Calculus Answers

Sketch Cardioid: r = a(1-cosθ)
Please explain me its solution step by step with graph...?
Please tell me as soon as possible...?

Find the derivative of the function.
y = int_{cos(x)}^{5 x} cos(u^4) du

Estimate the arc length of r(t) = (cos^2 t, sin 3t), t element of [0, pi] using trapezoidal rule with n = 6.

If f(x)=(1/x)[sup]x[/sup] , find f ``(1)

Evaluate lim x->1 [(1/1-x)-(1/1-x^2)]

Evaluate lim x->5 (sqrt(5x)-5)/(sqrt(2x-4)-sqrt(6).

Illustrate the molecular orbitals obtained by the combination of the following atomic orbitals:
(i) px – px
(ii) s – py
Comment on the nature of molecular orbitals obtain

The lowest wave number absorption line in the rotational spectrum of 14N16O molecule is found at 3.440 cm[sup]-1[/sup].
(a) Calculate the corresponding frequency of absorption.
(b) Which are the two energy levels involved in the transition?
(c) What is the value of the rotational constant (B) in m-1 unit?
(d) Calculate the moment of inertia for oxide molecule.
(f) Calculate the bond length (r) of nitric nitric oxide molecule.
(e) Calculate the reduced mass of nitric oxide molecule in pm unit.

Water is leaking out of an inverted conical tank at a rate of 10000 cm[sup]3[/sup]/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank.(cm[sup]3[/sup]/min)

I need to find the limit of a problem and I keep getting zero as the answer with different paths. The problem is the limit (x,y) approaches (0,0) of (6x^3)/(2x^4 + y^4).