Extrema's/Max. & Min.
Concavity/Inflection Points
Related Rates
Optimizing
100
y = 2x2 - 8x + 9
Minimum value is 1 at x = 2
100
y = 4x3 + 21x2 + 36x - 20
up (-7/4, ∞) , down (-∞, - 7/4)
100
The position equation for the movement of a particle is given by s=(t2-1)3 where s is in feet and t is in seconds. Find the acceleration of this particle at 2 seconds.
342 ft/s2
100
Find two positive numbers whose sum is 300 and whose product is a maximum.
x=150
y=150
200
y = (x2 - 1)1/2
Minimum value is 0 at x = -1, 1
200
y = 2x1/5 + 3
up (-∞, 00) , down (0,∞)
200
When a circular plate of metal is heated in an oven, its radius increases at the rate of .02cm/sec. At what rate is the plate's area increasing when the radius is 30cm.
32.77cm2/sec
200
Suppose C(x)= x^3-20x^2+20,000x is the cost of manufacturing x items. Find the production level that will minimize the average cost.
10 items
300
y = x/(x2+1)
Maximum value is 1/2 at x=1
Minimum value is -1/2 at x=-1
300
y = xex
(-2, -2/e2)
300
A plane is 750 meters in the air flying parallel to the ground at a speed of 100 m/s and is initially 2.5 kilometers away from a radar station. At what rate is the distance between the plane and the radar station changing (a) initially and (b) 30 seconds after it passes over the radar station?
97.0143
300
A farmer wishes to enclose a rectangular region adjacent to a river. No fencing is required along the river. The area of the region is to be 10,000 sq. meters. Find the dimensions and amount of fencing required for the field that requires the LEAST amount of fencing. (Write your functions and domain in terms of x).
x = 70.7m
y = 141.4m
Amount of fencing = 282.8m
400
Identify the critical point and determine the local extreme value:
y = x2/3(x+2)
crit. point: x=-4/5,0
Extremum: Max. 1.034, Min. 0
400
y = x1/3(x-4)
(0,0) , (-2, 6(21/3))
400
A tank of water in the shape of a cone is being filled with water at a rate of 12 m3/sec. The base radius of the tank is 26 meters and the height of the tank is 8 meters. At what rate is the depth of the water in the tank changing when the radius of the top of the water is 10 meters? Note the image below is not completely to scale….
3/25π
400
An open top rectangular box is constructed from a 10x16 inch piece of cardboard by cutting squares from each corner. Find the dimensions that will give maximum volume.
x = 2
500
Identify the critical point and determine the local extreme values:
y = x(4-x2)1/2
crit point: x = -2, -(21/2), (21/2), 2
Extremum: min. -2, max. 2
500
y = (x3 - 2x2 + x - 1)/(x - 2)
(1,1)
500
Water drains from a conical tank at the rate of 5 cubic feet per minute. How fast is the water level dropping when h=6 ft? The radius of the tank is 4ft and the height is 10ft.
.276ft/min
500
The profit for a certain company is given by P=230+20s-1/2s2 where s is the amount (in hundreds of dollars) spent on advertising. What amount of advertising gives the maximum profit?
2,000