Practice Exercises

In Questions A1–A10, a(t) denotes the acceleration function, v(t) the velocity function, and s(t) the position or height function at time t. (The acceleration due to gravity is −32 ft/sec².)

A1. If a(t) = 4t − 1 and v(1) = 3, then v(t) equals

(A) 2t² − t
(B) 2t² − t + 1
(C) 2t² − t + 2
(D) 2t² + 1

A2. If a(t) = 20t³ − 6t, s(−1) = 2, and s(1) = 4, then v(t) equals

(A) 5t⁴ − 3t² + 1
(B) 5t⁴ − 3t² + 3
(C) t⁵ − t³ + t + 3
(D) t⁵ − t³ + 1

A3. If a(t) = 20t³ − 6t, s(−1) = 2, and s(1) = 4, then s(0) equals

(A) 1
(B) 2
(C) 3
(D) 4

A4. A stone is thrown straight up from the top of a building with initial velocity 40 ft/sec and hits the ground 4 seconds later. The height of the building, in feet, is

(A) 88
(B) 96
(C) 112
(D) 128

A5. A stone is thrown straight up from the top of a building with initial velocity 40 ft/sec and hits the ground 4 seconds later. The maximum height is reached by the stone after

(A) 4/5 second
(B) 4 seconds
(C) 5/4 seconds
(D) 2 seconds

A6. If a car accelerates from 0 to 60 mph in 10 seconds, what distance does it travel in those 10 seconds? (Assume the acceleration is constant and note that 60 mph = 88 ft/sec.)

(A) 44 feet
(B) 88 feet
(C) 400 feet
(D) 440 feet

A7. A stone is thrown at a target so that its velocity is v(t) =100 − 20t ft/sec, where t is measured in seconds. If the stone hits the target in 1 second, then the distance from the sling to the target is

(A) 80 feet
(B) 90 feet
(C) 100 feet
(D) 110 feet

A8. A stone is thrown straight up from the ground. What should the initial velocity be if you want the stone to reach a height of 100 feet?

(A) 50 ft/sec
(A) 80 ft/sec
(A) 92 ft/sec
(C) 96 ft/sec

A9. If the velocity of a car traveling in a straight line at time t is v(t), then the difference in its odometer readings between times t = a and t = b is
(A) 
(B) 
(C) the net displacement of the car’s position from t = a to t = b
(D) the change in the car’s position from t = a to t = b

A10. If an object is moving up and down along the y-axis with velocity v(t) and s′(t) = v(t), then it is false that  gives

(A) s(b) − s(a)
(B) the total change in s(t) between t = a and t = b
(C) the shift in the object’s position from t = a to t = b
(D) the total distance covered by the object from t = a to t = b

A11. Solutions of the differential equation  are of the form

(A) x² − y² = C
(B) x² + y² = C
(C) x² − Cy² = 0
(D) x² = C − y²

A12. Find the domain of the particular solution to the differential equation  that passes through point (−2,1).

(A) x < 0
(B) −2 ≤ x < 0
(C) 
(D) 

A13. If  and y = 1 when x = 4, then

A14. If  and y = 0 when x = 1, then

(A) y = ln (x)
(B) y = ln (2 − x)
(C) y = −ln (2 − x)
(D) y = −ln (x)

A15. If  and y = 5 when x = 4, then y equals

A16. The general solution of the differential equation x dy = y dx is a family of

(A) circles
(B) hyperbolas
(C) parabolas
(D) none of these

A17. The general solution of the differential equation  is a family of

(A) parabolas
(B) lines
(C) ellipses
(D) exponential curves

A18. A function f(x) that satisfies the equations f(x)f′(x) = x and f(0) = 1 is

(A) 
(B) 
(C) f(x) = x
(D) f(x) = ex

A19. The curve that passes through the point (1,1) and whose slope at any point (x,y) is given by  has the equation

(A) 3x − 2 = y
(B) y³ = x
(C) y = x³
(D) 3y² = x² + 2

A20. Find the domain of the particular solution to the differential equation  that passes through the point (1,1).

(A) all real numbers
(B) |x| ≤ 1
(C) x ≠ 0
(D) x > 0

A21. If , where k is a constant, and if y = 2 when x = 1 and y = 4 when x = e, then, when x = 2, y equals

(A) 4
(B) ln 8
(C) ln 2 + 2
(D) ln 4 + 2

A22. 

The slope field shown above is for the differential equation

A23.

The slope field shown above is for the differential equation

 

A24. 

A solution curve has been superimposed on the slope field shown above. The solution is for the differential equation and initial condition

The slope fields below are for Questions A25–A29.

A25. Which slope field is for the differential equation ?

(A) I
(B) II
(C) III
(D) IV

A26. Which slope field is for the differential equation ?

(A) I
(B) II
(C) III
(D) IV

A27. Which slope field is for the differential equation ?

(A) I
(B) II
(C) III
(D) IV

A28. Which slope field is for the differential equation ?

(A) I
(B) II
(C) III
(D) IV

A29. A particular solution curve of a differential equation whose slope field is shown in II passes through the point (0,−1). The equation is

(A) y = −e^x
(B) y = −e^−x
(C) y = x² − 1
(D) y = −cos x

*A30. If you use Euler’s method with Δx = 0.1 for the d.e.  with initial value y(1) = 5, then, when x = 1.2, y is approximately

(A) 5.10
(B) 5.20
(C) 5.21
(D) 6.05

*A31. The error in using Euler’s method in Question A30 is

(A) 0.005
(B) 0.010
(C) 0.050
(D) 0.500

A32. 

Which differential equation has the slope field shown?

A33. 

Which function is a possible solution of the slope field shown?

(A) y = 1 − ln x
(B) y = 1 + ln x
(C) y = 1 + e^x
(D) y = 1 + tan x

A34.

Which differential equation has the slope field shown?

A35. 

Which differential equation has the slope field shown?

A36. 

Which differential equation has the slope field shown?

B1. If  and if s = 1 when t = 0, then, when , t is equal to

B2. If radium decomposes at a rate proportional to the amount present, then the amount R left after t years, if R₀ is present initially and k is the negative constant of proportionality, is given by

(A) R = R₀kt
(B) R = R₀e^kt
(C) R = e^R₀kt
(D) R = e^{R₀+ kt}

B3. The population of a city increases continuously at a rate proportional, at any time, to the population at that time. The population doubles in 50 years. After 75 years the ratio of the population P to the initial population P₀ is

B4. If a substance decomposes at a rate proportional to the amount of the substance present, and if the amount decreases from 40 g to 10 g in 2 hours, then the constant of proportionality is

B5. If (g′(x))² = g(x) for all real x and g(0) = 0, g(4) = 4, then g(1) equals

(A) 
(B) 
(C) 2
(D) 4

B6. The solution curve of  that passes through point (2,3) is

(A) y = e^x + 3
(B) y = 0.406e^x
(C) y = e^x − (e² + 3)
(D) y = e^x/(0.406)

B7. At any point of intersection of a solution curve of the d.e. and the line x + y = 0, the function y at that point

(A) is equal to 0
(B) is a local maximum
(C) is a local minimum
(D) has a point of inflection

B8. 

The slope field for  is shown above with the particular solution F(0) = 0 superimposed. With a graphing calculator,  to three decimal places is

(A) 0.886
(B) 0.987
(C) 1.000
(D) ∞

*B9. 

The graph displays logistic growth for a frog population F. Which differential equation could be the appropriate model?

*B10. The table shows selected values of the derivative for a differentiable function f.

Given that f(3) = 100, use Euler’s method with a step size of 2 to estimate f(7).

(A) 102.5
(B) 103
(C) 104
(D) 104.5

B11. A cup of coffee at temperature 180°F is placed on a table in a room at 68°F. The d.e. for its temperature at time t (in minutes) is ; y(0) = 180. After 10 minutes, the temperature (in °F) of the coffee is approximately

(A) 96
(B) 100
(C) 105
(D) 110

B12. A cup of coffee at temperature 180°F is placed on a table in a room at 68°F. The d.e. for its temperature at time t (in minutes) is ; y(0) = 180. Approximately how long does it take the temperature of the coffee to drop to 75°F?

(A) 15 minutes
(B) 18 minutes
(C) 20 minutes
(D) 25 minutes

B13. The concentration of a medication injected into the bloodstream drops at a rate proportional to the existing concentration. If the factor of proportionality is 30% per hour, in approximately how many hours will the concentration be one-tenth of the initial concentration?

*B14. Which of the following statements characterize(s) the logistic growth of a population whose limiting value is L and whose initial value is less than ?

I. The rate of growth increases at first.

II. The growth rate attains a maximum when the population equals .
III. The growth rate approaches 0 as the population approaches L.

(A) I only
(B) I and II only
(C) II and III only
(D) I, II, and III

*B15. Which of the following differential equations is not logistic?

*B16. Suppose P(t) denotes the size of an animal population at time t and its growth is described by the d.e. . If the initial population is 200, then the population is growing fastest

(A) initially
(B) when P = 500
(C) when P = 1000
(D) when 

B17. According to Newton’s law of cooling, the temperature of an object decreases at a rate proportional to the difference between its temperature and that of the surrounding air. Suppose a corpse at a temperature of 32°C arrives at a mortuary where the temperature is kept at 10°C. The differential equation is , where T is the temperature of the corpse (in °C) and t is hours. If the corpse cools to 27°C in 1 hour, then its temperature (in °C) is given by the equation

(A) T = 22e^0.205t
(B) T = 10e^1.163t
(C) T = 10 + 22e^−0.258t
(D) T = 32e^−0.169t