an over or under estimate and explain your reasoning
d) Is the right reimann sum of ∫ 𝑅(𝑡)𝑑𝑡8
0 an over or under estimate and explain your reasoning
e) Is the left reimann sum of ∫ 𝑅(𝑡)𝑑𝑡8
0 an over or under estimate and explain your reasoning
The average velocity (could ask for f or f’, requires different calculus)
‘Amount Function’ vs ‘Rate of Change function’
Riemann Sum FRQ
Riemann Sum FRQ
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Fundamental theorem of calculus (usually with f…but could be asked regarding f’) (2011c)
∫ 𝐻′(𝑡)𝑑𝑡10
2 ∫ 𝑅′′(𝑡)𝑑𝑡
8
0
Mean Value Theorem and/or Intermediate Value Theorem
a) What must be stated in problem in order to use the Intermediate Value Theorem?
b) What must be stated in the problem in order to use the Mean Value Theorem?
Using R(t)
c) Does the data support the conclusion that R(t) = 1000 liters per minute at some time t with 0 < t < 3. Give a reason for
you answer
d) Is there a time t, 0 < t < 8, at which R’(t) = 120? Justify your answer
(2 methods we can use…we will demonstrate both)
Using H(t)
e) Does the data support the conclusion that H(t) = 13 liters per minute at some time t with 2 < t < 10. Give a reason for
you answer
f) Is there a time t, 2 < t < 10, at which H’(t) = 2.5? Justify your answer
Riemann Sum FRQ
Riemann Sum FRQ
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2019 Question #2 (Calculator OK)
Riemann Sum FRQ
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Riemann Sum FRQ
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2018 Question #4 (No Calculator)
Riemann Sum FRQ
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Riemann Sum FRQ
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2017 Question #1 (Calculator OK)
Riemann Sum FRQ
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Riemann Sum FRQ
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2016 Question #1 (Calculator OK)
Riemann Sum FRQ
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Riemann Sum FRQ
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2015 Question #3 (No Calculator)
Riemann Sum FRQ
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Riemann Sum FRQ
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2014 Question #4 (No Calculator)
Riemann Sum FRQ
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Riemann Sum FRQ
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2013 Question #3 (No Calculator)
Riemann Sum FRQ
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Riemann Sum FRQ
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2012 Question #1 (Calculator OK)
Riemann Sum FRQ
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Riemann Sum FRQ
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2011 Question #2 (Calculator OK)
Riemann Sum FRQ
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Riemann Sum FRQ
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2010 Question #2 (Calculator OK)
Riemann Sum FRQ
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Riemann Sum FRQ
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The area under the curve has been 100% of the time a non-calculator question.
graph of 𝑓
The figure shows the piecewise linear function f. For −3 ≤ 𝑥 ≤ 7, the function g is defined as 𝑔(𝑥) = ∫ 𝑓(𝑡)𝑑𝑡𝑥
1. It is
also known that g(1) = 4.
A few things to do before starting the question during the AP test (these things will be unscored.
Finding values for g
a) Write and expression that involves an integral
to find the value for g at any x.
b) Find g(-1)
c) Find g(7)
d) Find g(-3)
e) Find g(4)
f) Find g(3)
Finding values for g’
a) Find g’(-2)
b) Find g’(3)
c) Find g’(-1)
d) Find g’(6)
Area under the Curve FRQ
Area under the Curve FRQ
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Finding relative minimums/maximums
a) Find all the x-coordinates in which g has a relative minimum. Give a reason for your answer
b) Find all the x-coordinates in which g has a relative maximum. Give a reason for your answer
Finding absolute minimums/maximums
a) Find all the x-coordinates in which g has an absolute minimum on the interval −3 ≤ 𝑥 ≤ 7. Give a reason for
your answer
b) Find all the x-coordinates in which g has an absolute maximum on the interval −3 ≤ 𝑥 ≤ 7. Give a reason for
your answer
Finding values for g’’
a) Find g’’(-2)
b) Find g’’(3)
c) Find g’’(-1)
d) Find g’’(6)
Area under the Curve FRQ
Area under the Curve FRQ
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graph of 𝑓
The figure shows the piecewise linear function f. For −3 ≤ 𝑥 ≤ 7, the function g is defined as 𝑔(𝑥) = ∫ 𝑓(𝑡)𝑑𝑡𝑥
1. It is
also known that g(1)=4.
Points of inflection and concavity
a) Provide all the values for x for which the graph of g has a point of inflection. Explain your reasoning.
b) Provide the intervals for which g is concave up. Explain
Increasing/decreasing and concave up/concave down
a) For −3 ≤ 𝑥 ≤ 7, on what open intervals, if any, is the graph of g both increasing and concave up?
b) For −3 ≤ 𝑥 ≤ 7, on what open intervals, if any, is the graph of g both decreasing and concave down?
c) For −3 ≤ 𝑥 ≤ 7, on what open intervals, if any, is the graph of g both increasing and concave down?
d) For −3 ≤ 𝑥 ≤ 7, on what open intervals, if any, is the graph of g both decreasing and concave up?
Area under the Curve FRQ
Area under the Curve FRQ
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Forcing you to demonstrate derivative/integral properties. **This is hard to anticipate what might be asked…so just a
few possibilities listed
a) The function r is defined by 𝑟(𝑥) = 𝑓(3𝑥 − 𝑥2). The the slope of the line tangent to the graph of r at the point
where x = 2
b) The function h is defined by ℎ(𝑥) =(𝑔(𝑥+1))4
32. What is the slop of the line tangent to h at the point x = 0?
c) The function b is defined by 𝑏(𝑥) =𝑔′(𝑥)
2𝑥. Find b’(3)
d) Let’s say that function g is defined and differentiable on the closed interval −8 ≤ 𝑥 ≤ 7
If ∫ 𝑔′(𝑥)𝑑𝑥 = 107
−8 , find the value of ∫ 𝑔′(𝑥)𝑑𝑥
−3
−8. Show the work that leads to your answer
e) Evaluate ∫ (3𝑓(𝑥) − 5)𝑑𝑥7
1
Area under the Curve FRQ
Area under the Curve FRQ
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2019 Question #3 (No Calculator)
Area under the Curve FRQ
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Area under the Curve FRQ
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2018 Question 3 (No Calculator)
Area under the Curve FRQ
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Area under the Curve FRQ
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2017 Question #3 (No Calculator)
Area under the Curve FRQ
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Area under the Curve FRQ
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2016 Question #3 (No Calculator)
Area under the Curve FRQ
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Area under the Curve FRQ
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2015 Question #5 (No Calculator)
Area under the Curve FRQ
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Area under the Curve FRQ
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2014 Question #3 (No Calculator)
Area under the Curve FRQ
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Area under the Curve FRQ
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2013 Question #4 (No Calculator)
Area under the Curve FRQ
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Area under the Curve FRQ
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2012 Question #3 (No Calculator)
Area under the Curve FRQ
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Area under the Curve FRQ
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2011 Question #4 (No Calculator)
Area under the Curve FRQ
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Area under the Curve FRQ
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2010 Question #3 (Calculator OK!)…this is the only #3 in this packet which is Calculator OK…after 2010, the collegeboard
switched the FRQ test to having 2 Calculator Questions (30 minutes), followed by 4 Non-Calculator Questions (60 minutes)
Area under the Curve FRQ
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Area under the Curve FRQ
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2010 Question #5 (No Calculator)
Area under the Curve FRQ
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Area under the Curve FRQ
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The context rate of change question is likely to be (but not guaranteed) a calculator question.
Typically the question has two functions. One will be an incoming/increasing amount function, while the other will be an
outgoing/decreasing amount function. This is not a steadfast rule of thumb as some years there is only an incoming
function (e.g. 2014), while other years there are two functions, but one of them will be a constant rate of change (e.g.
2018)
For these notes/examples we will work with two functions.
The example:
There is a small underground ant nest in the northwest corner of Johnsonville Park. Sally is watching the nest intently.
She determines the rate at which ants enter this nest is modeled by the function A, where 𝐴(𝑡) = 15cos (𝑡2
25) ants per
hour, t is measured in minutes and 0 ≤ 𝑡 ≤ 6. For the first 2 minutes no ants leave the nest. After 2 minutes Sally
determines the rate at which ants leave this nest is modeled by the function L, where 𝐿(𝑡) = 4𝑥 − 4(ln(𝑥2 + 0.1)) ants
per hour, t also measured in hours and 0 ≤ 𝑡 ≤ 6. She counted 300 ants in the nest at time t = 6.
Calculator comments
Understanding Value questions
a) Find A(4) and A’(4) and state the meaning for both in the context of this question.
b) Find L(4) and L’(4) and state the meaning for both in the context of this question.
c) Find ∫ 𝐴(𝑡)𝑑𝑡6
0 and express the meaning of this number
d) Write an integral which would show the total number of ants who have left the ants nest in the time interval
1 ≤ 𝑡 ≤ 4
Contextual Rate of Change FRQ
Contextual Rate of Change FRQ
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Comparing the arrival vs departure of the ants
a) Is the amount of ants in the nest increasing or decreasing at time t = 4 minutes? Give a reason for your answer
b) At what time interval is the amount of ants in the nest decreasing? Explain
c) The number of ants in the nest is constantly changing. What time is the rate at which the number of ants in the
nest is increasing the fastest? Justify
Contextual Rate of Change FRQ
Contextual Rate of Change FRQ
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d) What is the absolute maximum number of ants in the nest (to the nearest whole ant) in the time interval
0 ≤ 𝑡 ≤ 6? Justify
e) What is the absolute minimum number of ants in the nest (to the nearest whole ant) in the time interval
0 ≤ 𝑡 ≤ 6? Justify
f) How many ants were in the nest at time t = 0 (to the nearest whole ant)?
g) What is the average rate of change for the number of ants in the nest during the last 2 minutes (4 ≤ 𝑡 ≤ 6)
Sally studied the ant nest?
Contextual Rate of Change FRQ
Contextual Rate of Change FRQ
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IVT and/or MVT applications
Comments about the use of these theorems (conditions):
a) Sally’s friend Doug was studying the larger ant nest in the southeast corner of Johnsonville Park. Doug
determined the rate at which ants exited the nest to be a constant flow of 50 ants per minute. He did not
determine the rate at which the ants entered the nest but he did count the number of ants in this nest at 3
different times. At time t = 0 there were 400 ants, at time t = 3 there were 700 ants and at time t = 6 there were
1150 ants. Doug determines the rate at which ants entered the nest had to be exactly 100 ants per minute at
some time 0 ≤ 𝑡 ≤ 6. Doug also determined the rate at which ants entered the nest had to be at exactly 130
ants per minute at some point. Separately justify both statements.
b) Explain/Justify why there must be a time where the number of ants in the northwest nest (the nest Sally
studied) must be exactly 102.
Contextual Rate of Change FRQ
Contextual Rate of Change FRQ
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2019 Question #1 (Calculator OK)
Contextual Rate of Change FRQ
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Contextual Rate of Change FRQ
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2018 Question #1 (Calculator OK)
Contextual Rate of Change FRQ
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Contextual Rate of Change FRQ
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2017 Question #2 (calculator OK)
Contextual Rate of Change FRQ
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Contextual Rate of Change FRQ
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2015 Question 1 (Calculator OK)
Contextual Rate of Change FRQ
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Contextual Rate of Change FRQ
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2014 Question #1 (Calculator OK)
Contextual Rate of Change FRQ
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Contextual Rate of Change FRQ
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2013 Question #1 (Calculator OK)
Contextual Rate of Change FRQ
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Contextual Rate of Change FRQ
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2011-B Question #1 (Calculator OK)
Contextual Rate of Change FRQ
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Contextual Rate of Change FRQ
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2010 Question #1 (Calculator OK)
Contextual Rate of Change FRQ
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Contextual Rate of Change FRQ
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The Differential Function FRQ has been 100% of the time (for the last decade anyhow) a non-calculator question.
Three styles of this question you will encounter (in a typical AP test you will encounter one OR the other—except 2015))
**for the non-contextual questions there is likely to be a slope field component, though slope fields are not reviewed in these notes)
**these notes will focus on the non-contextual examples, though it should be noted you are just as likely to experience a contextual
problem in this year’s AP test. It should also be noted the skills required for the contextual problems would be the same as what
you will experience below…just with context .
What is meant by the term ‘differential’ function?
Examples of a differential function that is not separable
Second Derivative work
Consider the differential function 𝒅𝒚
𝒅𝒙= 𝟐𝒚 − 𝟓𝒙
a) Find 𝑑2𝑦
𝑑𝑥2 in terms of (only) x and y.
b) Determine the concavity for all solution curves in quadrant IV. Give a reason for your answer
Differential or Revolving Solid FRQ’s – AB topics only (Differential function notes)
Differential or Revolving Solid FRQ’s – AB topics only (Differential function notes)
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Second Derivative work … cont’d
Consider the differential function 𝒅𝒚
𝒅𝒙=
𝟐𝒚−𝒙
𝟓𝒙−𝒚𝟐
a) Evaluate 𝑑2𝑦
𝑑𝑥2 at
a. (2, 1) b. at the point on the curve where x = 0 and y = 2
Tangent line work
Consider the differential function 𝒅𝒚
𝒅𝒙= 𝟐𝒚 − 𝟓𝒙
a) Write the equation for the line tangent to the curve y at the point (-1 , 3)
b) Let y = f(x) be a particular solution to the given differential equation with the initial condition f(1) = -2. Write the
equation of the line tangent to the graph of y at x = 1. Use this equation to approximate f(1.2)
Consider the differential function 𝒅𝒚
𝒅𝒙=
𝟐𝒚−𝒙
𝟓𝒙−𝒚𝟐
b) Let y = f(x). Find the coordinates of all the points on the curve of f(x) where the tangent line is horizontal
c) Let y = f(x). Find the coordinates of all the points on the curve of f(x) where the tangent line is vertical
Differential or Revolving Solid FRQ’s – AB topics only (Differential function notes)
Differential or Revolving Solid FRQ’s – AB topics only (Differential function notes)
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Examples of a differential function that is separable
Tangent line work
Consider the differential function 𝒅𝒚
𝒅𝒙= 𝒆𝒚(𝟔𝒙𝟐 − 𝟐)
a) Let y = f(x) be the particular solution to the given differential equation with initial condition f (1)=0. Write an
equation for the tangent line to the graph of y at x = 1. Use this equation to approximate f(0.8).
b) Let y = f(x) be the particular solution to the given differential equation with initial condition f (-1)=0. Write an
equation for the tangent line to the graph of y at x = -1. Use this equation to approximate f(-1.2).
Finding particular solutions
Consider the differential function 𝒅𝒚
𝒅𝒙= 𝒆𝒚(𝟔𝒙𝟐 − 𝟐)
a) Find the particular solution 𝑦 = 𝑓(𝑥) to the given differential equation with the initial condition 𝑓(−1) = 0
b) Find the particular solution 𝑦 = 𝑓(𝑥) to the given differential equation with the initial condition 𝑓(2) = 0
Differential or Revolving Solid FRQ’s – AB topics only (Differential function notes)
Differential or Revolving Solid FRQ’s – AB topics only (Differential function notes)
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Finding particular solutions … cont’d
c) The difference between a general solution and a particular solution is:
**tie this into slope fields (note to teacher)
Consider the differential function 𝒅𝒚
𝒅𝒙=
−𝒚𝟐
𝟐𝒙−𝟒
d) Find the particular solution 𝑦 = 𝑓(𝑥) to the given differential equation with the initial condition 𝑓(3) = 1
Consider the differential function 𝒅𝒚
𝒅𝒙=
𝟏
(𝟐𝒙−𝟏)𝒄𝒐𝒔𝒚
e) Find the particular solution 𝑦 = 𝑓(𝑥) to the given differential equation with the initial condition 𝑓(1) = 𝜋
Differential or Revolving Solid FRQ’s – AB topics only (Differential function notes)
Differential or Revolving Solid FRQ’s – AB topics only (Differential function notes)
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This FRQ question is equally likely to be a non-calculator as it is a calculator question
AREA BETWEEN 2 CURVES
The shaded region is enclosed by the y-axis, the vertical line 2x and the graphs of 3( ) 5 6f x x x and the
horizontal line 1y .
a) Find the area of the shaded region Challenges: 1: 2: 3*:
Calculator notes and additional steps that likely will be required
Steps (other methods work…this is just what your teacher advises) 1. Identify which is the function ‘higher’ or ‘above’ the other 2. Determine the ‘interval’ (integration limits) needed 3. Find the volume under the ‘above’ function (integral) 4. Find the volume under the ‘below’ function (integral) 5. Subtract the volume of the ‘below’ function from the ‘above’ function
Differential or Revolving Solid FRQ’s – AB topics only (Revolving Solid notes)
Differential or Revolving Solid FRQ’s – AB topics only (Revolving Solid notes)
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Area of a solid Revolved around an axis
The shaded region is enclosed by the y-axis, the vertical line 2x and the graphs of 3( ) 5 6f x x x and the
horizontal line 1y .
First Type--axis is above (complete as a non-calculator example) Find the volume of the solid generated when the shaded region is rotated about the horizontal line y=8.
Steps When a solid is generated by rotating a figure around an axis *video Steps to find a volume of the solid: 1: Sketch/draw in the axis of rotation (what line are you rotating about?) 2: We will be determining the volume of 2 solids
1): 2):
3: Identify which is the function further from the axis of rotation (the outer curve).
a. This will be the ‘block of wood’ you are staring with before you ‘carve out’ the middle b. Notice how the radius changes as you move along the x-axis. Write the radius as a function of this outer curve Use this radius in the integral to determine the volume of the solid as a sum of an infinite number of cylinders, each infinitely thin.
4: Identify the function which is closer to the axis of rotation (this should be easy since you already identified the outer chunk ! (this will be the inner curve).
a. This will be the stuff you are to ‘carve out’ of our initial block of wood
b. b. Notice how this radius changes as you move along the x-axis. Write the radius as a function of this outer curve Use this radius in the integral to determine the volume of the solid as a sum of an infinite number of cylinders, each infinitely thin.
5: Subtract the ‘carved out’ volume from the ‘initial chunk’ volume.
Differential or Revolving Solid FRQ’s – AB topics only (Revolving Solid notes)
Differential or Revolving Solid FRQ’s – AB topics only (Revolving Solid notes)
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Area of a solid Revolved around an axis … cont’d
Second Type--axis is below (complete as a non-calculator example)
The shaded region is enclosed by the y-axis, the vertical line 2x and the graphs of 3( ) 5 6f x x x and the
horizontal line 1y .
Find the volume of the solid generated when the shaded region is rotated about the horizontal line y= -1 Third Type--axis of rotation is the y-axis (versus a horizontal line) … (complete as a calculator example)
The shaded region is enclosed by the graphs of ( ) xf x e and the horizontal line 1y ex .
Find the volume of the solid generated when the shaded region is rotated about the y-axis
Differential or Revolving Solid FRQ’s – AB topics only (Revolving Solid notes)
Differential or Revolving Solid FRQ’s – AB topics only (Revolving Solid notes)
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Areas of Volumes with known cross sections
The shaded region is enclosed by the graphs of 4 3( ) 3 6 6f x x x and the horizontal line 6y .
a) The shaded region is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of the solid. b) The shaded region is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are semi-circles. Find the volume of the solid.
c) The shaded region is the base of a solid. For this solid, the cross sections perpendicular to the x-axis is an isosceles right triangle with a leg in the shaded region. Find the volume of the solid.
d) The shaded region is the base of a solid. For this solid, the cross sections perpendicular to the x-axis is a rectangle whose height is 4 times the length of the base in the shaded region. Find the volume of the solid
Differential or Revolving Solid FRQ’s – AB topics only (Revolving Solid notes)
Differential or Revolving Solid FRQ’s – AB topics only (Revolving Solid notes)
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83
2017 Question #4 (No-Calculator)
Differential or Revolving Solid FRQ’s – AB topics only
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Differential or Revolving Solid FRQ’s – AB topics only
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85
2016 Question #5 (No-Calculator)
Differential or Revolving Solid FRQ’s – AB topics only
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Differential or Revolving Solid FRQ’s – AB topics only
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87
2015 Question #4 (no Calculator)
Differential or Revolving Solid FRQ’s – AB topics only
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Differential or Revolving Solid FRQ’s – AB topics only
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89
2012 Question #5 (No Calculator)
Differential or Revolving Solid FRQ’s – AB topics only
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Differential or Revolving Solid FRQ’s – AB topics only
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91
2011 Question #5 (No Calculator)
Differential or Revolving Solid FRQ’s – AB topics only
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Differential or Revolving Solid FRQ’s – AB topics only
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93
2010 Question #4 (No Calculator)
Differential or Revolving Solid FRQ’s – AB topics only
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Differential or Revolving Solid FRQ’s – AB topics only
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These problems tend to draw from a wide spectrum of calculus topics and is difficult to predict what the question will entail. These notes (only 2 pages) will focus on a quick review of the BC components that are likely to be seen. Some parts of the problem will focus on BC topics (1/3 to 1/2) while the rest of the problem (1/2 to 2/3) is going to be AB topics.
Decomposition of Fractions
Euler’s method
Improper Integrals
Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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Logistical Functions
Series Convergence or Divergence
Writing a taylor Polynomial
Length of a curve (parametric)
Length of a curve (standard function)
Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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97
2019 Question #5 (No Calculator)
Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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99
2018 Question #2 (Calculator OK)
Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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2017 Question #5 (No Calculator)
Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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103
2016 Question #4 (No Calculator)
Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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105
2015 Question #5 (No Calculator)
Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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107
2014 Question #5 (No Calculator)
Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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109
2013 Question #5 (No Calculator)
Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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111
2012 Question #4 (No Calculator)
Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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2011 Question #3 (No Calculator)
Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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115
2010 Question #5 (No Calculator)
Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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Mixed AB and BC topics -- Random / Revolving / Differential FRQ’s
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The parametric FRQ is typically a Calculator question but has also been a non-calculator FRQ
For each of the topics to follow we will use the the following scenario (and complete using a calculator)
For t > 0 , a particle is moving along a curve so that its position is ( x(t) , y(t) ). At time t = 4, the particle is at
position ( -2, 6). It is also known 𝒙′(𝒕) = (𝟐𝒕)𝒍𝒏(𝒕 + 𝟐) 𝒂𝒏𝒅 𝒚′(𝒕) = −𝟎. 𝟓 + 𝒄𝒐𝒔𝟐(𝒕)
Find the position of the particle
Find the position of the particle at time t = 1
Find the position of the particle at time t = 7
Slope of tangent lines
What is the slope of the tangent line at time t = 3?
At what time is the slope of the tangent line to the curve equal to 15?
Parametric Functions FRQ’s
Parametric Functions FRQ’s
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Finding the speed
What is the speed of the particle at time t = 1
What is the speed of the particle at time t = 5
What time t is the speed of the particle equal to 8
What time t is the speed of the particle equal to 4
Total distance traveled by the particle in a time interval
Parametric Functions FRQ’s
Parametric Functions FRQ’s
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For t > 0 , a particle is moving along a curve so that its position is ( x(t) , y(t) ). At time t = 4, the particle is at
position ( -2, 6). It is also known 𝒙′(𝒕) = (𝟐𝒕)𝒍𝒏(𝒕 + 𝟐) 𝒂𝒏𝒅 𝒚′(𝒕) = −𝟎. 𝟓 + 𝒄𝒐𝒔𝟐(𝒕)
What was the total distance traveled from time t = 0 to time t=1
What was the total distance traveled from time t = 1 to time t =2
What was the total distance traveled from time t = 0 to time t = 2
Position vs velocity vs acceleration
Find the acceleration vector of the particle at time t = 4
When the horizontal acceleration is equal to 2, what is the vertical acceleration?
Parametric Functions FRQ’s
Parametric Functions FRQ’s
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Second Derivative
Particle Movement
Describe the movement of the particle at time t = 3
Describe the movement of the particle at time t =9
At the time t which the particle first changes vertical direction, what is the horizontal direction?
Parametric Functions FRQ’s
Parametric Functions FRQ’s
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2016 Question 2 (Calculator OK)
Parametric Functions FRQ’s
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Parametric Functions FRQ’s
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2015 Question 2 (Calculator OK)
Parametric Functions FRQ’s
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Parametric Functions FRQ’s
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2012 Question 2 (Calculator OK)
Parametric Functions FRQ’s
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Parametric Functions FRQ’s
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2011 Question 1 (Calculator OK)
Parametric Functions FRQ’s
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Parametric Functions FRQ’s
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2010 Question #3 (Calculator OK)…reminder, in 2010 (and years prior) there were 3 calc frq’s…not there are only 2
Parametric Functions FRQ’s
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Parametric Functions FRQ’s
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Polar functions…what are they? What are their parts?
Graphing ‘coordinate points’ (not on AP specifically…but key to understanding the polar graphs)
Graph: a) (1,𝜋
6) Graph: c) (1, −
𝜋
2) Graph: e) (−1,
𝜋
3)
b) (1.6,𝜋
6) d) (2.2, −
5𝜋
4) f) (−1.4,
−4𝜋
3)
g) (2,17𝜋
6) h) (1, −
11𝜋
3) i) (−0.8,
−5𝜋
2)
j) (0,𝜋
6)
Graphing Polar functions
𝑟 = 𝑠𝑖𝑛2𝜃 𝑟 = 𝑠𝑖𝑛3𝜃
Polar Functions FRQ’s
Polar Functions FRQ’s
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Converting a polar graph to rectangular functions…𝑥(𝜃) and 𝑦(𝜃)
𝑟 = 2 + 3𝑠𝑖𝑛𝜃 𝑟 = 1 + 2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃
Position of a particle
A particle moves along the polar curve 𝑟 = 2 + 3𝑠𝑖𝑛𝜃 so that at time t seconds, = 2𝑡 . Find the times t in the interval
0 ≤ 𝑡 ≤ 4 where x-coordinate of the particle’s position is x = 1
Slope of polar functions
Find the slope of the tangent line to 𝑟 = 2 + 3𝑠𝑖𝑛𝜃 at 𝜃 = 𝜋
For the curve 𝑟 = 2𝜃 + 𝑠𝑖𝑛𝜃, find the value of 𝑑𝑦
𝑑𝜃 at 𝜃 =
5𝜋
6
Vertical and horizontal tangent lines
For both examples use the interval 0 ≤ 𝜃 ≤ 2𝜋
At what value of 𝜃 does the curve of 𝑟 = 2 + 3𝑠𝑖𝑛𝜃 have horizontal tangent lines
At what value of 𝜃 does the curve of 𝑟 = 2 + 3𝑠𝑖𝑛𝜃 have vertical tangent lines
Polar Functions FRQ’s
Polar Functions FRQ’s
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Finding the area of a region in a system of polar equations.
The graphs of the polar curves 2r and 2(1 cos )r
for 0 is shown to the right.
a) Find the area of the shaded region
The graphs of the polar curves 2r and 2(1 cos )r
for 0 is shown to the right.
b) Find the area of the shaded region
The graphs of the polar curves 2r and 2(1 cos )r
for 0 is shown to the right.
c) The ray 𝜃 = 𝑘, where 𝜋
2≤ 𝑘 ≤ 𝜋, divides the shaded
region into 2 equal areas. Write, but do not solve, an
equation involving one or more integrals whose
solution gives the value of k
Polar Functions FRQ’s
Polar Functions FRQ’s
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Finding the Distance between two curves
The graphs of the polar curves 2r and 2(1 cos )r
for 0 is shown to the right.
Find the distance between the two curves at 𝜃 =5𝜋
6
Find the distance between the two curves at 𝜃 =𝜋
3
Finding the rate of change of the distance between two curves
The graphs of the polar curves 2r and 2(1 cos )r
for 0 is shown to the right.
a) Find the rate at which the distance between the two
curves is changing when 𝜃 =5𝜋
6
b) What is the maximum distance from the origin the
curve 2(1 cos )r attains in the interval 𝜋
6≤ 𝜃 ≤
3𝜋
4?
Polar Functions FRQ’s
Polar Functions FRQ’s
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2019 Question 2 (Calculator OK)
Polar Functions FRQ’s
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2018 Question 5 (No Calculator)
Polar Functions FRQ’s
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2017 Question 2 (Calculator OK)
Polar Functions FRQ’s
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2014 Question 2 (Calculator OK)
Polar Functions FRQ’s
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2013 Question 2 (Calculator OK)
Polar Functions FRQ’s
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2019 Question #6 (No Calculator)
Taylor Polynomial FRQ’s
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Taylor Polynomial FRQ’s
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2018 Question #2 (Calculator OK)
Taylor Polynomial FRQ’s
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Taylor Polynomial FRQ’s
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2017 Question #5 (No Calculator)
Taylor Polynomial FRQ’s
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2016 Question #6 (No Calculator)
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2013 Question #2 (Calculator OK)
Taylor Polynomial FRQ’s
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Taylor Polynomial FRQ’s
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2012 Question #6 (No-Calculator)
Taylor Polynomial FRQ’s
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2011 Question #1 (Calculator OK)
Taylor Polynomial FRQ’s
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Taylor Polynomial FRQ’s