Calculus Answers

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.

Let D be a convex region in R2 and let L be a line segment of length I that connects points on the boundary of D. As we move one end of L around the boundary, the other end will also move about this boundary, and the midpoint of L will trace out a curve within D that bounds a (smaller) region R. Using the corollary to Green’s Theorem for finding area, find an expression that relates the area of R to the area of D in terms of the length I of the line segment. [You might start with some simple regions, but you must show this generally.] 

Locate the critical points and identity the stationary points of 4x^4-16x^2+17