x - Epsilen

!
A. Limits - L’Hopital’s Rule
What you are finding: L’Hopital’s Rule is used to find limits of the form
f ( x)
lim where lim f ( x)
= limg( x)
= 0 or lim f ( x)
= limg( x)
= # .
x "c g( x)
x "c
x "c
x "c
x "c
0 "
How to find it: Try and find limits by traditional methods (plugging in). If you get or
0 "
!
, apply
f ( x)
f # ( x)
L’Hopital’s rule, which says that lim = lim . L’Hopital’s rule can be applied whenever plugging
x "c g( x)
x "c g # ( x)
0
in creates an indeterminate form:
0
!
!
,"
" ,0# "," $ ",1" ,0 0 , and " 0 . A limit involving 0" # or # $ # is found by
creating a quotient out of that expression. A limit involving exponents ( 1
!
!
" ,0 0 , or " 0 ) involves taking a
natural log of the expression to “move the exponent down.”
e
1. Find lim
x "0
x + cos x # x # 2
x 4 # x 3
!
A.
" 1
30
!
B.
" 1
24
!
C.
" 1
6
D. 0 E. nonexistent
_______________________________________________________________________________________
!
2. Find
lim
x "1 + x 2 ( #1)ln(
x #1)
A. 0 B.
!
"1
2
!
C.
1
2
D. -1 E. nonexistent
© 2011 www.mastermathmentor.com - 7 - Demystifying the BC Calculus MC Exam

!
!
3. Find
lim
x "2
e t 2 x
#1
$ dt
2
x 3 # 4x
A. 0 B.
!
e 4
12
!
C.
e 3
8
© 2011 www.mastermathmentor.com - 8 - Demystifying the BC Calculus MC Exam
!
D.
"e
6
E. nonexistent
_______________________________________________________________________________________
4. A particle moves in the xy-plane so that the position of the particle at any time t is given by
x( t)
= cost and y( t)
= sin 2 dy
4t. Find lim
t "0 dx .
A. -32 B. -16 C. -8 D. -4 E. 0
!
_______________________________________________________________________________________
5. A population of bacteria is growing and at any time t, the population is given by 500 1+ 2
t
" %
$ ' . Find the
# t &
maximum limit of the population.
A. 500 B. 500 + e C. 500e D. 2 + ln500 E.
!
!
!
!
!
500e 2

!
!
B1. Integration by Parts
What you are finding: When you attempt to integrate an expression, you try all the rules you have been
given to that point - typically power, substitution, and the like. But if these don’t work, integration by parts
may do the trick. Integration by parts is usually used when you are need to find the integral of a product.
How to find it: Integration by parts states that # u dv = uv " # v du + C.
To perform integration by parts,
u = v =
set up:
. You need to fill in the u and the dv from the original problem. Determine
du = dv =
du and v, then substitute into the formula. You are replacing one integration problem with another that
might more easily be done with simple ! methods. The trick is to determine the u and the dv. Functions that
can be “powered down” are typically the u and functions that have repetitive derivatives (exponential and
!
trig) are typically the dv.
!
!
!
6.
"
2x cos4x dx =
A.
x
sin4x + cos4x + C B.
2
!
© 2011 www.mastermathmentor.com - 9 - Demystifying the BC Calculus MC Exam
x
2
1
sin4x + cos4x + C
8
C. 8x sin4x + 32cos4x + C D. 8x sin4x " 32cos4x + C
E.
x
sin4x " cos4x + C
2
!
_______________________________________________________________________________________
!
!
7.
"
x 2 e 2x dx =
A. e 2x 2x 2 ( " 8x +16)
+ C B. e
!
2x 2x 2 ( " 8x + 2)
+ C
C. e 2x x 2 # x 1&
% " " ( + C
$ 2 2 2'
D. e 2x x 2 # x 1 &
% " + ( + C
$ 2 2 4 '
E.
!
!
" %
$ ' + C
# 3 &
e 2x x 3

!
!
!
8. Let R be the region bounded by the graph of y = 2x ln x, the x-axis and
the line x = e, as shown by the figure to the right. Find the area of R.
e
A.
!
2 +1
2 B.
e 2 "1
2
e
C.
2
2 D. 2e E. 2e +1
!
!
_______________________________________________________________________________________
!
!
9. The shaded region between the graph of y = 2tan "1 x and the x-axis for
0 ≤ x ≤ 1 as shown in the figure is the base of a solid whose cross-sections
perpendicular to the x-axis are squares. Find the volume of the solid.
"
A. + ln2 #1
4
!
B. " + e # ln2
C. " # ln2
!
"
D. # ln2
4
"
E. # ln2
2
!
_______________________________________________________________________________________
10. The function f is twice-differentiable and its derivatives are continuous. The table below gives the
value of
x f x
f , f " and f " for x = 0 and x = 2. Find the value of
( ) f " ( x)
f " ( x)
0 3 6 2
!
2 #5 #1 4
© 2011 www.mastermathmentor.com - 10 - Demystifying the BC Calculus MC Exam
!
!
# x f " ( x)
dx .
A. -10 B. -8 C. 6 D. 14 E. 16
!
2
0

!
!
!
B2. Integration using Partial Fractions
What you are finding: When you attempt to integrate a fraction, typically you let u be the expression in the
denominator and hope that du will be in the numerator. When this doesn’t happen, the technique of partial
dx
fractions may work. One form of this type of problem is
x
!
2 " where x
+ mx + n
!
2 + mx + n factors into two
non-repeating binomials.
dx
How to find it: Use the “Heaviside method.” Factor your denominator to get "
. You need to
( x + a)
( x + b)
1
write
( x + a)
( x + b)
!
as
x + a
!
+ . To find the numerator of the x + a expression, cover up the x + a in
x + b
1
expression, and plug in x = "a. To find the numerator of the x + b expression, cover up the
( x + a)
( x + b)
1
x + b in
expression, and plug in x = "b. From there, each expression can be integrated.
( x + a)
( x + b)
!
!
!
11.
"
A.
4x + 2
x 2 dx =
+ 4x + 3
1
2 ln x 2 + 4 x + 3 + C B.
D. ln x +1 " 5ln x + 3 + C E. ln
!
!
!
2ln x 2 + 4x + 3 + C C. 5ln x + 3 " ln x +1 + C
5
x + 3
+ C
x +1
_______________________________________________________________________________________
!
!
12. Use the substitution
u = cos x to find
sin x
#
cos x( cos x "1)
A. lncos x ! "1 + C B. "lncos x "1 + C C.
1" cos x
D. ln + C
cos x
!
!
cos x
E. ln + C
cos x "1
!
!
© 2011 www.mastermathmentor.com - 11 - Demystifying the BC Calculus MC Exam
dx.
!
cos x "1
"ln + C
cos x

!
!
!
!
13.
#
x 3
x 2 dx =
"1
x
A.
2
2 + 2ln x 2 "1 + C
x
B.
!
2 x +1
+ ln
x "1
+ C
2
C.
x
D.
!
2 x "1
+ ln
x +1
+ C
2
5
x + 3
E. ln + C
x +1
!
x 2
2 " 2ln x 2 "1 + C
_______________________________________________________________________________________
14. Region R is defined as the region between the graph of
9
y =
x 2 , x = 2 and the x-axis as shown in the
+ x " 2
figure to the right. Find the area of region R.
!
!
A. ln12 B. 1+ 3ln4
C.
3ln 1
4
!
E. infinite
!
D. 6ln2
© 2011 www.mastermathmentor.com - 12 - Demystifying the BC Calculus MC Exam

!
C. Improper Integrals
What you are finding: An improper integral is in the form
be in the form
continuous.
b
!
# f ( x)
dx or # f ( x)
dx or # f ( x)
dx. It also can
" f ( x)
dx where there is at least one value c such that a ≤ c ≤ b for which
a
"
b
!
How ! to find it: Improper integrals are just limit problems in disguise: # f ( x)
dx = lim # f ( x)
dx or
b
"#
b
a %"#
a
b
© 2011 www.mastermathmentor.com - 13 - Demystifying the BC Calculus MC Exam
"
a
a
a
$"
"
$"
b $"
a
f ( x)
is not
$ f ( x)
dx = lim $ f ( x)
dx . In the case where there is a discontinuity at x = c, the improper integral is split
into two pieces:
!
" f ( x)
dx = lim " f ( x)
dx + lim " f ( x)
dx. Improper integrals usually go hand-in-hand
a
k #c $
with area and volume problems.
!
15. Which of the following are convergent?
!
I.
"
1
# dx II.
x
1
!
k
a
k #c +
b
k
1
1
" dx III.
x
0
!
"
1
1
# dx
A. I only B. II only C. III only D. II and III only E. I, II and III
_______________________________________________________________________________________
!
16.
#
$
0
A.
xe
"4 x
" 1
16
dx =
!
B.
1
16
C. -16 D. 16 E. infinite
x 3

!
17. The region bounded by the graph of
Find the volume of the solid.
y = 4
, the line x = 4 and the x - axis is rotated about the x-axis.
x
A. π B. 2π C. 4π D. 16π E. infinite
!
_______________________________________________________________________________________
!
18.
"
#
1
A.
1
dx =
x ( x +1)
"
4
!
!
B.
"
2
C. π D. 2π E. infinite
_______________________________________________________________________________________
4
1
19. To the right is a graph of f ( x)
= . Find the value of
3 # f ( x)
dx .
( x "1)
"2
A. 0
1
B.
3
!
!
C. " 1
3
1
D.
6
E. Divergent
!
!
© 2011 www.mastermathmentor.com - 14 - Demystifying the BC Calculus MC Exam

!
D. Euler’s Method
What you are finding: Euler’s Method provides a numerical procedure to approximate the solution of a
differential equation with a given initial value.
How to find it: 1) Start with a given initial point (x, y) on the graph of the function and a given "x = dx.
2) Calculate the slope using the DEQ at the point.
3) Calculate the value of dy using the fact that dy "
!
dy
#x .
dx
4) Find the new values of y and x: ynew = yold + dy and xnew = xold + "x
5) Repeat the process at step 2).
There are calculator programs available to perform Euler’s
!
Method. Typically, Euler Method problems occur
in the non-calculator section where only one ! or two steps of the method need to be performed.
!
20. Let
y = f ( x)
be the solution to the differential equation
( ) = "1. What is the approximation for
f 3
size of 0.5?
f 4
dy
dx = x + y 2 with the initial condition that
( ) if Euler’s Method is used, starting at
!
A. 2.25 B. 3.25 C. 4.5 D. 5.5
!
!
E. 12.5
x = 3 with a step
_______________________________________________________________________________________
dy
21. Let y = f ( x)
be a solution to the differential equation
dx
!
!
= y x with initial condition f ( 0)
= k ,
k a constant, k " 0. If Euler’s method with 3 steps of equal size starting at x = 0 gives the
approximation f ( 3)
" 0, find the value of k.
!
A. ! "
!
1
2
B.
1
C. 1 D. -1 E. "
2 3
2
!
!
© 2011 www.mastermathmentor.com - 15 - Demystifying the BC Calculus MC Exam
!

dy y
22. Consider the differential equation = with initial condition f ( 2)
= "4. Find the difference between
dx x
the exact value of f ( 8)
and an Euler approximation of f ( 8)
using a step of 0.5.
A. 0 B. 1
!
C. 2 ! D. 25 E. 50
!
!
_______________________________________________________________________________________
dy
23. (Calc) Consider the differential equation = cos x with initial condition f ( 0)
= 0. Find the difference
dx
between the exact value of f
!
!
" #
%
$ 2
&
( and an Euler approximation of f
'
" #
%
$ 2
&
( using two equal steps.
'
A. 0 B. 0.230 C. 0.341 D. 0.555 E. 0.707
!
© 2011 www.mastermathmentor.com - 16 - Demystifying the BC Calculus MC Exam
!

!
E. Logistic Curves
What you are finding: Logistic curves occur when a quantity is growing at a rate
proportional to itself and the room available for growth. This room available is called
the carrying capacity. The curve has a distinctive “S-shape” where the initial stage of
growth is exponential, then slows, and eventually the growth essentially stops.
How to find it: Logistic growth is signaled by the differential equation
( ) =
C
© 2011 www.mastermathmentor.com - 17 - Demystifying the BC Calculus MC Exam
dP
dt
= kP( P " t).
While this DEQ
can be solved into P t
, students are not responsible for that equation. They need to know how to
"Ckt
1+ de
d
determine the time when the logistic growth is the fastest. This ! is accomplished by
!
2 P
= 0. Also students
2
dt
need to know that the curve has a horizontal asymptote meaning limP(
t)
= C ( the carrying capacity).
t "#
24. A population of students having contracted the flu in a school year ! is modeled by a function P that
dP P # P &
satisfies the logistic differential equation with ! = % 2 " ( . If P( 0)
=100, find limP(
t).
dt 600 $ 800'
t "#
A. 400 B. 800 C. 1,600 D. 2,400 E. 4,800
!
_______________________________________________________________________________________
25. A population is modeled by a function G that satisfies the logistic differential equation
dG G # G &
= % 1" ( . If G( 0)
=1, for what value of G is the population growing the fastest?
dt e $ 4e'
A. 4 B. e C. 2e D. 4e E. 4e 2
!
_______________________________________________________________________________________
dy
26. Consider the differential equation = ky( L " y).
Let y = f ( x)
be the particular solution to the
dx
differential equation with f ( 0)
= 2. If x " 0, find the range of f ( x).
A. ( 0,L)
B. ( 0,2)
C. ( L,2]
D. [ 2,L)
E. [ 2,kL)
!
!
! !
!
!
!
!
!
!
!
!

F. Arc Length
What you are finding: Given a function on an interval [a, b], the arc length is defined as the total length of
the function from x = a to x = b. For this section, we will only concentrate on curves that are defined in
function form. Functions defined parametrically, in polar or in vector-valued forms have their own formulas.
How to find it: The arc length of a continuous function
b
!
f ( x)
over an interval [a, b] is given by
L = 1+ [ f " ( x)
] 2
# dx. Most problems involving arc length need calculators because of the difficulty of
a
integrating the expression.
27. (Calc) An ant walks around the first quadrant region R bounded by the
y-axis, the line y = 2x and the curve f ( x)
= 6 " 4x 3 2 as shown in the
figure to the right. Find the distance the ant walked.
!
!
A. 4.149 B. 10.312 C. 11.485 D. 11.881 E. 12.385
_______________________________________________________________________________________
28. If the length of a curve from
!
!
x = 2 to x = 8 is given by
point (-1, 4), which of the following could be the equation for the curve?
1+ 81x 4
dx , and the curve passes through the
!
A. y =13" 9x
!
2 B. y = 4 " 3x 3 C. y = 7 + 3x 3
D. y = "1" 3x 3 E. y = 9x 2 " 5
!
!
© 2011 www.mastermathmentor.com - 18 - Demystifying the BC Calculus MC Exam
8
"
2
!

29. (Calc) The yellow bird in the popular game
Angry Birds flies along the path y = 4 + 3x "
!
1 2
x
2
when x ≥ 0. When x = 4 (the point on the figure to the
right), the player touches the screen and the bird leaves the
path and travels along the line tangent to the path at that
point. If the bird crashes into the x-axis, find the total
distance the bird flies.
A. 6.800 B. 11.314 C. 12.000
D. 15.461 E. 18.114
_______________________________________________________________________________________
30. (Calc) The graphs of i) y = x 2 , ii) y = x and iii) y = 2 x "1 all pass through the points (0,0) and
(1,1). Find the difference in arc length from the largest arc length to the shortest arc length of these
functions on the interval [0,1].
A. 0.058 ! B. 0081 !
C. ! 0.284 D. 0.361 E. 0.419
_______________________________________________________________________________________
31. Find the arc length of the graph of
x = 1
3 y 2 3 2
( + 2)
for
0 " y " 3.
A. 12 B. 10 C. 6 D. 3 3 E.
!
!
© 2011 www.mastermathmentor.com - 19 - Demystifying the BC Calculus MC Exam
!
!
3

!
!
G. Parametric Equations
What you are finding: Parametric equations are continuous functions of t in the form x = f ( t)
and y = g( t).
Taken together, the parametric equations create a graph where the points x and y are independent of each
other and both dependent on the parameter t (which is usually time). Parametric curves when graphed do not
have to be functions. Typically, it is necessary to take derivatives of parametrics. Since the study of vectors
parallels the study of parametrics, in this section we will only analyze the very ! few problems that are not
associated with motion in the plane.
How to find it: If a smooth curve C is given by the parametric equations x = f ( t)
and y = g( t),
then the
dy
slope of C at the point (x, y) is given by
dx
!
!
=
dy
dt ,dx
dx dt
" 0.
dt
The 2 nd d
derivative of the curve is given by
!
2 d " dy %
$ '
y d " dy % dt # dx &
= 2 $ ' = .
dx dx # dx & dx
dt
2
" dx %
The arc length is given by L = $ ' +
# dt &
!
dy
t= b
2
" %
( $ ' dt. The curve must be smooth and may not intersect itself.
# dt &
t= a
32. What is the area under the curve described by the parametric equations x = 2cost and y = 3sin 2 t for
0 " t " #?
A. 4 B. 8 C.
!
11
4
© 2011 www.mastermathmentor.com - 20 - Demystifying the BC Calculus MC Exam
!
! D.
11
2
_______________________________________________________________________________________
33. A position of a particle moving in the xy-plane is given by x = t 3 " 6t 2 + 9t +1 and
y = 2t 3 " 9t 2 " 2. For what values of t is the particle at rest?
A. 0 only B. 1 only C. 3 only
!
D. 0 and 1 only E. 0, 1 and 3
!
E.
4
3

!
34. A curve C is defined by the parametric equations x = t 2 " t " 4 and y = t 3 " 7t " 2. Which of the
following is the equation of the line tangent to the graph of C at the point (2, 4) ?
!
!
A. y = 6 " x B. x " 4y +14 = 0 C. 5x " 3y + 2 = 0
!
D.
y = 4x " 4 E. No tangent line at (2, 4)
!
_______________________________________________________________________________________
35. Describe the behavior of curve C defined by the parametric equations
at t = 1.
A. Increasing, concave up B. Decreasing, concave up
C. Increasing, concave down D. Decreasing, concave ! down
E. Increasing, no concavity
© 2011 www.mastermathmentor.com - 21 - Demystifying the BC Calculus MC Exam
!
x =1+ 1
t and y = t 3 + 2t 2 " t "1
_______________________________________________________________________________________
36. Find the expression which represents the length L of the path described by the parametric equations
!
!
!
x = sin 2 ( 2t)
and y = cos( 3t)
for
# 2
0 " t " #
2 .
A. L = $
0
2sin2t cos2t " 3sin3t dt
!
B. L = 4 sin 2 4t 2 + 9sin 2 9t 2 #
0
dt
C. L = 16sin 2 4t 2 cos 2 4t 2 + 9sin 2 9t 2 " 2
# dt
D. L = 4sin 2 2t cos 2 2t + 9sin 2 " 2
#
3t dt
E.
0
" 2
#
L = 16sin 2 2t cos 2 2t + 9sin 2 3t dt
0
!
!
" 2
0

H. Vector-Valued Functions
What you are finding: While concepts like unit vectors, dot products, and angles between vectors are
important for multivariable calculus, vectors in BC calculus are little more than parametric equations in
disguise.
How to find it: Typically, you will be given a situation where an object is moving in the plane. You could be
given either its position vector x( t)
and y( t),
its velocity vector x " ( t)
and y " ( t)
or its acceleration vector
x " ( t)
and y " ( t)
and use the basic derivative or integral relationships that have been taught in AB calculus to
find the other vectors. The one formula that students should know is that the speed of the object is defined
as the absolute value ! of the velocity: v t
[ ] 2
( ) = x " ( t)
[ ( ) ] 2
+ ! y " t
. The speed is a scalar, not a vector.
37. A particle moves on a plane curve such that at any time t > 0, its x-coordinate is
y-coordinate is 2 " ! t 2 ( ) 2
. Find the magnitude of the particle’s acceleration at t = 1.
A. 4 B.
!
!
2 C. 2 2 D. 3 2 ! E. 4 2
!
t " t 2 + t 3 while its
_______________________________________________________________________________________
38. The position of an object moving in the xy-plane with position function
t ≥ 0. What is the maximum speed attained by the object?
A. 1 B.
!
© 2011 www.mastermathmentor.com - 22 - Demystifying the BC Calculus MC Exam
!
2 C. 2 D. 4 E. 2 2
!
!
r( t)
= 1+ sint,t + cost ,
_______________________________________________________________________________________
39. A xy-plane has both its x and y-coordinates measured in inches. An ant is walking along this plane with
its position vector as 2t 3 2 ,3t "1 , t measured in minutes. What is the average speed of the ant
measured in inches per minute from t = 0 to t = 3 minutes?
A. 1 ! B.
!
14
3
!
C.
3 "1 D. 3 E. 9
!

40. An object moving in the xy-plane has position function
the motion of the object.
r( t)
=
1
( t +1)
2 ,t 2 " 6ln t + 2
( ) , t ≥ 0. Describe
A. Left and up B. Left and down C. Right and up
D. Right and down E. Depend ! on the value of t
_______________________________________________________________________________________
41. An object moving along a curve in the xy-plane has position ( x( t),y
( t)
) at time t with
dx
dy
= 8t + 2 and = sin2t for t " 0.
dt dt
At time t = 0, the object is at position (5, π). Where ! is the object at t =
!
!
"
2 ?
A. " 2 ( + " + 5," ) B. " 2 + " + 5," +1
D. 2", 1 # &
% ( E. ( 2" + 5," )
$ 2'
!
!
!
!
( ) C.
© 2011 www.mastermathmentor.com - 23 - Demystifying the BC Calculus MC Exam
!
( 2" + 5," +1)
_______________________________________________________________________________________
( ) at time t with
42. (Calc) An object moving along a curve in the xy-plane has position x( t),y
( t)
dx
dt
!
= x 2 + 3x +1 and dy
dt = et 2 "1
for t # 0.
At time t = 0, the object is at position (-6, -7). Find the position of the object at t = 2.
A. (4.667, 13.053) B. (-3.683, 12.718) C. (2.317, 19.718)
D. (4.427, 6.053) !
E. (-1.573, -0.947)