Stewart_Calc_7ET ch03sec07
MULTIPLE CHOICE
1. The equation of motion is given for a particle, where s is in meters and t is in seconds. Find
the acceleration after seconds.
a.
b.
c.
d.
e.
2. Find the average rate of change of the area of a circle with respect to its radius r as r
changes from to .
a.
b.
c.
d.
e.
3. The mass of the part of a metal rod that lies between its left end and a point x meters to the
right is
.
Find the linear density when x is m.
a.
b.
c.
d.
e.
4. Suppose the daily total cost (in dollars) of manufacturing x televisions is
What is the marginal cost when x = 300? What is the actual cost incurred in manufacturing
the 301st television?
a.
$195.33, $195.42
b.
$220.00, $220.28
c.
$195.33, $195.98
d.
$220.00, $220.73
5. Find the rate of change of y with respect of x at the indicated value of x.
t = csc x – 18 cos x;
a.
b.
c.
d.
6. Water flows from a tank of constant cross-sectional area 50 through an orifice of
constant cross-sectional area located at the bottom of the tank. Initially, the height of
the water in the tank was 20 ft, and t sec later it was given by the equation
How fast was the height of the water decreasing when its height was 2 ft?
a.
ft/sec
h
b.
ft/sec
c.
ft/sec
d.
ft/sec
7. In an adiabatic process (one in which no heat transfer takes place), the pressure P and
volume V of an ideal gas such as oxygen satisfy the equation
,
where C is a constant. Suppose that at a certain instant of time, the volume of the gas is 2L,
the pressure is 100 kPa, and the pressure is decreasing at the rate of 5 kPa/sec. Find the rate
at which the volume is changing.
a.
L/sec
b.
L/sec
c.
L/sec
d.
L/sec
8. The quantity Q of charge in coulombs C that has passed through a point in a wire up to time
t (measured in seconds) is given by
.
Find the current when .
a.
b.
c.
d.
e.
2
25
14
14
1
14
1
14
9. The mass of part of a wire is kilograms, where x is measured in meters from
one end of the wire. Find the linear density of the wire when x = m .
a.
b.
c.
d.
e.
10. The height (in meters) of a projectile shot vertically upward from a point m above
ground level with an initial velocity of 25.48 m/s is after t seconds.
a) When does the projectile reach its maximum height?
b) What is the maximum height?
a.
b.
c.
d.
e.
NUMERIC RESPONSE
1. The position function of a particle is given by
When does the particle reach a velocity of m/s?
2. Refer to the law of laminar flow. Consider a blood vessel with radius 0.01 cm, length 3 cm,
pressure difference and viscosity = 0.028.
Find the velocity of the blood at radius r =
3. A spherical balloon is being inflated. Find the rate of increase of the surface area
with respect to the radius r when r = ft.
4. A company makes computer chips from square wafers of silicon. It wants to keep the side
length of a wafer very close to mm. The area is A(x). Find ( ).
5. Newton’s Law of Gravitation says that the magnitude F of the force exerted by a body of
mass m on a body of mass M is .
Find .
6. If a tank holds 5000 gallons of water, and that water can drain from the tank in 40 minutes,
then Torricelli’s Law gives the volume V of water remaining in the tank after t minutes as
.
Find the rate at which water is draining from the tank after minutes.
SHORT ANSWER
1. Suppose that f and g are functions that are differentiable at x = 2 and that f (2) = –1,
(2) = 3, g(2) = 3, and (2) = –4. Find .
2. s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in
seconds, where . Find the position, velocity, and speed of the body at the indicated
time.
; t = 3
3. s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in
seconds, where . Find the position, velocity, and speed of the body at the indicated
time.
; t = 1
4. s(t) is the position of a body moving along a coordinate line, where , and s(t) is
measured in feet and t in seconds.
a. Determine the time(s) and the position(s) when the body is stationary.
b. When is the body moving in the positive direction? In the negative direction?
c. Sketch a schematic showing the position of the body at any time t.
5. In calm waters, the oil spilling from the ruptured hull of a grounded tanker spreads in all
directions. Assuming that the polluted area is circular, determine how fast the area is
increasing when the radius of the circle is 20 ft and is increasing at the rate of ft/sec.
Round to the nearest tenth if necessary.
6. Find the differential of the function at the indicated number.
;
7. Two chemicals react to form another chemical. Suppose that the amount of chemical formed
in time t (in hours) is given by
1
6
where is measured in pounds.
a. Find the rate at which the chemical is formed when Round to two decimal places.
b. How many pounds of the chemical are formed eventually?
8. Suppose that f and g are functions that are differentiable at x = –3 and that f (–3) = 3, (–
3) = –5, g (–3) = 3, and (–3) = 3. Find .