Calculus Answers

y=(u-2)³ and u=1/2x=1

use Lagrange’s Multiplier method to find the maximum and minimum of f(x,y)= y^2 - x^2 subjected to the constraint of 1/4 x^2 + y^2 = 1

A market research company has derived the profit function for a Chackzy Inc as TP = -Q 2 + 60Q + 100

However, Mr John, the chairman of Chackzy Inc. needs more detailed information. You are assigned by the company to answer his questions which are as follows:

(a) At what level of output will my company maximize the profit?

(b) What will my company’s maximum profit be?

(c) What quantities will give my firm a breakeven state?

(d) Sketch the ‘TP’ graph, with clear indication of the maximum point as well as the breakeven points. Label the graph accordingly

A tank having a capacity of 1000 liters, initially contains 400 liters of sugar water having a concen-

tration of 0.2 Kg of sugar for each liter of water. At time zero, sugar water with a concentration of

50 gm of sugar per liter begins pumped into the tank at a rate of 2 liter per minute. Simultaneously,

a drain is opened at the bottom of the tank so that the volume of the sugar-water solution in the

tank reduces 1 liter per minute. Determine the following:

Clairaut's theorem holds that Uxy=Uyx,show that the following equations obey clairaut's theorem.

1.u=in(x+2y).

2.u=e^xy siny.

Laplace's equation holds that Uxx+Uyy=0,verify that the second derivative of the following equations are laphace's equation.

1.u=in√x²+y².

2.u=x²-y².

Determine the first order partial derivative of the following functions:

1.z=in(x+t²).

2.f(x,y,z)=xy²e^-xz.

Given the total cost, TC = 4 +Q, and the demand function is P = 10 – 2Q:

(a) Obtain the total revenue function.

(b) Find the profit function in terms of Q.

(c) Find the quantity when the profit is maximized. What is the maximum profit?

1. Determine the equation of the family of the curves whose slope is

3x – 5.

Also find the equation of the member whose curve passes through point

(1, 1).

2. A body is thrown vertically upward from the ground with an initial velocity

of 64 ft/sec. Find the maximum height attained by the body. Regardless of

frictional force, determine the total time the body returns to the ground.

(note: a = 32ft/sec2)

The half-life of a radioactive substance is defined to be the amount of time it takes for the substance to decay 50% of its amount. If substance X has a half-life of 3,600 years, what part of substance X will remain after 4,500 years?