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    Calculus and Analysis

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    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

    Differential Prime Functions

    Assume that f/g (x) = x^2 + 2x, where f and g are differentiable functions such that f(2) = 2 and f'(2)= 3. Find g'(2).

    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

    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

    Orthogonal Unit Vectors

    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.

    Parallel or Perpendicular Planes

    Question: Find if the planes are parallel, perpendicular or neither. If they are not parallel then find the equation for the line of intersection. z = x = y , 2x - 5y -z = 1 Verify that your answer is indeed a line of intersection.

    Identical Parallel Planes

    Which of the following planes are parallel? Are any of them identical? P1: 4x - 2y - 6z = 3 P2: 4x - 2y - 2z = 6 P3: -6x + 3y -9z = 5 P4: z = 2x - y - 3 please explain each step in detail

    Equation of a Surface

    Describe in words the surface whose equation is given [note r - cylindrical coordinate, &#961; - spherical coordinate] a) r = 3 b) &#961; = 3 c) &#966; = &#960;/2 d) &#952; = &#960;/3 Give detailed explanation.

    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