Mr. Benson
BC Calculus
Chapter 4&5 Practice Problems Answers
1)
1
1
dx = 4 arc tan x | = 4 arc tan 1 - 4 arc tan (-1) = 4( ) - 4(- ) = 2
4
4
1 + x2
-1
-1
4
2) For what value of x does the function f(x) = x3 - 9x2 - 120x + 6 have a local minimum?
f '(x) = 3x2 - 18x - 120 = 3(x2 - 6x - 40) = 3(x - 10)(x + 4) = 0
-4 10
+ | - | +
min at x = 10
x = -4, 10
3) The acceleration of a particle moving along the x-axis at time t is given by a(t) = 4t - 12. If the velocity is 10
when t = 0 and the position is 4 when t = 0, then at what values of t is the particle changing direction?
v(t) =
4t - 12 dt = 2t2 - 12t + C
a(t) dt =
v(0) = 10
C = 10
2
v(t) = 2t - 12t + 10 = 2(t2 - 6t + 5) = 2(t - 5)(t - 1) = 0
t = 1, 5
4) The average value of the function f(x) = (x - 1)2 on the interval from x = 1 to x = 5 is what?
Avg =
1
b- a
b
f(x) dx =
a
e3 ln x + e3x dx =
5)
/4
6)
0
sin x dx +
0
1
5-1
5
(x - 1)2 dx =
1
x3 + e3x dx =
1 4 1 3x
x + e +C
4
3
cos x dx = -cos x
- /4
1 64
16
( )=
4 3
3
/4
0
| + sin x | = - cos + cos 0 + sin 0 - sin (- ) = 1
4
4
0
- /4
7) Boats A and B leave the same place at the same time. Boat A heads due north at 18 km/hr. Boat B heads due east
at 24 km/hr. After 2.5 hours, how fast is the distance between the boats increasing?
a2 + b2 = c2
a = 18 · 2.5 = 45
b = 24 · 2.5 = 60
da
db
= 18,
= 24
dt
dt
a2 + b2 = c2
a
da
db
dc
+b
=c
dt
dt
dt
45(18) + (60)(24) = (75)
1
dc
dt
dc
= 30
dt
100
8) If
100
f(x) dx = A and
30
f(x) dx = B , then
50
50
f(x) dx = A - B
30
9) Find the following for the graph of y = x3 - 4x2 + 4x + 2:
a) Critical points
b) Identity types of Extrema
c) Increasing and decreasing intervals
d) Inflection points
e) Concave up and concave down intervals
y = x3 - 4x2 + 4x + 2
y' = 3x2 - 8x + 4 = (3x - 2)(x - 2) = 0
x = 2/3, 2
2/3
2
max at 2/3 and min at 2, inc: ( , 2/3) (2, ) and dec: (2/3, 2)
+ | - | +
y'' = 6x - 8 = 0
x = 4/3
4/3
- | +
inflection pt at 4/3, up: (4/3, ) and down: ( , 4/3)
10) Using your calculator, what is the average value of the function f(x) = ln2x on the interval [2, 4]?
Avg =
11)
d
dx
1
4-2
3x
4
2
1
ln2 x dx = 2.4086 = 1.204
2
cos t dt = 3 cos 3x
0
3
12) If the definite integral
1
error?
3
x2 + 1 dx is approximated by using the Trapezoidal Rule with n = 4, what is the
x2 + 1 dx =
1
3
1 3
4 32
x + x | = 12 - =
= 10.6
3
3
3
1
x
y
1
1.5
2
2.5
3
2
3.25
5
7.25
10
.5
Trap4 = (2 + 2(3.25) + 2(5) + 2(7.25) + 10) = 10.75
2
error = .083 =
1
12
2
13) A particle moves along the x-axis so that its acceleration at any time t > 0 is given by a(t) = 12t - 18. At time t =
1, the velocity of the particle is 0 and its position is 9.
a) Write an expression for the velocity of the particle v(t).
b) Write an expression for the position x(t) of the particle.
a) v(t) =
a(t) dt =
12t - 18 dt = 6t2 - 18t + C
b) x(t) =
v(t) dt =
6t2 - 18t + 12 dt = 2t3 - 9t2 + 12t + C
v(1) = 6 - 18 + C = 0
C = 12
x(1) = 2 - 9 + 12 + C = 9
v(t) = 6t2 - 18t + 12
C=4
x(t) = 2t3 - 9t2 + 12t + 4
14) Water is draining at the rate of 48 ft3 /min from the vertex at the bottom of a conical tank whose diameter at its
base is 40 feet and whose height is 60 feet.
a) Find an expression for the volume of water in the tank in terms of its radius at the surface of the water.
b) At what rate is the radius of the water in the tank shrinking when the radius is 16 feet?
c) How fast is the height of the water in the tank dropping at the instant that the radius is 16 feet?
V=
1 2 dV
r h,
= -48 , radius of 20 and height of 60
3
dt
a) V =
1 2
r h
3
V=
1 2
r (3r) = r3
3
dV
dr
= 3 r2
dt
dt
b) V = r3
c) 3r = h
3
x
15) Let F(x) =
dr dh
=
dt
dt
cos
0
3r = h
dr
-48 = 3 (16)2
dt
dr
1
=ft/min
dt
16
dh
3
=ft/min
dt
16
t
3
+
2
2
dt on the closed interval [0, 4 ].
a) Find F'(2 )
b) Find the average value of F'(x) on the interval [0, 4 ].
a) F'(x) = cos
16) Find
x
3
+
2
2
x 3x dx =
F'(2 ) = cos( ) +
3x3/2 dx =
3
1
=
2
2
2 3 5/2
x
+C
5
17) Write the integral that gives the area of the region above the x-axis and below the curve y = 8 + 2x - x2?
y = 8 + 2x - x2 = (-x + 4)(x + 2) = 0
4
Area =
8 + 2x - x2 dx
-2
x = -2, 4
3
18) Find a positive value of c, for x, that satisfies the conclusion of the Mean Value Theorem for Functions for f(x) =
6x - 5 on the interval [2, 5].
MVT: Instantaneous = Average
19) The average value of f(x) =
20)
d
dx
1
e- 1
x2
e
1
1
b- a
21 7
c=
=
6
2
6c - 5 = 16
Avg =
f(c) =
b
f(x) dx
f(c) =
a
1
5-2
5
6x - 5 dx =
2
1
(48) = 16
3
1
from x = 1 to x = e is what?
x
e
1
1
1
dx =
ln(x) | =
-1
x
e-1
e
1
sin2t dt = 2xsin2(x2 ) - 2sin2 (2x)
2x
21) A 20 foot ladder slides down a wall at 5 ft/sec. At what speed is the bottom sliding out when the top is 10 feet
from the floor?
x2 + y2 = z 2
x = 10, z = 20
dy
= -5
dt
y=
x2 + y2 = z 2
x
300
dx
dy
dz
+y
=z
dt
dt
dt
10
dx
+
dt
300(-5) = 20(0)
dx
=
dt
300
=5 3
2
22) Find two non-negative numbers x and y whose sum is 100 and for which x 2 y is a maximum.
x + y = 100
y = 100 - x
2
2
P = x y = x (100 - x) = 100x2 - x3
P' = 200x - 3x2 = 0
x = 0,
0 200/3 100
X | + | - | X
200
3
max at x = 200/3
y = 100/3
23) Using your calculator, find the distance traveled from t = 1 to t = 5 seconds, for a particle whose velocity is given
by v(t) = t + ln t.
5
distance traveled =
t + ln(t) dt = 16.047
1
4
24) A rectangle is to be inscribed between the parabola y = 4 - x2 and the x-axis, with its base on the axis. A value
of x that maximizes the area of the rectangle is what?
Area of the rectangle = length · width = (2x)(4 - x2 ) = 8x - 2x3
A' = 8 - 6x2 = 0
x = 1.155
25) A data-recording thermometer recorded the soil temperature in a field every 2 hours from noon to midnight, as
shown in the following table. Use the Trapezoidal Rule to estimate the average temperature for the 12-hour
period.
Time
Noon
2
4
6
8
10
Midnight
Temp (°F)
67
68
70
70
69
69
68
Trap =
2
(67 + 2(68) + 2(70) + 2(70) + 2(69) + 2(69) + 68) = 827
2
Avg =
1
(827) = 68.917
12
5