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Calculus and Analysis

Differentiation: s = ut + 1/2at^2

This equation represents displacement of a body(s) against time (t) where (u) is the initial velocity and (a) is the acceleration. Differentiate to derive the equation for instantaneous velocity, which would be represented by the gradient of a graph. s = ut + 1/2at^2

Minimization, maximization (Calculus)

An ant is walking around the outside of the cube in "straight" paths (where we define a straight path in this case as one formed by the edges of a cross section created by a plane slicing through the cube). For example, to get from point Q to point R in the picture above on the right, the ant walks along the red path. There are

Comparing Graphs : First and Second Derivatives

X -1.5 -1.0 -0.5 0 0.5 1.0 1.5 f(x) -1 -4 -6 -7 -6 1.0 -7 f'(x) -7 -5 -3 0 3 5 7 Let f be a function that is differentiable for all real numbers. The table above gives the values of f and its derivative f' for selected points x in the closed interv

Proportionality and Rate of Change Word Problem

A container has the shape of an open right circular cone. The height of the container is 10cm and the diameter of the opening is 10cm. Water in the container is evaporating so that its depth h is changing at the constant rate of -3/10 cm/hr. Show that the rate of change of the volume of water in the container due to evaporati

Word Problem : Intermediate Value Theorem

At 8am on Saturday, a man begins running up the side of a mountain to his weekend campsite. On Sunday at 8am, he runs back down the mountain. It takes him 20 minutes to run up, but only 10 minutes to run down. At some point on his way down, he realizes that he passed the same exact place at exactly the same time on Saturday.

Limitation

Let X(n)=Sum{1/(n+i), i=1->n}, find the limit of X(n) as n tends to infinity

Work Done to Pump Water out of a Tank

A tank is in the shape of an inverted cone (pointy at the top) 6 feet high and 8 feet across at the base. The tank is filled to a depth of 3 feet. How much work is done in emptying the tank through a hole at the top? (Weight density of water is 62.4 lb/ft^3). *I found the distance to be 6-y and used the disk method to solve

Lower Estimate, Upper Estimate?

Please see the attached file for the fully formatted problems. 14) A power plant generates electricity by burning oil. Pollutants produced as a result of the burning process are removed by scrubbers in the smokestacks, Over time, the scrubbers become less efficient and eventually they must be replaced when the amount of pol

Solving Equations : Newton's Method

Please see the attached file for the fully formatted problem. The equation x2 - 3x + 1 = 0 has a solution for x>= 0. Give the third approximationby using Newton's method. Your first approximation is to be 1.

Surface of Revolution

Y=2*sqrt[x] y=0, x=3 We are using the following formula 2pi intergal of r(x) times the sqrt of 1 + (f'(x))^2.

Differential equations functions

Define: f(x) = (x^2)sin(1/x)+x if x doesn't equal 0 f(x) = 0 if x=0 Prove that the function f:R-> R is differentiable and that f'(0)=1. Also prove that there is no neighbourhood I of 0 such that the function f:I->R is increasing.

Level curves of a function

Describe the level curves of the function. Sketch the level curves for the given values of c. f(x,y) = x^2 + 2y^2, c = 0,1,2,3,4

8 calculus problems

Please answer eight (8) calculus problems. Please show as much works as possible for every problem. The problems are posted in the following website: http://www.netprofitspro.com/math.html

Real Analysis: Differential Equations (Leibnitz Formula)

Let I be an open interval and n be a natural number. Suppose that both f:I->R and g:I->R have n derivatives. Prove that fg:I->R has n derivatives, and we have the following formula called Leibnitz's formula: (fg)^n(x) = the sum as k=0,1,2,...n of(n choose k)f^k(x)g^(n-k)(x) for all x in I. Write the formula out explicitly

Fundamental differential equation analysis

If I be an open interval containing the point x. (x0) and suppose that the function f:I->R has two derivatives. Prove that lim as h->0 (f(x.+h) - 2f(x.) + f(x.-h))/ h^2 = f"(x.)

Limits of Functions

For which real values alpha does lim {x -> 0+} x^alpha sin(1/x) exist? It is easy to show using the epsilon - delta definition below that this limit exists for all real alpha >= 1. In fact the limit is zero in this case. The case alpha equals zero is also quite simple and the limit does not exist. Consider the two sequence

Projectile motion

With this problem im seeking a detailed explanation. I already have some of the answers however i dont see how they were obtained. Can you please solve this and show me how each answer was obtained. My numbers are way off from what they should be. Thanks The quarterback of a football team releases a pass at a heigh

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

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

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.

Interval of Convergence

Find the open interval of convergence and test the endpoints for absolute and conditional convergence (x+3)^n / n!

Convergence test

Sigma, infinite above and n=1 below, times (1+1/n)^n test for convergence or divergence, absolute or conditional