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Calculus and Analysis

Calculus - Chain Rule

Use an appropriate form of the chain rule to find dz / dt for z = 3 cos x - sin x y ; x = 1/t, y = 3t .

Calculus Questions - Automotive Examples

See the attached file for full description. 40) The value of a certain automobile purchased in 1997 can be approximated by the function v(t)=25(0.85)^t , where t is the time in years, from the date of purchase, and v is the value, in thousands of dollars. (a) Evaluate and interpret v(4). (b) Find an expression for v1(t) inclu

Sampling Distribution of Estimators

Can someone please help solve and understand this problem. The answer to the question are (a) All Values; (b) alpha = m/(m+4n). See attached file for full problem description.

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 ∂^2 u / ∂ x^2 = ∂^2 u / ∂ t^2) and the heat equation (c ∂^2 u / ∂

Solving Differential Equations and Modeling

(a) Find the solution of the initial-value problem? dy/dx = cos(4x) / 3 − sin(4x), y = 6 when x = 0. (b) (i) Find, in implicit form, the general solution of the differential equation dy/dx = −e^x + e^−x / y(e^x − e^−x + 1)^3 (y > 0). (ii) Find the corresponding explicit form of this g

Total Differentials

I have 2 functions I would like to take the total differential of: I= I (R-PI) and C=C(Y-T, R-PI)) Where PI is Pi. If you can walk me through the process, that will help me better understand it and be able to work on other problems

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

Solving Differential Equations

I am having problems solving this linear equation. I think it's the sin that is throwing me off. Can you show me how to solve this? dy/dt = 2y + sin 2t

Differential Equations : Phase Lines

Suppose you wish to model a population with a differential equation of the form dP/dt = f(p), where P(t) is the population at time t. Experiments have been performed on the population that give the following information: ? The population P = 0 remains constant. ? A population close to 0 will decrease. ? A population of P =

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

Parametric Equations

Find the parametric equations that correspond to the given equation: ^ ^ ^ r = t e^(-t) i - 5t^(2) k

Statistics and Probability : 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

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 &#952; satisfies the differential equation: solve the equation for &#952; given that when t=0, &#952;=0 and d&#952;/dt=0 See a

Differential Equations

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.