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    Calculus and Analysis

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    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

    Calculus Problems

    Given the table above, graph the function, identify the graph of the function (line, parabola, hyperbola, or exponential), explain your choice, and give the domain and range as shown in the graph, and also the domain and range of the entire function. See attached file for full problem description.

    Calculus: The Length of a Side, Distance and Finding the Hypotenuse

    1. The volume of a cube is given by V = s^3. Find the length of the side of a cube if the Volume is 729 cm^3. 2. Suppose you travel north for 35 kilometers then travel east 65 kilometers. How far are you from your starting point? North and east can be considered the directions of the y- and x-axis respectively. Round to the

    Solving Differential Equations: Particular Solutions

    1. Find the particular solution of the following differential equations: A) Given that when t=0, y=3 and dy/dt=0.5 B) Given that when x=0, y=0 and dy/dx=1 2. In a galvanometer the deflection θ satisfies the differential equation: solve the equation for theta given that when t=0, theta=0 and d(theta)/dt=0 See attac

    The integrating factor method

    1. Solve the differential equation Given that y=2.5 when x=1 2. Apply the integrating factor method to solve this equation analytically I think the answer should be around 3.4119 See attached file for full problem description.

    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.

    Calculus - Vectors - Vector Equation and Cartesian Equation

    Question (i) Find the vector equation of plane which passes through the points (2,1,1) , (1,-1,-1) and (-1,1,2). Question (2) Obtain the Cartesian equation of a plane passing through three points (3,6,5) , (4,5,2) and (2,3,-1). The question file is attached.

    Differential Equations : RLC Series Circuit

    A mass-spring system with damping consists of a 7- kg mass, a spring with spring constant of 3 N/m. a frictional component with damping constant 2 N-sec/m. and an external force given by f(t)=10 cos 10t N. Using a 10-ohm resistor construct an RLC series circuit that is the analog of this mechanical system in the sense that the t

    Calculus - Vectors - Finding Unit Vectors

    Q.2(a) Find the value of Lamda if Lamda(2i- 4j+4k) is a unit vector. (b) If a,b and c are any vectors such that a = ( 2 , 4 , - 5), b = ( 1 , 1 , 1 ), c = ( 0 , 1 , 2 ) find the unit vectors parallel to (a+b+c). (c)If r = xi+yj+zk then write the unit vector in the direction of r. (d)Determi

    Solve: Graphing Parabolas

    Please answer the following questions: 1. Sketch the graph of the parabola y= 3x^2 - 2x + 1. 2. Sketch the graph of the function y = sqrt(16 - 4x^2).

    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.

    Particular Solution that Satisfies the Initial Condition

    1.Find the particular solution that satisfies the initial condition. Differential Equation Initial Condition 2 2xy'-1nx =0 y(1)=2 2.Write and solve the differential equation that models the verbal statement. The rate of change of P with respect to t is proportional to 10-t.

    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

    Principal and Interest

    The Board of Governors at the Federal Reserve Bank set the prime interest rate. These rates trickle down through the economy at various rates and affect consumer spending and saving habits. The economists at the Federal Reserve follow sample households in order to extrapolate information which they use in forecasting. For exampl

    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

    Derivatives and Rate of Change Word Problems

    1.) A deposit of S dollars that earns 100r% annual interest compounded continuously leaves a balance of P = 'Se' power of 'n' or ( 'Se' to the 'n') dollars after "t" years. a) What will an amount of $ 5000 grow to after 15 years at 10% annual interest compounded continuously? b) Determine the rate at which P is growi

    Calculus - Linear Operators - Vectors - Equation of a plane

    Question (1): Find the equation of the plane containing the vector a = (3 , - 2 , - 1 ) and parallel to the vectors b = (1, - 2 , 4 ) and c = (3 , 2 , - 5 ) Question (2) : Find the equation of a plane passing through A and B whose position vectors are 4i + j - 2k , 5i + 2j + k respectively and parallel to the vector 3i

    Variation of Parameters

    Use variation of parameters to find the general solutions of the following differential equations: #(t^2 -1)x'' - 2tx' + 2x (t^2 - 1)^2 if two solutions to the associated homogeneous equation are known to be e^t and 1/t

    Differential Equations : Variation of Parameters

    #11.14) Use variation of parameters to find the general solutions of the following differential equations. y'' + (1/x)y' - 1/(x^2)y = lnx if two solutions to the associate homogeneous problem are known to be x and 1/x

    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