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Calculus and Analysis

Extreme Values, Differentials and Maximizing Areas

1) Find the absolute extreme values of the function f(x,y) = x^2 + xy - x - 2y + 4 on the region D enclosed by y= -x, x=3, y=0 2) Given a circle of radius R. Of all the rectangulars inscribed in the circle, find the rectangular with the largest area. 3) a) Find the differential df of f(x,y)= x(e^y) b) use the differenti

Differential equations to describe infection rates

Use models to describe the population dynamics of disease agents. Total population is a Constant (T). A small group of infected individuals are introduced into a large population. Describe spread of infection within population as a function of time. This disease which, after recovery, confers immunity. The population can be

Taylor Polynomial

Find the Taylor polynomial of degree 4 at c=1 for the equation f(x)=ln x and determine the accuracy of this polynomial at x=2.

Angle between a cube's diagonal

Find the angle between a cube's diagonal and one of its sides. (use the vector calculus to get your answer) give detailed response. explain each step.

Cylindrical and spherical coordinates.

Write the equations i) x^2 - y^2 - 2z^2 = 4 and ii) z = x^2 - y^2 in a) cylindrical coordinates b) spherical coordinates give detailed explanation for each step of the solutions.

Moivre-Laplace Formula

Moivre-Laplace formula exp(ix) = cos(x) + i sin(x), where i = (-1)^(1/2) , and which is widely used in different items of mathematics is usually deduced from the Maclaurin expansions of the functions involved. But the theory of Taylor (Maclaurin) expansions is a part of more general theory developed in the course of the fun

Evaluation of a Function

A certain rational function f(x) contains quadratic functions in both its numerator and denominator. Aside from that, we also know the folliwing things about f: f has a vertical asymptote at x=5 f has a single x-intercept of x=2 f is removably discontinous at x=1, lim as (x)approaches 1 of f(x)= -1/9 evaluate lim of f(

Differentiation: Word problem - rate of change

A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30cm wide at the bottom, 80 cm wide at the top, and has a height of 50 cm. If the trough is being filled with water at a rate of 0.2m3/min, how fast is the water level rising when the water is 30cm deep?

Rate of Change Word Problem

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1m higher than the bow of the boat. If the rope is pulled in at a rate of 1m/s, how fast is the boat approaching the dock when it is 8m from the dock?

Volume of water poured in to bowl with iron ball.

A bowl is shaped like a hemisphere with radius R centimeters. An iron ball with radius R/2 centimeters is placed in the bowl and water is poured in to a depth of 2R/3 centimeters. How much water was poured in?

Working with growth and decay rates and decay rate expressions.

A crude-oil refinery has an underground storage tank which has a fixed volume of 'V' liters. Due to pollutants, it gets contaminated with 'P(t)' kilograms of chemical waste at time 't' which is evenly distributed throughout the tank. Oil containing a variety of pollutants with concentration of 'k' kilograms per liter enters

Find the maximum area of a window.

A special window has the shape of a rectangle surrmounted by an equilateral triangle. If the perimeter is 16 feet, what dimensions will admit the most light? (hint: Area of equilateral triangle = the square root of 3/4 times x squared.)

Newton's Law of Cooling relating to differential equations.

At 10:00 AM, an object is removed from a furnace and placed in an environment with a constant temperature of 68 degrees. Its core temperature is 1600 degrees. At 11:00 AM, its core temperature is 1090 degrees. Find its core temperature at 5:00 PM on the same day.

Sample Question: Word problems

1. The manager of a large apartment complex knows from experience that 80 units will be occupied if the rent is 320 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 8 dollar increase in rent. Similarly, one additional unit will be occupied for each 8 dollar decreas

Calculus

1. A weather balloon is rising vertically at a constant rate of 4 ft/s directly above a straight and level road. When the balloon is 75 ft above the road, a car moving at 55 ft/s passes directly under the balloon. Based on this information find: a. the rate the distance between the balloon and the car is changing 3 sec after t

A minimization fencing problem.

A rectangular field is going to be enclosed and divided into two separate rectangular areas. (Areas do not have to be equal). Find the minimum fencing that is required if the total area of the field is 1200m2.

Rate of production of eggs

Eggs are produced at a rate of R(t)eggs per hour,where t=0 represents 12:00 midnight and R(t)(in thousands of eggs) is :- R(t)= -10cospi/12t+10 a)how many eggs are produced in one day. b)When are the eggs produced at the fastest rate c)A machine can produce eggs at a constant rate. At the end of 1 week the same

Tan line and velocity problems

The parabola y = (x^2) + 3 has two tangents which pass through the point (0, -2). One is tangent to the to the parabola at (A, A^2 + 3) and the other at (-A, A^2 + 3). Find (the positive number) ? If a ball is thrown vertically upward from the roof of 64ft foot building with a velocity of 96 ft/sec, its height after t seconds