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# Calculus and Analysis

### New stock price after the increase of beta.

A stock currently trades with a beta of 1. Company management is considering a bold new venture that will greatly increase the stock's perceived risk. If beta increases to 2, would you expect the stock price to be cut in half, if other variables are held constant?

### Partial Differential Equations : Heat Equations

1) Let A(x,y) be the area of a rectangle not degenerated of dimensions x and y, in a way that the rectangle is inside a circle of a radius of 10. Determine the domain and the range of this function. 2) The wave equation (c^2 &#8706;^2 u / &#8706; x^2 = &#8706;^2 u / &#8706; t^2) and the heat equation (c &#8706;^2 u / &#8706;

### Ordinary Differential Equations Fourth Order Runge Kutta Method

Question Use Runge-Kutta method of order four to approximate the solution to the given initial value problem and compare the results to the actual values. y'=e^(t-y) , 0 <=t <=1 , y(0)=1 with h = 0.5(Interval) Actual solution is y(t)= In((e^t+e-1). For full description of the problem, please see the attached question

### Geometry Questions

1. Consider the graph of y = tan x. (a) How does it show that the tangent of 90 degrees is undefined? (b) What are other undefined x values? (c) What is the value of the tangent of angles that are close to 90 degrees (say 89.9 degrees and 90.01 degrees)? (d) How does the graph show this? 2. A nautical mile depends on

### Four problems are solved in this posting. One is involved Taylor Series expantion about x - 0, the second is involved finding Partial Derivatives, the third is involved Double Integral and the fourth is involved finding Divergence and Curl of a given vector field. For complete description of the questions, please see the posted questions.

Question (1) Write the Taylor series with center zero for the function f(x) = In(1 + x^2 ) Question (2) Compute the first-order partial derivatives of f(x, y) = 2x/(x-y) Question (3) Evaluate the double integral (1 to 3)(0 to 1) of (2x-3y)dx dy Question (4) Calculate the divergence and curl of the vector field F(x,

### Solving Inseparable Differential Equations

Consider the following very simple model of blood cholesterol levels based on the fact that cholesterol is manufactured by the body for use in the construction of cell walls and is absorbed from foods containing cholesterol: Let C(t) be the amount (in milligrams per deciliter) of cholesterol in the blood of a particular person a

### Random Variables and Limit State Functions

Consider the following two collections of data that represent realizations of two random variables X1 and X2: X1: 18.9 21.1 17.8 20.2 16.0 19.0 20.9 19.1 22.5 18.7 15.:3 17.5 22.1 19.8 20.76 X2: 2:3.9 17.8 20.7 20.6 20.0 21.6 25.0 21.9 21.5 20.6 22.0 20.4 2:3.2 21.5 2:3.0 2:3.:3 21.8 2:3.8 26.6 2:3.0 22.0 2:3.8 22.1 (a) Es

### Modeling Data With Linear functions, Trends and Forecasting

1. America creates more garbage than any other nation. According to Denis Hayes, president of Seattle's nonprofit Bullitt Foundations and a founder of Earth Day, "We need to be an Heirloom Society instead of a Throw-Away Society." The EPA estimates that, on average, we each produce 4.4 pounds of garbage daily (Source: Take out

### Lagrange Multipliers

Use the method of Lagrange multipliers to find the extreme valus of 3x - 4y +12z on the spherical surface with equation x^2+y^2+z^2=1.

### Solving Differential Equations, Initial Value Problems and Circles and Ellipses

1 First find the general solution of the differential equation dy/dx = 3y. Then find the particular solution the satisfies the initial condition that y(1) = 4. 2 Solve the initial value problem dy/dx = y^3 , y(0) = 1 3 Find the center and radius of the circle described in the equation 2x^2+2y^2-6x+2y=3. 4

### Initial Value Problems:Laplace Transforms & Convolution Theorem

1. For the problem given below use the convolution theorem to write a formula for the solution of the I.V. problem in terms of f(t) y''-5y'+6y=f(t) y(0) = y'(0)=0 2. Use Laplace Transforms to solve the following equation t^2 y'-2y = 2 (no IC's)

### Sturm-Liouville Problem

This has to be converted to a Sturm-Liouville equation and then solved y'' + 2*y' + (1+k)*y = 0 BC: y(0) = y(1) =0 I need all the steps taken to convert to an SL Eqn and the the solution of that eqn.

### Calculus - Numerical Analysis - Approximations

Use Modified Euler's Method to approximate the solution to the initial value problem and compare the results to the actual values y'=1+(t-y)^2 , 2 <=t<=3 , y(2) = 1 with h = 0.5 Actual solution y(t)=t+1/(1-t) For full description of the question, please see the attached question file

### Finding Zeros of Functions

Find the zero of the linear function f(x)=3x-12 Find the zeros of f(x)=x^2-2x-3 Find the vertex of f(x)=x^2-2x+4 Find the axis of symmetry of f(x)=x^2-2x+4 Find the zeros and state the multiplicity of each for f(x)=x^2(x+3)(x+1)^4 Find the zeros of f(x)=x^2-8x+12

### Solving differential equation using modified Euler's method.

Use the modified Euler's method to solve y' = -y +x + 2; y(0) =2 on the interval [0,1] with h = 0.1 Carry all computations to three decimal places.

### Differential Equations with Boundary Conditions : Eigenvalues, Eigenfunctions and Sturm-Liouville Problems

Consider the equation: y'' + k*y = 0 with BC: y(0) = 0 , y(&#61552;) = a &#61625;0 Answer the following: 1. What are the restrictions on k such that there is a nontrivial solution? 2. Find a solution using eigenfunction expansion on [0,&#61552;] 3. Find a solution to the differential equation using any method 4. Comp

### Calculus - Vectors - Area of Triangle, Volume of Parallepipe

Question (1) a = (3 , 1 , 2 ) , b = ( - 1 , 1 , 0 ) , c = ( 0 , 0 , - 4 ) , then show that a × ( b × c ) &#8800; (a × b) × c Question(2) Given P ( 2 , 1 , 5 ), Q = ( - 1 , 3 , 4 ) and R = ( 3 , 0 , 6 ), then find (a) a vector orthogonal to the plane through the points P,Q and R (b) Find the area of the triangle PQR

### Determine whether the given vectors are orthogonal, parallel or neither

Determine whether the given vectors are orthogonal, parallel or neither. <-5,3,7> and <6,-1,2> <4,6> and <-3,2> -i + 2j + 5k and 3i + 4j - k 2i + 6j - 4k and -3i -9j +6k Find a unit vector that is orthogonal to both i+j and i+k. If a = <3,0,-1>, find a vector b such that comp_a b = 2 (component of b in the a direction

### Eigenvalues, Eigenfunctions and Differential Equations

Consider the equation: y'' + k*y = 0 with BC: y(0) = 0 , y(&#61552;) = a &#61625;0 Answer the following: 1. What are the restrictions on k such that there is a nontrivial solution? 2. Find a solution using eigenfunction expansion on [0,&#61552;] 3. Find a solution to the differential equation using any method 4. Co

### Eigenvalues, Eigenfunctions and Sturm-Liouville Problems

1. y'' + k*y = 0 BC: y'(0) = 0 y'(L) = 0 2. y'' + k*y = 0 BC: y(0) = y(&#61552;) y'(0) = y'(&#61552;) 3. y'' + k*y = 0 BC: y(0) = 0 y(&#61552;) +2*y'(&#61552;) = 0 4. y'' + 2*y' + (1+k)*y = 0 BC: y(0) = y(1) =0 Please see the attached file for

### Calculus: Three-dimensional space

4. What are the projections of the point (2, 3, 5) on the xy, yz-, and xz-planes? Draw a rectangular box with the origin and (2, 3, 5) as opposite vertices and with its face parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box. 8. Find the lengths of the sides of the tr

### Numerical, Graphical, and Analytic Analysis

An open box of maximum volume is to be made from a square piece of material 24 inches on a side, by cutting equal squares from the corners and turning up the sides. See attached file for full problem description.

### Inverse Function

4. A retailer you spoke with in New York City's fashion district imports haute couture from European designers. One of the accommodations which must be considered when importing fashion from other countries is the difference in the size charts. A function that will convert dress sizes in the United States to those in Italy is

### Finding X, Y and Z Intercepts of 3-D Graphs

How do you find the x, y, and z-intercepts of a 3-Dimensional graph step by step? I am trying to interpret some graphs and would like this information to assist me in the interpretation. keywords: 3D, 3Dimensional

### Volume of a solid

Find the volume of the solid that is generated by rotating the region formed by the graphs of y = 2x^2 and y =4x about the line x = 3 .

### Current and Charge in the Series RLC Circuit

Please see attached file. 13. A series circuit consists of a resistor with R = 20 ohm, an inductor with L = 1 H, a capacitor with C = 0.002 F, and a 12-V battery. If the initial charge and current are both 0, find the charge and current at time t. 14. A series circuit contains a resistor with R = 24 ohm an inductor with L = 2

### Solve Differential Equation Using Power Series

Solve differential equation using power / Taylor series. Need problems 2,4,6,8,10. See attached file for full problem description. 2. y' = xy 4. (x- 3)y' + 2y = 0 6. y'' = y 8. y'' = xy 10. y'' + x^2y = 0, y(0) = 1, y'(0) = 0.

### Simple harmonic oscillation driven by an external force

2. A spring with a 4-kg mass has natural length 1 m and is maintained stretched to a length of 1.3 m by a force of 24.3 N. If the spring is compressed to a length of 0.8 m and then released with zero velocity, find the position of the mass at any time t. 10. As in Exercise 9, consider a spring with mass m, spring constant k,

### Calculus - Double Integrals - volume of rectangular box

A box with its base in the xy-plane has its four upper vertices on the surface with equation z = 48 - 3x^2 - 4y^2 . What is the maximum possible volume.

### Differential Equations : Solving Boundary-Value Problems

3. Solve the boundary-value problem, if possible. a. y''-6y'+9y =0, y(0) =1 and y(1) = 0 b. 9y''-18y'+10y = 0 , y(0) =0 and y(pie) = 1 4. If a, b and c are all positive constants and y(x) is a solution of the differential equation ay''+by'+cy = 0, show that lim x->infinity y(x) = 0