Applications of Differentiation

Find the particle at rest. t^3 - 11t^2 + 24t

(4/3) and 6

Find the critical points of the function F(x)= (x^4) − 6x^2

0, √3, -√3

Find the inflection points of the function F(x)= (x^4) - (4x^3)

0,2

A company has started selling a new type of smartphone at the price of $160 - 0.05x where x is the number of smartphones manufactured per day. The parts for each smartphone cons $60 and the labor and overhead for running the plant costs $7000 per day. How many smartphones should the company manufacture and sell per day to maximize profit?

1000

A supermarket employee wants to construct an open-top box from a 14 by 30 in piece of cardboard. To do this, the employee plans to cut out squares of equal size from the four corners so the four sides can be bent upwards. What size should the squares be in order to create a box with the largest possible volume?

3

Find the relative minima of the function F(x)= (x-2)((x-3)^2)

3

Find the relative maxima of the function F(x)= (x^4) - (x^5)

4/5

A rancher wants to construct two identical rectangular corrals using 300 feet of fencing. The rancher decides to build them adjacent to each other, so they share fencing on one side. What dimensions should the rancher use to construct each corral so that together they will enclose the largest possible area?

50

Find the particle at rest. t^3 - 4t^2 + 18

8/3

A cryptography expert is deciphering a computer code. To do this, the expert needs to minimize the product of a positive rational number and a negative rational number, given that the positive number is exactly 8 greater than the negative number. What final product is the expert looking for?

positive: 4 negative: -4

Find the particle at rest. s(t) = t^3 - 13t^2

t= 26/3

A particle moves along a horizontal line. Its position function is s(t) for t≥0. For each problem, find the velocity function v(t), the acceleration function a(t), the times t when the particle changes directions, the intervals of time when the particle is moving left and moving right, and the times t when the acceleration is 0. s(t) = -t^3 + t^2 = 56t

v(t) = -3t^2 + 2t + 56 a(t) = -6t + 2 t= 1/3 Changes Direction at t= 14/3 Moving right at (0, 14/3) Moving left at (14/3, ∞)

Find the velocity, acceleration, and speed at a given time. s(t) = -t^3 + 15t^2 at t= 5

v(t) = -3t^2 + 30t a(t) = -6t + 30 speed = 25

Find the velocity, acceleration, and speed at a given time. s(t)= -t^3 + 22t -121t t= 2

v(t) = -3t^2 + 44t -121 a(t) = -6t + 44 speed = 45

A particle moves along a horizontal line. Its position function is s(t) for t≥0. For each problem, find the velocity function v(t), the acceleration function a(t), the times t when the particle changes directions, the intervals of time when the particle is moving left and moving right, and the times t when the acceleration is 0, and the intervals of time when the particle is slowing down and speeding up. s(t) = t^3 - 24t^2 + 144t

v(t) = 3t^2 - 48t + 144 a(t) = 6t - 48 When acceleration is at 0, t=8 Changes direction at t= 4,12 Moving right (0,4) U (12, ∞) Moving left (4,12) Speeding up (4,8) U (13, ∞) Slowing down (0,4) U (8,12)

A particle moves along a horizontal line. Its position function is s(t) for t≥0. For each problem, find the velocity function v(t), the acceleration function a(t), the times t when the particle changes directions, the intervals of time when the particle is moving up and moving down, and the times t when the acceleration is 0, and the intervals of time when the particle is slowing down and speeding up. s(t) = t^4 - 10t^3

v(t) = 4t^3 - 30t^2 a(t) = 12t^2 - 60t When acceleration is at 0 t= 5 Changes direction at t= 15/2 Moving up (15/2, ∞) Moving down (0, 15/2) Speeding up (0,5) U (15/2, ∞) Slowing down (5, 15/2)