Practice 4 Math 242
Free Response
1. Find an equation of the tangent line to the curve.
6. Determine where f is discontinuous.
ÔÏÔ
ÔÔ !x
if
ÔÔ
ÔÔ
ÔÔ
f(x) = ÌÔ 3 ! x
if
ÔÔ
ÔÔ
ÔÔ (3 ! x) 2 if
ÔÔ
ÔÓ
x
y=
at (4, 0.2)
x+6
2. Newton's Law of Gravitation says that the
magnitude F of the force exerted by a body of
GmM
mass m on a body of mass M is F =
.
2
r
x<0
0"x<3
x>3
7. Find the interval on which the curve
x
Find
dF
(6).
dr
F(x) =
# 8 dt
+ 8t
is concave downward.
0
3. Use Newton's method with the specified initial
approximation x 1 to find x 3 , the third
8. Find the limit.
ÊÁ
Á
lim ÁÁÁ
Á
x$%Ë
approximation to the root of the given equation.
(Give your answer to four decimal places.)
2
x + ax !
ˆ˜
2
˜
x + bx ˜˜˜
˜
¯
Ê -1 ˆ &
x
9. If f(x) = 5 + 3x + e , find ÁÁÁÁ f ˜˜˜˜ (6).
Ë ¯
4
x ! 13 = 0, x 1 = 2
4. Two carts, A and B, are connected by a rope 40
ft long that passes over a pulley (see the figure
below). The point Q is on the floor 10 ft directly
beneath and between the carts. Cart A is being
pulled away from Q at a speed of 5 ft/s. How fast
is cart B moving toward Q at the instant when
cart A is 8 ft from Q?
10. If a bacteria population starts with 150 bacteria
and doubles every three hours, then the number
t/3
of bacteria after t hours is n = f(t) = 150(2 ).
When will the population reach 45,000?
11. Find an equation of the tangent line to the curve
at the given point.
2x
y = 7e cos 'x, (0, 7)
5. Use the linear approximation of the function
f(x) = 7 ! x at a = 0 to approximate the
number
7.1 .
1
12. A television camera is positioned 4,600 ft from
the base of a rocket launching pad. The angle of
elevation of the camera has to change at the
correct rate in order to keep the rocket in sight.
Also, the mechanism for focusing the camera has
to take into account the increasing distance from
the camera to the rising rocket. Let's assume the
rocket rises vertically and its speed is 680 ft/s
when it has risen 2,600 ft. If the television
camera is always kept aimed at the rocket, how
fast is the camera's angle of elevation changing at
this moment? Round the result to the nearest
thousandth.
13. Find the solution of the equation correct to four
decimal places.
x
ln(e ! 2) = 7
Multiple Choice
Identify the choice that best completes the statement or answers the question.
14. Consider the following problem: A farmer with
890 ft of fencing wants to enclose a rectangular
area and then divide it into four pens with fencing
parallel to one side of the rectangle. What is the
largest possible total area of the four pens?
16. Find the dimensions of the rectangle of largest
area that can be inscribed in an equilateral triangle
of side L = 9 cm if one side of the rectangle lies
on the base of the triangle. Round the result to
the nearest tenth.
Select the correct answer.
2
a. 19,825.5 ft
2
b. 19,802.5 ft
2
c. 19,801.5 ft
2
d. 19,902.5 ft
2
e. 19,791.5 ft
15. Evaluate the integral.
Select the correct answer.
a. 5.5 cm, 4.4 cm
b. 4 cm, 3.91 cm
c. 7.5 cm, 2.9 cm
d. 4.5 cm, 3.9 cm
e. 4.5 cm, 4 cm
2
2
y
x
17. Find the tangent line to the ellipse
+
=1
16
4
Ê
ˆ
at the point ÁÁÁÁ 2, ! 3 ˜˜˜˜ .
Ë
¯
3
# ||| 2x ! x
!1
2
|
|| dx
Select the correct answer.
a. y = 1.29x ! 2.31
b. y = !0.71x ! 1.31
c. y = 0.29x ! 3.31
d. y = 0.288x ! 2.308
e. none of these
Select the correct answer. The choices are
rounded to the nearest hundredth.
a. 19.33
b. 22.00
c. 14.67
d. 4.00
e. 32.67
2
ID: A
Practice 4 Math 242
Answer Section
FREE RESPONSE
1
(x ! 4) + 0.2
200
!2GmM
216
1.8989
3.36
2.66
at 0 and 3
(!1, %)
a!b
2
1/4
24.69
y = 14x + 7
0.112
x = 7.0018
1. y =
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
MULTIPLE CHOICE
14.
15.
16.
17.
B
D
D
D
1