Applications: rate of change monotonicity maxima and minima - Question Bank

1. Find the rate of change of the diagonal of a square when the side length is 10 cm and the side is increasing at 2 cm/s.
A) sqrt(2) cm/s
B) 2*sqrt(2) cm/s
C) 10*sqrt(2) cm/s
D) 20*sqrt(2) cm/s
2. A cylindrical can is to have a volume of 1000 cm^3. Find the dimensions (radius and height) that minimize the surface area.
A) r = 10/cbrt(pi), h = 10*cbrt(pi)
B) r = 5*cbrt(2/pi), h = 5*cbrt(2/pi)
C) r = 10/sqrt(pi), h = 10*sqrt(pi)
D) r = 5, h = 40/pi
3. Find the interval where the function f(x) = x^2 * ln(x) is concave down.
A) (0, 1/sqrt(e))
B) (0, sqrt(e))
C) (1/sqrt(e), inf)
D) (0, 1)
4. A particle's position is given by s(t) = t^3 - 9t^2 + 15t. Find the time intervals when the particle is moving to the left.
A) (1, 5)
B) (-inf, 1) U (5, inf)
C) (1, inf)
D) (-inf, 1)
5. Find the local minimum value of f(x) = x^4 - 4x.
A) -3
B) 0
C) 3
D) 4
6. The area of a circle is increasing at a rate of 10 cm^2/min. Find the rate at which the radius is increasing when the area is 25*pi cm^2.
A) 1/pi cm/min
B) 2/pi cm/min
C) 1/(2*pi) cm/min
D) 10/pi cm/min
7. Find the absolute maximum value of f(x) = x^3 - 3x^2 + 5 on the interval [0, 3].
A) 5
B) 0
C) 2
D) -1
8. The rate of change of population is proportional to the current population P. If dP/dt = kP, find the population after time t, given P(0) = P0.
A) P(t) = P0 * e^(kt)
B) P(t) = P0 + kt
C) P(t) = P0 * (1 + kt)
D) P(t) = P0 / (1 + kt)
9. A rectangular field is to be enclosed by a fence. If 1000 m of fencing is available, find the dimensions of the field that will maximize the area.
A) 250m x 250m
B) 200m x 300m
C) 100m x 400m
D) 500m x 500m
10. Find the point of inflection for f(x) = x^4 - 6x^2.
A) (sqrt(3), -9) and (-sqrt(3), -9)
B) (0, 0)
C) (sqrt(3), 0) and (-sqrt(3), 0)
D) (3, -27) and (-3, -27)
11. Find the interval where the function f(x) = x^3 - 6x^2 + 5 is decreasing.
A) (0, 4)
B) (-inf, 0) U (4, inf)
C) (0, 2)
D) (2, 4)
12. The radius of a sphere is increasing at a rate of 0.1 cm/s. How fast is the volume changing when the radius is 5 cm?
A) 10*pi cm^3/s
B) 2.5*pi cm^3/s
C) 5*pi cm^3/s
D) 20*pi cm^3/s
13. Find the local maximum value of f(x) = x*e^(-x^2).
A) 1/sqrt(2e)
B) 0
C) -1/sqrt(2e)
D) e
14. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled at a rate of 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock?
A) 1/8 m/s
B) 1/10 m/s
C) 8/10 m/s
D) 10/8 m/s
15. Find the interval where the function f(x) = x/(x^2 + 1) is increasing.
A) (-1, 1)
B) (-inf, -1) U (1, inf)
C) (-inf, 0)
D) (0, inf)
16. The length of a rectangle is decreasing at a rate of 2 cm/s and the width is increasing at a rate of 3 cm/s. Find the rate of change of the area when the length is 10 cm and the width is 5 cm.
A) 5 cm^2/s
B) -5 cm^2/s
C) 10 cm^2/s
D) -10 cm^2/s
17. Find the absolute minimum value of f(x) = x^3 - 3x + 1 on the interval [-2, 3].
A) -1
B) 1
C) -2
D) 19
18. A balloon is being inflated at a rate of 100 cm^3/sec. How fast is the radius of the balloon increasing when the radius is 10 cm?
A) 1/(10*pi) cm/sec
B) 1/(2*pi) cm/sec
C) 1/(20*pi) cm/sec
D) 1/(4*pi) cm/sec
19. Find the local extrema of f(x) = x^2 * e^x.
A) x = 0 (min), x = -2 (max)
B) x = 0 (max), x = -2 (min)
C) x = 0 (min), x = 2 (max)
D) x = 0 (max), x = 2 (min)
20. The height of a triangle is increasing at a rate of 2 cm/min, while the base is decreasing at a rate of 3 cm/min. Find the rate of change of the area when the height is 10 cm and the base is 5 cm.
A) -5 cm^2/min
B) 5 cm^2/min
C) -10 cm^2/min
D) 10 cm^2/min
21. Find the interval of concavity for the function f(x) = sin(x) in the interval [0, 2*pi].
A) (0, pi)
B) (pi, 2*pi)
C) (pi/2, 3*pi/2)
D) (0, 2*pi)
22. Two cars start moving from the same point. One travels north at 30 km/h and the other travels east at 40 km/h. At what rate is the distance between them increasing after 2 hours?
A) 50 km/h
B) 60 km/h
C) 70 km/h
D) 80 km/h
23. Find the value of x where f(x) = x^3 - 3x^2 + 4 has a local minimum.
A) x = 0
B) x = 1
C) x = 2
D) x = 3
24. If the perimeter of a square is increasing at a rate of 4 cm/s, at what rate is the area increasing when the side length is 10 cm?
A) 40 cm^2/s
B) 80 cm^2/s
C) 20 cm^2/s
D) 10 cm^2/s
25. Find the absolute maximum value of f(x) = 3x^4 - 4x^3 on the interval [-1, 2].
A) 16
B) 0
C) 7
D) -7/3
26. The tip of the minute hand of a clock is 10 cm from the center. How fast is the tip moving when the time is 3:00 PM?
A) 0 cm/min
B) 5*pi/3 cm/min
C) 10*pi/3 cm/min
D) 20*pi/3 cm/min
27. Find the interval where the function f(x) = x^5 - 5x^4 is strictly decreasing.
A) (0, 4)
B) (-inf, 0) U (4, inf)
C) (4, inf)
D) (-inf, 0)
28. A car is traveling at 30 m/s when the brakes are applied. The distance it travels before stopping is given by s(t) = 30t - 5t^2. Find the time it takes to stop.
A) 2 seconds
B) 3 seconds
C) 5 seconds
D) 6 seconds
29. Find the point of local minimum for the function f(x) = x^3 - 12x + 1.
A) (2, -15)
B) (-2, 17)
C) (2, 17)
D) (-2, -15)
30. The volume of a cylinder is increasing at a rate of 10 cm^3/sec. If the radius is fixed at 5 cm, how fast is the height changing?
A) 10/(25*pi) cm/sec
B) 10/(5*pi) cm/sec
C) 10/(50*pi) cm/sec
D) 10/(pi) cm/sec
31. Find the local maximum value of f(x) = -x^3 + 3x^2 - 5.
A) -5
B) -7
C) -3
D) -2
32. A particle moves along the curve y = x^3. If the x-coordinate is increasing at a rate of 2 units/sec, at what rate is the y-coordinate changing when x = 3?
A) 6 units/sec
B) 18 units/sec
C) 54 units/sec
D) 27 units/sec
33. Find the interval where the function f(x) = x^4 - 2x^2 is concave up.
A) (-1, 1)
B) (-inf, -1) U (1, inf)
C) (-inf, 0)
D) (0, inf)
34. The area of a rectangle is increasing at a rate of 5 cm^2/s. If the length of the rectangle is kept constant at 10 cm, at what rate is the width changing?
A) 0.5 cm/s
B) 0.25 cm/s
C) 1 cm/s
D) 0.1 cm/s
35. Find the absolute minimum value of f(x) = x^4 - 2x^2 + 3 on the interval [-2, 2].
A) 3
B) 1
C) 2
D) 0
36. If the distance traveled by an object is given by s(t) = t^3 - 6t^2 + 5t, find the time when the velocity is zero.
A) t=1, t=5
B) t=2, t=3
C) t=0, t=6
D) t=1, t=6
37. Find the point of inflection for the function f(x) = x^3 - 6x^2 + 12x - 5.
A) (2, 3)
B) (1, 2)
C) (2, 5)
D) (3, 4)
38. Water is leaking out of a conical tank at a rate of 3 m^3/min. The tank has a height of 10 m and a radius of 5 m. How fast is the water level falling when the water is 8 m deep?
A) 15/(16*pi) m/min
B) 3/(4*pi) m/min
C) 5/(8*pi) m/min
D) 3/(16*pi) m/min
39. Find the local minimum value of the function f(x) = x^3 - 6x^2 + 5.
A) 5
B) -27
C) -11
D) 0
40. The radius of a circle is increasing at a rate of 2 cm/s. What is the rate of change of its area when the radius is 5 cm?
A) 10*pi cm^2/s
B) 20*pi cm^2/s
C) 5*pi cm^2/s
D) 25*pi cm^2/s
41. Find the interval of concavity for the function f(x) = x^4 - 4x^3.
A) (-inf, 0)
B) (0, inf)
C) (-inf, 0) U (0, inf)
D) (0, 2)
42. A ladder 10 m long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 0.1 m/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 m from the wall?
A) 0.075 m/s
B) 0.15 m/s
C) 0.05 m/s
D) 0.1 m/s
43. Find the value of x for which the function f(x) = 2x^3 - 9x^2 + 12x + 1 has a local maximum.
A) x = 1
B) x = 2
C) x = 3
D) x = 0
44. The volume of a cube is increasing at a rate of 10 cm^3/min. How fast is the surface area of the cube increasing when the edge length is 5 cm?
A) 4 cm^2/min
B) 8 cm^2/min
C) 10 cm^2/min
D) 2 cm^2/min
45. Find the absolute maximum value of f(x) = x^3 on the interval [-1, 2].
A) -1
B) 8
C) 1
D) 0
46. Determine the interval where the function f(x) = sin(x) is decreasing in the interval [0, 2*pi].
A) (0, pi)
B) (pi, 2*pi)
C) (pi/2, 3*pi/2)
D) (0, pi/2)
47. If the side length of a square is increasing at a rate of 0.5 cm/s, what is the rate of change of its area when the side length is 4 cm?
A) 2 cm^2/s
B) 4 cm^2/s
C) 8 cm^2/s
D) 1 cm^2/s
48. Find the local maximum value of the function f(x) = x^3 - 3x + 1.
A) 1
B) 3
C) -1
D) 0
49. The rate of change of the volume of a sphere with respect to its radius r is given by:
A) 4*pi*r
B) 4*pi*r^2
C) 4/3*pi*r^3
D) 8*pi*r
50. Find the point on the curve y = x^2 that is closest to the point (2, 0).
A) (1, 1)
B) (2, 4)
C) (0, 0)
D) (1, -1)
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