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Calculus and Analysis

Multivariable calculus

Question 1) Find the direction from (-3, 1, 2), in which g(x,y,z)=x^2*y*z-2*z^3 decreases fastest. question 2) Follow the line in the direction you found in part 1) to estimate, using linear approximation, the location of the point closest to the coordinates (-3,1,2) at which g=1;Do not use a calculator; Express the answer us

Multivariable calculus, Partial derivative

Let x = (1/2)(u^2-v^2), y=uv, and f=f(x,y) [QUESTION 1] use the chain rule to derive the change of variables formula in matrix form: (fu,fv)=A*(fx,fy) {actually it is vertical , so fu is at the top and fv is at the bottom. Same for fx and fy: fx is under fy; sorry for the notation I cant do it another way} [QUESTION 2]

Multivariable calculus

Consider the function : f(x,y) = x(x-1)(x-2) + (y-1)(x-y) [QUESTION 1]find the maximum and the minim values of the directional derivative (df/ds)]u at ( 1 , 3/2 ) as u varies . ( (df/fs)]u : I can't write the symbol clearly but it means : the derivative of f according to s on the directi

Multivariable calculus

Consider a triangle in the plane, with angles , a, b , c. Assume that the radius of its circle is equal to 1. 1) by decomposing the triangle into six right triangles having the incenter as a common vertex, express the area A of the triangle in term of a, b , c ( the answer should be a symmetric expression). Then use the resu

Exact equation

The following differential equation is exact. Find a function F(x,y) whose level curves are solutions to the differential equation: ydy-xdx=0 "F(x,y) such that the solutions are F(x,y)=c for an arbitrary constant c".

Steady state deflection differential equation

The steady state deflection is given by: y''''+c^4*y=f(x) calculate and plot the deflection for a load: f = 1 for |x|<10, f=0 everywhere else. using Fourier transform. Plot the deflection for various values of c.

Solve the following word problem.

A closed box with a square base is to have a volume of 1,500 cubic inches. Express its surface area as a function of the length of its base.

Multivariable Calculus

Please see the attached file for full problem description. --- Let S denote the closed cylinder with bottom given by z=0, top given by z=4, and lateral surface given by the equation x^2 + y^2 = 9. Orient S with outward normals. Determine the indicated scalar and vector surface integrals.

Differenetial equation

I am having difficulties gaining a solution for the following differential equation of x'=2t3 -6t2 + t1/2. Could I please get assistance with detailing my solution.

The Method of Lagrange Multipliers

Use the method of Lagrange multipliers to find the indicated extremum. You may assume the extremum exists. Find the maximum and minimum values of f(x,y,z) = x + 3y - z subject to z = 2x^2 + y^2

Exponential order

Show that equation is of exponential order and not of exponential Order. (please see attachment for details)

Laplace transforms

Solve (see in attachment) without using partial fraction decompositions. Thanks you