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Calculus and Analysis

Use the Uniqueness Theorem for the Initial Value Problem

4. For the initial value problem dy/dx = 3y^(2/3), y(2) = 0, (a) does existence uniqueness Theorem 1 imply the existence of a unique solution? Explain. (b) Which of the following functions are solutions to the above differential equation? Explain. (b_1) y(x) = 0 (b_2) y(x) = (x - 2)^3 (b_3) y(x) = (x - alpha)^3, x <

Derivative Question - Algebraic Function

This is from a Trig/Calculus course...Explain FULLY: If F(x) = x^4 - 2x^3 + 4x^2 - 9 Note: ^ indicates exponant. Find F prime of x. It will be a derivative. I need every step explained clearly as I have a bet riding on this! I need to be able to show every step in order to win my bet.

Assorted Differentiation and Tangent to Curve Problems

A. i) Differentiate the equations given as items 21 and 22 on your worksheet. ii) Refer to the formula given as item 23 of your worksheet. The equation relates to one particular machine in an engineering workshop. The machine sots C pounds to lease each week according to the formula and 't' is the number of hours per wee

Laplace Transform Solutions

Use the laplace transform to solve the ODE y"+3y = cos(2t), y(0)=0 , y'(0)=0 Show all details related to using the inverse transform.

Horse Velocity Using Derivatives

The problem is in JPEG, thank you. Quarter horses race a distance of 440 yards (a quarter mile) in a straight line. During a race the following observations where made. The top line gives the time in seconds since the race began and the bottom line gives the distance (in yards) the horse has traveled from the starting line.

Population growth differential equation.

The birth rate in a state is 2% per year and the rate is 1.3% per year. The population of the state is now 8,000,000. a) At what rate are babies being born in the state now? with units b) At what rate are people dying in the state now? c) Write a differential equation that the population of the state satisfies. include

Application of Stokes Theorem

Use Stokes' Theorem to evaluate int (F.dr) over C where F = x^2*y i +x/3 j +xy k and C is the curve of intersection of hyperbolic paraboloid z= y^2-x^2 and teh cylinder x^2+y^2=1 oriented counterclockwise.

The Existence Theorem for Nonlinear Differential Equations

Please see the attached file for the fully formatted problems. Let g(x,y) be Lipschitz continuous. Let ? (x) = y , and for n > 0 define ? (x) = y + Prove that ? (x) &#61614; ?(x) on [x - , x + ], for some > 0, where ?(x) solves the ODE ?'(x) = g(x, ?(x)), and ?(x ) = y

The Mean Value Theorem and Directional Derivatives

Please see the attached file for the fully formatted problems. Let F: R^n --> R be continuously differentiable. Show that at each point x E R^n there is a direction hx so that the directional derivative is 0, i.e., df/dhx (x) = 0. Is hx unique? Give a method for determining hx.

Implicit Function Theorem

Please see attachment. Thank you. Use the Implicit Function Theorems to show that the system of equations: