BC Calculus
2020 Exam Practice
FR #1 (25 minutes: 15 points)
Name______________________________
Set a timer for 25 minutes to complete this problem. You may use your notes, textbooks, or any
materials I gave you throughout the year. You are not expected to use a calculator, but you
may use one if you would like. You should show all your steps as if you did not have a
calculator. I am guessing that the 25-minute problem will be worth 15 points and the 15-minute
problem will be worth 10 points for a total of 25 points. The college board has said that the
25-minute problem will be worth 60% and the 15-minute problem will be worth 40%, so that is
my best guess at how it may be broken down this year. Please show all appropriate
mathematics: no bald answers!
1 The graph of 𝑓′, consisting of 3 line segments, a quarter circle, and a portion of the graph of
%&
𝑦 = ' ( , is shown below. It is known that 𝑓(0) = 5.
Graph of 𝑓′
a)
On the interval [-4,6], find all x-values at which 𝑓(𝑥) has relative maxima and relative
minima. Give a reason for your answers. [3 points]
b)
On the interval [-4,6], find all x-values at which 𝑓(𝑥) has points of inflection. Give a
reason for your answer. [3 points]
The graph of 𝑓′, consisting of 3 line segments, a quarter circle, and a portion of the graph of
%&
𝑦 = ' ( , is shown below. It is known that 𝑓(0) = 5.
Graph of 𝑓′
c)
Write an expression for 𝑓(𝑥) that includes an integral. Use that expression to find the
values of 𝑓(1) and 𝑓(3). [3 points]
d)
On the interval [-4,6], find the absolute maximum and absolute minimum values of
𝑓(𝑥). Justify your answers. [3 points]
e)
Find the area of the region bounded by the x-axis, the vertical line 𝑥 = 4, and the
%&
portion of the graph of 𝑦 = ' ( . (Note that there is no upper bound, so this will be an
improper integral.) [3 points]
BC Calculus
2020 Exam Practice
FR #2 (25 minutes: 15 points)
Name______________________________
Set a timer for 25 minutes to complete this problem. You may use your notes, textbooks, or any
materials I gave you throughout the year. You are not expected to use a calculator, but you
may use one if you would like. You should show all your steps as if you did not have a
calculator. I am guessing that the 25-minute problem will be worth 15 points and the 15-minute
problem will be worth 10 points for a total of 25 points. The college board has said that the
25-minute problem will be worth 60% and the 15-minute problem will be worth 40%, so that is
my best guess at how it may be broken down this year. Please show all appropriate
mathematics: no bald answers!
The function 𝑓(𝑥) is given by this accumulation function:
a)
Find all intervals on which 𝑓(𝑥) is increasing or decreasing. Give reasons for your
answers. [3 points]
b)
Find all intervals on which 𝑓(𝑥) is concave up and concave down. Give reasons for your
answer. [3 points]
c)
Find 𝑓(𝑥) by completing the definite integral. Show all steps. [3 points]
The function 𝑓(𝑥) is given by this accumulation function:
d)
On the interval [-1,4], find the absolute maximum and absolute minimum values of
𝑓(𝑥). Justify your answers. [3 points]
e)
Find the equation of the line tangent to 𝑓(𝑥) at the point when 𝑥 = 2. Use the
equation of the tangent line to approximate the value of 𝑓(4). [3 points]
BC Calculus
2020 Exam Practice
FR #3 (25 minutes: 15 points)
Name______________________________
Set a timer for 25 minutes to complete this problem. You may use your notes, textbooks, or any
materials I gave you throughout the year. You are not expected to use a calculator, but you
may use one if you would like. You should show all your steps as if you did not have a
calculator. I am guessing that the 25-minute problem will be worth 15 points and the 15-minute
problem will be worth 10 points for a total of 25 points. The college board has said that the
25-minute problem will be worth 60% and the 15-minute problem will be worth 40%, so that is
my best guess at how it may be broken down this year. Please show all appropriate
mathematics: no bald answers!
&
As shown in the graph below, Region R is bounded by the function 𝑓(𝑥) = ' ( )*')&+ , the
&
horizontal line 𝑦 = 0.5, and the vertical line 𝑥 = 0. Region S is bounded by 𝑓(𝑥) = ' ( )*')&+ ,
the horizontal line 𝑦 = 0, and the vertical lines 𝑥 = −2 and 𝑥 = 0.
a)
Using the method of Partial Fractions, set up and evaluate a definite integral to find the
area of Region S. [3 points]
b)
Show the setup of an integral that could be used to find the area of Region R. You DO
NOT need to evaluate the integral. [2 points]
&
As shown in the graph below, Region R is bounded by the function 𝑓(𝑥) = ' ( )*')&+ , the
&
horizontal line 𝑦 = 0.5, and the vertical line 𝑥 = 0. Region S is bounded by 𝑓(𝑥) = ' ( )*')&+ ,
the horizontal line 𝑦 = 0, and the vertical lines 𝑥 = −2 and 𝑥 = 0.
c)
Show the setup of the integral that would yield the volume of the solid formed when
Region S is rotated about the x-axis. You DO NOT need to evaluate the integral.
[2 points]
d)
Region R is the base of a solid with cross-sections perpendicular to the x-axis that are
squares. Show the setup of the integral that would yield the volume of the solid. You
DO NOT need to evaluate the integral. [2 points]
e)
Show the setup of the integral that would yield the volume of the solid formed when
Region R is rotated about the y-axis. You DO NOT need to evaluate the integral.
[2 points]
f)
Write an expression including an integral that would yield the perimeter of Region S.
You DO NOT have to evaluate the integral portion of the expression. [4 points]
BC Calculus
2020 Exam Practice
FR #4 (25 minutes: 15 points)
Name______________________________
Set a timer for 25 minutes to complete this problem. You may use your notes, textbooks, or any
materials I gave you throughout the year. You are not expected to use a calculator, but you
may use one if you would like. You should show all your steps as if you did not have a
calculator. I am guessing that the 25-minute problem will be worth 15 points and the 15-minute
problem will be worth 10 points for a total of 25 points. The college board has said that the
25-minute problem will be worth 60% and the 15-minute problem will be worth 40%, so that is
my best guess at how it may be broken down this year. Please show all appropriate
mathematics: no bald answers!
Consider the function 𝑓(𝑥) = 2 ln(𝑥 − 3), which passes through the point (4,0).
a)
Find the equation of the line tangent to 𝑓(𝑥) at the point (4,0). Use that equation to
find an approximation for 𝑓(4.2). [3 points]
b)
Is the approximation you found in part (a) greater than or less than the actual value of
𝑓(4.2)? Justify your answer. [2 points]
c)
Starting at the point (4,0), use Euler’s Method to approximate the value of 𝑓(4.2) using
two steps of equal size. [4 points]
Consider the function 𝑓(𝑥) = 2 ln(𝑥 − 3), which passes through the point (4,0).
d)
Find the third-degree Taylor Polynomial for 𝑓(𝑥), centered at 𝑥 = 4. Use the
polynomial to approximate the value of 𝑓(4.2). [4 points]
e)
Show the setup of a definite integral that will yield the length of the curve of 𝑓(𝑥)
from 𝑥 = 4 to 𝑥 = 4.2. You DO NOT have to evaluate the integral. [2 points]
𝑥
0
3
5
9
𝑔(𝑥)
−2
−1
−
1
8
−
𝑔′(𝑥)
4
√8
√3
3
4
1
20
𝐁𝐂 𝟏: The functions 𝑓 and 𝑔 are twice differentiable. Selected values of 𝑔(𝑥) and 𝑔′ (𝑥) are given in the
3𝑥
table above. The function 𝑓 is defined by 𝑓(𝑥) = 2 + ∫ 𝑔(𝑡)𝑑𝑡.
0
1
(A) Explain why there must be a number 𝑐, for 0 < 𝑐 < 9, such that 𝑔′ (𝑐) = .
3
3
(B) Evaluate ∫ 40𝑥𝑓 ′′′ (𝑥)𝑑𝑥.
0
(C) Using a right Riemann sum with three subintervals indicated in the table above, approximate
the length of the curve of 𝑔(𝑥) from 𝑥 = 0 to 𝑥 = 9.
(D) Let 𝑃𝑛 (𝑥) denote the 𝑛th degree Taylor polynomial for 𝑓 about 𝑥 = 0. Find 𝑃2 (𝑥).
∞
(E) Consider the geometric series ∑ 𝑎𝑟 𝑛 whose first three terms are defined by the polynomial 𝑃2 (𝑥)
𝑛=0
1
found in part (𝐷). Find the sum of this series when 𝑥 = , or show that the series diverges.
6
Created by Bryan Passwater
bryanpasswater1@gmail.com
BC Calculus
2020 Exam Practice
FR #6 (25 minutes: 15 points)
Name______________________________
Set a timer for 25 minutes to complete this problem. You may use your notes, textbooks, or any
materials I gave you throughout the year. You are not expected to use a calculator, but you
may use one if you would like. You should show all your steps as if you did not have a
calculator. I am guessing that the 25-minute problem will be worth 15 points and the 15-minute
problem will be worth 10 points for a total of 25 points. The college board has said that the
25-minute problem will be worth 60% and the 15-minute problem will be worth 40%, so that is
my best guess at how it may be broken down this year. Please show all appropriate
mathematics: no bald answers!
Two scientists, Gill and Jeff, are studying the stinkbug population in the upstairs hallway. At the
beginning of their study (𝑡 = 0) the stinkbug population is 1000. Both scientists decide to come
up with differential equations that each believes will accurately model the population of
$%
'
stinkbugs. Gill decides on the model $& = ( 𝐺, with 𝐺(0) = 1, where 𝑡 is measured in days and
𝐺(𝑡) is measured in thousands of stinkbugs. Jeff comes up with the alternate model
$'
=
𝐽(4 − 𝐽), with 𝐽(0) = 1, where 𝑡 is measured in days and 𝐽(𝑡) is measured in thousands
$&
(
of stinkbugs.
a)
Scientist Gill draws his slope field as shown below. Draw in the particular solution that
passes through the point (0,1). [2 points]
b)
Scientist Jeff draws his slope field as shown below. Draw in the particular solution that
passes through the point (0,1). [2 points]
Two scientists, Gill and Jeff, are studying the stinkbug population in the upstairs hallway. At the
beginning of their study (𝑡 = 0) the stinkbug population is 1000. Both scientists decide to come
up with differential equations that each believes will accurately model the population of
$%
'
stinkbugs. Gill decides on the model $& = ( 𝐺, with 𝐺(0) = 1, where 𝑡 is measured in days and
𝐺(𝑡) is measured in thousands of stinkbugs. Jeff comes up with the alternate model
$'
= ( 𝐽(4 − 𝐽), with 𝐽(0) = 1, where 𝑡 is measured in days and 𝐽(𝑡) is measured in thousands
$&
of stinkbugs.
$%
'
$-
'
c)
Find the particular solution to Scientist Gill’s differential equation $& = ( 𝐺 with the
initial condition that 𝐺(0) = 1. [5 points]
d)
Find the particular solution to Scientist Jeff’s differential equation $& = ( 𝐽(4 − 𝐽) with
the initial condition that 𝐽(0) = 1. (This solution will require the use of Partial Fractions
decomposition.) [6 points]