Calculus Answers

Calculate dy/dx for the following equations:
a) cos(x)cos(y)=sin(y)
b) y=x^sin(x)

Solve the following by Banoulli equation?

dr/d(theta) + r tan(theta) = cos^2 (theta)

r ( pi/4) = 1

Solve the following with the help of Banoulli equation?

x(2+x) dy/dx + 2(1+x) y = 1 + 3x^2
y(-1)=1

Please, kindly prove to me the first derivatives of all trigonometric rations and their inverses

i) Determine which of the vector fields given below is conservative
a) F = (((y)/(x^2+y^2))-1)i - ((x)/(x^2+y^2))j
a) F = (((x)/(x^2+y^2))-1)i - ((y)/(x^2+y^2))j

ii) Show the points A(pi /6, pi /6) and B(–pi, pi) on the x,y-plane

iii) Calculate the work done (in dimensionless units) by the conservative field between these points, while presenting answer in exact form and then calculate up to four significant figures

1)Use spherical coordinates to calculate the triple integral of f(x,y,z) over the given region D
a.f(x,y,z)=y ,D: x^(2)+y^(2)+z^(2)<=1,x,y,z<=0
b.F(x,y,z)=(x^(2)+y^(2)+z^(2))^(-3/2),D:2<=x^(2)+y^(2)+z^(2)<=4
2)Find a potential function for (yz^(2),xz^(2),2xyz)

1)Find the total mass of the square disk 0<x,y<1 if the mass density is f(x,y)= x^(2)+y^(2)
2)Calculate the average value of the function f (x,y) = e^(x+y) on the square [0,1]x[0,1]
3)Calculate the average height above the x-axis of a point in the region 0<=x<=1,0<=y<=x^(2)

00 is equal to ?

1)find the volume ò the region bounded by
Z=8-y^(2),y=8-x^(2),x=0,y=0,z=0
2)a)find curl(z-y^(2),x+z^(3),y+x^(2))
B)curl(x/y,y/z,z/x)

Find the Fourier transform of the following function:
f(x)= 1 , |x | < a
0 , |x | > a

Hence evaluate ∫∞ sinax/x dx
.-∞