Enm2600 - calculate the integral and present your answer

Question

Question 1:

A surface is defined by the following equation:

z (x, y) = √(12(sec² x/y -1)) + ln [9/10(x²/Π² + y²/4)]

a) Find the equation of the tangent plane to the surface at the point P(Π/3, 2).

b) Find the total differential of function z (x, y) at that point.

In the approximate calculations keep your answers up to four significant figures places.

Question 2:

Find all critical points of the function

z (x, y ) = e^{5(y +1)} (4x² -16x + 5y + 5)

and determine their character, that is whether there is a local maximum, local minimum, saddle point or none of these at each critical point.

Question 3:

Consider the double integral V = ∫∫ xy² dx.dy over the domain in polar coordinates where r varies from 3 to 5, and θ varies from Π /6 to Π /2.

a) Sketch the region over which the integration is being performed.

b) Reduce the integral to polar coordinates and indicate limits of integration.

c) Calculate the integral and present your answer with four significant figures.

Question 4:

Find the solution of the Cauchy problem for the differential equation

y" - 4y' + 4y = 2 sin (2x) subject to the initial conditions: y(0) = 0, y'(0) = -1.

Question 5:

Using the Maclaurin series expansion of exp(x²) and cos x²,

a) Determine the Maclaurin series of function f (x) = (e^x2 - cosx²)/x² up to the term of x⁸ inclusive.

b) Calculate then the approximate value of the integral  I = ₀∫^0.5 f(x)dx up to three significant figures.

Question 6:

Determine which of the differentials (or maybe both?) is the total differential:

a) (e^y + ye^x + 3) dx + (xe^y + e^x - 2) dy;

b) (2 cos x - sin x) exp (2x + 3y)dx + 3 exp (2x + 3y) sin x dy.

Find the potential function in the case of the total differential subject to the condition:

U(1, 1) = 2e.

Question 7:

Calculate the total electric charge Q of a wire which shape is described parametrically in 3D space as x = 3 cos t, y = 3 sin t, z = 3 tan t, where 0 ≤ t ≤ Π /2 and length is in m, if the line density of electric charge is given by the formula q (x, y, z) = y x5/81 C/m. Calculate you answer up to four significant figures and present in the dimensional form.

*Question 8:

A periodic function f (t) is defined on its period by the formula:

               - cos t, - Π/t ≤ t < 0; 2 .
f (t ) = {

               cos t, 0 < t ≤ Π/2.
a) Plot the function f (t) on the interval -2Π ≤ t ≤ 2Π.

b) Determine the period of the function.

c) Determine, whether the function is odd or even?

d) Find the average value of the function on its period.

e) Present the function by the Fourier series using the symbol.

f) Present first four terms of the Fourier series in the explicit form.