super fun book of frq’s - mr. hanson's math classes · 2019. 8. 13. · super fun book of...

A.P. Calculus BC

Super Fun

book of FRQ’s

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

Derivatives you must know

Integrals you must know

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

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.

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

a) with a right riemann sum

b) with a left riemann sum

c) with a trapezoidal sum

Riemann Sum FRQ

Approximate the value of ∫ 𝐻(𝑡)𝑑𝑡10

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

8∫ 𝐻(𝑡)𝑑𝑡

2 ∫ |𝑅(𝑡)|𝑑𝑡

Riemann Sum FRQ

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

2 an over or under estimate and explain your reasoning

c) Is the left reimann sum of 1

d) Is the right reimann sum of ∫ 𝑅(𝑡)𝑑𝑡8

e) Is the left reimann sum of ∫ 𝑅(𝑡)𝑑𝑡8

The average velocity (could ask for f or f’, requires different calculus)

Riemann Sum FRQ

Fundamental theorem of calculus (usually with f…but could be asked regarding f’) (2011c)

∫ 𝐻′(𝑡)𝑑𝑡10

2 ∫ 𝑅′′(𝑡)𝑑𝑡

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

2019 Question #2 (Calculator OK)

Riemann Sum FRQ

2018 Question #4 (No Calculator)

Riemann Sum FRQ

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

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)

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?

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

−8. Show the work that leads to your answer

e) Evaluate ∫ (3𝑓(𝑥) − 5)𝑑𝑥7

2018 Question 3 (No Calculator)

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)

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.

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

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

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?

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.

2017 Question #2 (calculator OK)

2015 Question 1 (Calculator OK)

2011-B Question #1 (Calculator OK)

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)

Second Derivative work … cont’d

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

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)

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

Examples of a differential function that is separable

Tangent line work

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

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

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)

𝒅𝒙=

−𝒚𝟐

𝟐𝒙−𝟒

d) Find the particular solution 𝑦 = 𝑓(𝑥) to the given differential equation with the initial condition 𝑓(3) = 1

𝒅𝒙=

(𝟐𝒙−𝟏)𝒄𝒐𝒔𝒚

e) Find the particular solution 𝑦 = 𝑓(𝑥) to the given differential equation with the initial condition 𝑓(1) = 𝜋

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)

Area of a solid Revolved around an axis

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.

Area of a solid Revolved around an axis … cont’d

Second Type--axis is below (complete as a non-calculator example)

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

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

2017 Question #4 (No-Calculator)

Differential or Revolving Solid FRQ’s – AB topics only

2015 Question #4 (no Calculator)

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

Logistical Functions

Series Convergence or Divergence

Writing a taylor Polynomial

Length of a curve (parametric)

Length of a curve (standard function)

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

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

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?

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?

2010 Question #3 (Calculator OK)…reminder, in 2010 (and years prior) there were 3 calc frq’s…not there are only 2

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, −

2) Graph: e) (−1,

b) (1.6,𝜋

6) d) (2.2, −

4) f) (−1.4,

−4𝜋

g) (2,17𝜋

6) h) (1, −

11𝜋

3) i) (−0.8,

−5𝜋

j) (0,𝜋

Graphing Polar functions

𝑟 = 𝑠𝑖𝑛2𝜃 𝑟 = 𝑠𝑖𝑛3𝜃

Polar Functions FRQ’s

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 =

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

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

b) Find the area of the shaded region

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

Finding the Distance between two curves

Find the distance between the two curves at 𝜃 =5𝜋

Find the distance between the two curves at 𝜃 =𝜋

Finding the rate of change of the distance between two curves

a) Find the rate at which the distance between the two

curves is changing when 𝜃 =5𝜋

b) What is the maximum distance from the origin the

curve 2(1 cos )r attains in the interval 𝜋

6≤ 𝜃 ≤

2018 Question 5 (No Calculator)

Taylor Polynomial FRQ’s

super fun book of frq’s - mr. hanson's math classes · 2019. 8. 13. · super fun book of...

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