Show:
Basic Derivatives
Product and Quotient Rule
Chain Rule
Optimization
Related Rates
100
What is the derivative of y = 3x2 ?
y' = 6x
100
Find the derivative of y = 3x2 sin x
3x2 cos x + 6x sin x
100
Find y' if y = (3x - 9)4
12(3x - 9)3
100
Build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will maximize the total area of the pen ?
x=50 ft. and y=125 ft.
100
A screen saver displays the outline of a 3 cm by 2 cm rectangle and then expands the rectangle in such a way that the 2 cm side is exanpanding at the rate of 4 cm/sec and the proportions of the rectangle never change. How fast is the area of the rectangle increasing when its dimensions are 12 cm by 8 cm?
96 cm2/sec.
200
Find the derivative of f(x) = 5
0
200
f(x) = (3x - 2x2)(5 + 4x)
-24x2 + 4x + 15
200
Find the derivative of f(x) = sin 4x
4 cos 4x
200
An open rectangular box with square base is to be made from 48 ft.2 of material. What dimensions will result in a box with the largest possible volume ?
x=4 ft. and y=2 ft.
200
A ladder 10 feet long is resting against a wall. If the bottom of the ladder is sliding away from the wall at a rate of 1 foot per second, how fast is the top of the ladder moving down when the bottom of the ladder is 8 feet from the wall?
What is 4/3 feet per second.
300
Find the derivative of f(x) = 6x3 + x - 2
18x2 + 1
300
Find the derivative of y = (5x - 2)/(x2 + 1)
(-5x2 + 4x + 5)/(x2 + 1)2
300
Find y' if y = (9x2 + 4)1/3
(6x)/(9x2 + 4)2/3
300
A container in the shape of a right circular cylinder with no top has surface area 3 ft.2 What height h and base radius r will maximize the volume of the cylinder ?
r=1 ft. and h=1 ft.
300
The radius of a sphere is increasing at a rate of 2 meters per second. At what rate is the volume increasing when the radius is equal to 4 meters?
128Π cubic meters per second.
400
Find the derivative of f(x) = 3sinx - 2cosx
3cosx + 2sinx
400
Find the derivative of f(x) = (2x + 5)/(x)1/2
(2x - 5)/(2x3/2)
400
f(x) = sin3 4x Find f'(x)
12sin2 4x cos 4x
400
A cylindrical can is to hold 20 m.3 The material for the top and bottom costs $10/m.2 and material for the side costs $8/m.2 Find the radius r and height h of the most economical can.
r=2 m. and h=5 m.
400
David and Angela start at the same point. At time t = 0 , Angela starts running 30ft/sec north, while David starts running 40ft/sec east. At what rate is the distance between them increasing when they are 100 feet apart?
50 feet per second
500
The derivative of 13x4 + 3x2 - 6x +3
What is 52x3 + 6x - 6
500
The derivative of 3xsin3x
What is 3sin3x +9xcos3x
500
The derivative of sin(3x)^2
What is 2sin3x*3cos3x
500
Car B is 30 miles directly east of Car A and begins moving west at 90 mph. At the same moment car A begins moving north at 60 mph. What will be the minimum distance between the cars and at what time t does the minimum distance occur ?
t=13.8 min
500
A baseball diamond is 90 feet square, and the pitcher's mound is at the center of the square. If a pitcher throws a baseball at 100 miles per hour, how fast is the distance between the ball and first base changing as the ball crosses home plate?
71 miles per hour