Applications: rate of change monotonicity maxima and minima - One Line Questions

1. Find the absolute maximum value of f(x) = x^3 on the interval [-1, 2]. — 8
2. Find the absolute minimum value of f(x) = x^3 - 3x + 1 on the interval [-2, 3]. — -2
3. Find the local minimum value of f(x) = x^4 - 4x. — 3
4. Find the local maximum value of f(x) = -x^3 + 3x^2 - 5. — -7
5. 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. — -5 cm^2/min
6. Find the interval where the function f(x) = x^4 - 2x^2 is concave up. — (-inf, -1) U (1, inf)
7. Find the interval where the function f(x) = x/(x^2 + 1) is increasing. — (-1, 1)
8. Find the interval of concavity for the function f(x) = x^4 - 4x^3. — (0, inf)
9. Find the interval where the function f(x) = x^2 * ln(x) is concave down. — (0, 1/sqrt(e))
10. If y = x^3 - 6x^2 + 5, find the interval where the function is strictly increasing. — (-inf, 0) U (4, inf)
11. Find the interval where the function f(x) = x^5 - 5x^4 is strictly decreasing. — (0, 4)
12. Find the interval where the function f(x) = x^3 - 6x^2 + 5 is decreasing. — (0, 4)
13. Determine the interval where the function f(x) = sin(x) is decreasing in the interval [0, 2*pi]. — (pi, 2*pi)
14. Find the interval of concavity for the function f(x) = sin(x) in the interval [0, 2*pi]. — (pi, 2*pi)
15. Find the point on the curve y = x^2 that is closest to the point (2, 0). — (1, 1)
16. 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. — (1, 5)
17. Find the point of local minimum for the function f(x) = x^3 - 12x + 1. — (2, -15)
18. Find the point of inflection for the function f(x) = x^3 - 6x^2 + 12x - 5. — (2, 3)
19. Find the point of inflection for f(x) = x^4 - 6x^2. — (sqrt(3), -9) and (-sqrt(3), -9)
20. 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? — 10*pi/3 cm/min
21. 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? — 0.075 m/s
22. 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? — 0.5 cm/s
23. Find the local maximum value of the function f(x) = x^3 - 3x + 1. — 1
24. 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? — 1/(10*pi) cm/sec
25. 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? — 1/10 m/s
26. 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. — 1/(2*pi) cm/min
27. Find the local maximum value of f(x) = x*e^(-x^2). — 1/sqrt(2e)
28. 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? — 20*pi cm^2/s
29. 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? — 5*pi cm^3/s
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? — 10/(25*pi) cm/sec
31. 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? — 3/(16*pi) m/min
32. Find the absolute maximum value of f(x) = 3x^4 - 4x^3 on the interval [-1, 2]. — 16
33. 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? — 8 cm^2/s
34. 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. — 6 seconds
35. 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. — 250m x 250m
36. Find the absolute minimum value of f(x) = x^4 - 2x^2 + 3 on the interval [-2, 2]. — 1
37. 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? — 8 cm^2/min
38. The rate of change of the volume of a sphere with respect to its radius r is given by: — 4*pi*r^2
39. 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? — 40 cm^2/s
40. Find the local minimum value of the function f(x) = x^3 - 6x^2 + 5. — -11
41. Find the absolute maximum value of f(x) = x^3 - 3x^2 + 5 on the interval [0, 3]. — 5
42. 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. — -5 cm^2/s
43. 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? — 50 km/h
44. 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? — 54 units/sec
45. 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. — P(t) = P0 * e^(kt)
46. A cylindrical can is to have a volume of 1000 cm^3. Find the dimensions (radius and height) that minimize the surface area. — r = 10/cbrt(pi), h = 10*cbrt(pi)
47. 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. — 2*sqrt(2) cm/s
48. 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. — t=1, t=5
49. Find the value of x where f(x) = x^3 - 3x^2 + 4 has a local minimum. — x = 2
50. Find the local extrema of f(x) = x^2 * e^x. — x = 0 (min), x = -2 (max)
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