Calculus Answers

Poiseuille’s law asserts that the speed of blood that is r centimeters from the central axis of an artery of radius R is S(r) = c(R^2 − r^2), where c is a positive constant. Where is the speed of the blood greatest?

Let E be the solid bounded by y = x^2, z = 0, y + 2z = 4. Express the integral

Given

u=rcos(wt)i+rsin(wt)j where r and w are constants, obtain the speed of the movement

Show that the transformation x = z −"\\frac{b}{3a}" converts the cubic equation ax³ + bx² + cx + d= 0 into one of the form z³ + 3Hz + G = 0, i.e. the x² term goes away.

If F_n is the n-th Fibonacci number, show that lim "\\lim_{n \\to \\infty} \\frac { F_{n+1}}{F_n} = \\frac{1+\\sqrt{5}}{2}"

Give an example of a function of two variables such that f(0, 0) = 0 but f is NOT continuous at (0, 0). Explain why the function f is NOT continuous at (0, 0).

The displacement, 

y(m), of a body in damped oscillation is y=2e^-t sin sin 3t . 

The task is to:

1.  Use the Product Rule to find an equation for the velocity of the object if v=dy/dt.

Prove that integral of 0 to pie √x e^-x3 dx = √pie/3

Two industrial plants, 𝐴𝐴 and 𝐵𝐵, are located 15 miles per apart and emit 75 ppm (parts per

million) and 300 ppm of particular matter, respectively. Each part is surrounded by a restricted

area of radius 1 mile in which no housing is allowed, and the concentration of pollutant arriving

at any other point 𝑄𝑄 from each plant decreases with the reciprocal of the distance between that

plant and 𝑄𝑄. Where should a house be located on a road joining the two plants to minimize the

total pollution arriving from both plants?

A cylindrical can is to be constructed to hold a fixed volume of liquid. The cost of the

material used for the top and bottom of the can is 3 cents per square inch, and the cost of the

material used for the curved side is 2 cents per square inch. Use calculus to derive a simple

relationship between the radius and height of the can that is the least costly to construct.