Practice Exercises

A1. A particle moves along a line in such a way that its position at time t is given by s = t³ – 6t² + 9t + 3. Its direction of motion changes when

(A) t = 1 only
(B) t = 2 only
(C) t = 3 only
(D) t = 1 and t = 3

A2. A body moves along a straight line so that its velocity v at time t is given by v = 4t³ + 3t² + 5. The distance the body covers from t = 0 to t = 2 equals

(A) 34
(B) 49
(C) 24
(D) 44

A3. A particle moves along a line with velocity v = 3t² – 6t. The total distance traveled from t = 0 to t = 3 equals

(A) 0
(B) 4
(C) 8
(D) 9

A4. A particle moves along a line with velocity v(t) = 3t² – 6t. The net change in position of the particle from t = 0 to t = 3 is

(A) 0
(B) 4
(C) 8
(D) 9

A5. The acceleration of a particle moving on a straight line is given by a = cos t, and when t = 0 the particle is at rest. The distance it covers from t = 0 to t = 2 is

(A) sin 2
(B) 1 – cos 2
(C) 2 – sin 2
(D) –cos 2

A6. During the worst 4-hour period of a hurricane the wind velocity, in miles per hour, is given by v(t) = 5t – t² + 100, 0 ≤ t ≤ 4. The average wind velocity during this period (in mph) is

(A) 100
(B) 102
(C) 
(D) 

*A7. The acceleration of an object in motion is given by the vector . If the object’s initial velocity was , which is the velocity vector at any time t?

A8. A stone is thrown upward from the ground with an initial velocity of 96 ft/sec. Its average velocity (given that a(t) = –32 ft/sec²) during the first 2 seconds is

(A) 16 ft/sec
(B) 32 ft/sec
(C) 64 ft/sec
(D) 80 ft/sec

A9. Assume that the density of vehicles (number of cars per mile) during morning rush hour, for the 20-mile stretch along the New York State Thruway southbound from the Governor Mario M. Cuomo Bridge, is given by f(x), where x is the distance, in miles, south of the bridge. Which of the following gives the number of vehicles (on this 20-mile stretch) from the bridge to a point x miles south of the bridge?

A10. The center of a city, that we will assume is circular, is on a straight highway (along the diameter of the circle). The radius of the city is 3 miles. (NOTE: The equation for a circle centered at the origin is x² + y² = r², where r is the radius of the circle.) The density of the population, in thousands of people per square mile, is given by f(x) = 12 – 2x at a distance x miles from the highway. The population of the city (in thousands of people) is given by the integral

B1. A car accelerates from 0 to 60 mph in 10 seconds, with constant acceleration. (Note that 60 mph = 88 ft/sec.) The acceleration (in ft/sec²) is

(A) 1.76
(B) 5.3
(C) 6
(D) 8.8

*For Questions B2–B4, use the following information: The velocity v of a particle moving on a curve is given, at time t, by v = 〈t,t – 1〉. When t = 0, the particle is at point (0,1).

*B2. At time t the position vector R is

*B3. The acceleration vector at time t = 2 is
(A)〈1,1〉
(B)〈1,–1〉
(C)〈1,2〉
(D)〈2,–1〉

*B4. The speed of the particle is at a minimum when t equals

(A) 0
(B) 
(C) 1
(D) 1.5

CHALLENGE *B5. A particle moves along a curve in such a way that its position vector and velocity vector are perpendicular at all times. If the particle passes through the point (4,3), then the equation of the curve is

(A) x² + y² = 5
(B) x² + y² = 25
(C) x² + 2y² = 34
(D) x² – y² = 7

*B6. The velocity of an object is given by . If this object is at the origin when t = 1, where was it at t = 0?

(A) (–2,–4)
(B) (2,4)
(C) 
(D) 

B7. Suppose the current world population is 8 billion and the population t years from now is estimated to be P(t) = 8e^0.0105t billion people. On the basis of this supposition, the average population of the world, in billions, over the next 25 years will be approximately

(A) 8.096
(B) 8.197
(C) 8.827
(D) 9.148

B8. A beach opens at 8 A.M. and people arrive at a rate of R(t) = 10 + 40t people per hour, where t represents the number of hours the beach has been open. Assuming no one leaves before noon, at what time will there be 100 people there?

(A) 9:45
(B) 10:00
(C) 10:15
(D) 10:30

B9. Suppose the amount of a drug in a patient’s bloodstream t hours after intravenous administration is . The average amount in the bloodstream during the first 4 hours is

(A) 6.0 mg
(B) 11.0 mg
(C) 16.6 mg
(D) 24.0 mg

B10. A rumor spreads through a town at the rate of R(t) = t² + 10t new people per day t days after it was first heard. Approximately how many people hear the rumor during the second week (from the 7th to the 14th days) after it was first heard?

(A) 359
(B) 1535
(C) 1894
(D) 2219

B11. Oil is leaking from a tanker at the rate of L(t) = 1000e^–0.3t gal/hr, where t is given in hours. The total number of gallons of oil that will leak out during the first 8 hours is approximately

(A) 1271
(B) 3031
(C) 3161
(D) 4323

CHALLENGE B12. The population density of Winnipeg, which is located in the middle of the Canadian prairie, drops dramatically as distance from the center of town increases. This is shown in the following table:

Using a left Riemann Sum, we can calculate the population living within a 10-mile radius of the center to be approximately

(A) 36,000
(B) 226,200
(C) 691,200
(D) 754,000

B13. If a factory continuously dumps pollutants into a river at the rate of  tons per day, then the amount dumped after 7 weeks is approximately

(A) 0.07 ton
(B) 0.90 ton
(C) 1.55 tons
(D) 1.27 tons

B14. A roast at 160°F is put into a refrigerator whose temperature is 45°F. The temperature of the roast is cooling at time t at the rate of R(t) = –9e^–0.08t°F per minute. The temperature, to the nearest degree F, of the roast 20 minutes after it is put into the refrigerator is

(A) 45°
(B) 70°
(C) 81°
(D) 90°

B15. How long will it take to release 9 tons of pollutant if the rate at which the pollutant is being released is P(t) = te^–0.3t tons per week?

(A) 10.2 weeks
(B) 11.0 weeks
(C) 12.1 weeks
(D) 12.9 weeks

B16. What is the total area bounded by the curve f(x) = x³ – 4x² + 3x and the x-axis on the interval 0 ≤ x ≤ 3?

(A) –2.25
(B) 2.25
(C) 3
(D) 3.083

B17. Water is leaking from a tank at the rate of W(t) = –0.1t² – 0.3t + 2 gal/hr. The total amount, in gallons, that will leak out in the next 3 hours is approximately

(A) 2.08
(B) 3.13
(C) 3.48
(D) 3.75

B18. A bacterial culture is growing at the rate of B(t) = 1000e^0.03t bacteria in t hours. The total increase in bacterial population during the second hour is approximately

(A) 1015
(B) 1046
(C) 1061
(D) 1077

B19. A website went live at noon, and the rate of hits (visitors/hour) increased continuously for the first 8 hours. The graph of R(t), the rate of hits for the website in visitors/hour, is shown below, where t is the number of hours since the website went live.

Using a midpoint or a trapezoidal sum, approximate the time when the 200th visitor clicked on the site.

(A) between 2 and 3 P.M.
(B) between 3 and 4 P.M.
(C) between 4 and 5 P.M.
(D) after 5 P.M.

B20. An observer recorded the velocity of an object in motion along the x-axis for 10 seconds. Based on the table below, use a trapezoidal approximation, with 5 subintervals indicated in the table, to estimate how far from its starting point the object came to rest at the end of this time.

(A) 0 units
(B) 1 unit
(C) 3 units
(D) 4 units

CHALLENGE B21. An 18-wheeler is traveling at a speed given by  mph at time t hours. The fuel economy for the diesel fuel in the truck is given by f(v) = 4 + 0.01v miles per gallon. The amount, in gallons, of diesel fuel used during the first 2 hours is approximately

(A) 20
(A) 21.5
(A) 23.1
(A) 24