Calculus Answers

A storage tank is a right circular cylinder 20 ft long and 8 ft in diameter with

its axis horizontal. If the tank is half full of olive oil weighing 57 lb/ft3, find the

work done in emptying it through a pipe that runs from the bottom of the

tank to an outlet that is 6 ft above the top of the tank.

A spring, whose unstretched length is 2.0 m is compressed to a length of 1/2 m

when a force of 10 N is applied. Find the work required to compress the spring

to a length of 1.0 m.

Consider the function g defined by:

g(x,y) = cos (xπ "\u221ay" ) + 1/log₃(x - y)

Determine ∂²g/∂y∂x

Use the Lagrange multipliers to solve the following:

Maximize f(x,y,z) = 4x + 2y + z

subject to x² + y + z² = 1

For what values of 𝑐 does the curve 𝑓(𝑥) = 2𝑥^3 + 𝑐𝑥^2 + 2𝑥 have maximum and minimum points?

Show that the minimum and maximum points of every curve in the family of polynomials

𝑓(𝑥) = 2𝑥^3 + 𝑐𝑥^2 + 2𝑥 lie on the curve 𝑦 = 𝑥 − 𝑥^3.

Show that the minimum and maximum points of every curve in the family of polynomials

𝑓(𝑥) = 2𝑥^3 + 𝑐𝑥^2 + 2𝑥 lie on the curve 𝑦 = 𝑥 − 𝑥^3

For what values of 𝑐 does the curve 𝑓(𝑥) = 2𝑥^3 + 𝑐𝑥^2 + 2𝑥 have maximum and minimum points?

Show that the minimum and maximum points of every curve in the family of polynomials

𝑓(𝑥) = 2𝑥^3 + 𝑐𝑥^2 + 2𝑥 lie on the curve 𝑦 = 𝑥 − 𝑥^3.

Suppose 𝑓 is differentiable on ℝ and has two roots. Show that 𝑓′ has at least one root.

Find the area between f(x)=x²+4x and g(x)=x²-6x+5 over the interval 0 less or equal to x or x is less or equal to 1

Use substitution rule to integrate

1.) Integral x³(x⁴+16)³dx

2.) Integral 3x²(x³+5)⁷dx

3.) Integral x³/√1-x⁴ dx

4.) Integral sin x/(cos x)³ dx

Deadline May 9 2022

Hope i will get full explanation to how you solve this, because i need the understandable, because i am absent in that class, thanks bro.

Activity 2

Directions :Solve the logistics problem. show your solution.

The population of a certain bacteria follows the logistic growth pattern, initially, there are 10 g of bacteria present in the culture. Two hours later, the culture weighs 25 g. The maximum weight of the culture is 100g.

a.Write the corresponding logistic model for the bacterial growth?

b. What is the weight of the culture after 5 hours?

c. When will the culture's weight be 75 g?