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Calculus and Analysis

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.

Taylor Approximation Related Problem

Please see the attached file for the fully formatted problems. Questions pertain to Second order Taylor approximations and integrals for two first order differential equations.

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.

Analysis/total differentials

Evaluate f(1,2) and f(1.05, 2.1) for the function Æ'(x,y)=x/y. a) calculate Δz b) use the total differential dz to approximate Δz

Chain rule evaluation

You are given the function w=yz/x, where x=θ^2, y=r+θ and z=r-θ. Find ∂w/∂θ. a) using the appropriate chain rule b) converting w to a function of r,θ before differentiating. Which of the above is quicker?

Gradient Plane Functions

A metal plate is located in an xy-plane such that the temperature T at (x,y) is inversely proportional to the distance from the origin, and the temperature at point P(3,4) is 100 (i.e. the temperature at any point (x,y) is described by the function T(x,y) = 500/(x^2 + y^2)^1/2 a) in what direction does

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

Set up integral for surface area.

Please see the attached file for the fully formatted problems. Use the formulas to set up an integral for the surface area of the first octant portion of the sphere p=a, do not evaluate. See attachment

Proof: Upper and lower limits

Please see the attached file for the fully formatted problems. Let be a sequence of real numbers. We define and I'm having trouble with the following three proofs: 1) Show that 2) Show that if the limit of only exists when , then . 3) Show that if , then the limit exists, and .

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

Mass and centroid of a Plane Lamina

Please see the attached file for the fully formatted problem. Find the mass and centroid of a plane lamina with the given shape and density delta, the region bounded by y = x2 and x = y2 delta(x,y) = x2 + y2.

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.)

Position and Velocity Functions for the Ball

A ball is dropped from the top of a building which is 1000 feet tall. GIVEN (s(t)=-16t^2+v(initial)t+s(initial)) A. Write the position and velocity functions for the ball. B. Find the instantaneous velocity went t = 2 seconds. C. How long does it take the ball to reach the ground. Please solve using calculus (derivativ

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

First and Second Derivative of a Cross Product

Find the vectors T, N, and B at the given point. r(t)=<e^t , e^t sint , e^t cost> , (1,0,1) I'm having problems with this one because it requires lots of calc. I and II and i just can't remember how to do some of this. I can take the first derivative and the second but doing the cross product is giving me trouble.

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