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Calculus and Analysis

Inequalities

Prove that if x>0, then 1+x/2-x^2/8<(1+x)^(1/2)<1+x/2

Analysis/total differentials

Evaluate f(1,2) and f(1.05, 2.1) for the function Æ'(x,y)=x/y. a) calculate &#916;z b) use the total differential dz to approximate &#916;z

Chain rule

You are given the function w=yz/x, where x=&#952;^2, y=r+&#952; and z=r-&#952;. Find &#8706;w/&#8706;&#952;. a) using the appropriate chain rule b) converting w to a function of r,&#952; before differentiating. Which of the above is quicker?

Gradients

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

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

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.

Word problem derivatives

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

Calc. III vectors

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

Find all differentiable functions.

Find all differentiable functions f : R ---> R such that (f composed f) = f. R = All Real Numbers (f composed f) = the function f composed with itself

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.

Orthogonality

Find a unit vector that is orthogonal to both i + j and i + k give detailed explanation for each step

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.