Calculus Answers

Consider the amusement ride consisting of two circular motions. There is motion on a circle of radius 100 centered at (0,0,100) in the y, z-plane, going counter-clockwise, taking 2 pi seconds to complete a revolution. Attached to this circle of radius 5, in a plane parallel to the x,y-plane, going counter-clockwise, taking pi seconds to complete a revolution. At t=0, you enter the ride on this smaller circle at location (5,100,100). At t = pi/2, you are ejected tangentially and subject to gravitational acceleration a(t) = (0,0,-32). Where should the ground based rescue team place an air mattress? Assume the ground is the x,y-plane, z-axis up

Use cylindrical coordinates to find the volume inside the cylinder x^2+y^2=1 above the plane z=0, and below the surface z=2-y^2

For the function f(x), f(x) = x + a + integral from 0 to 1 of {f(t)}^2 dt

is true. What is the maximum value for the constant a?

a) -3/4
b) -1/2
c) -1/3
d) 1/3
e) 1/2

If y = e^ax cos^3x sin^2x find dy/dx

f(x)=x/square root x

A particle is moving along the curve y=\sqrt{x}. As the particle passes through the point (4,2), its x-coordinate increases at a rate of 3\,cm/s. How fast is the distance from the particle to the origin changing at this instant?

Sketch solution curves for the following equation:
y' = 0.5y-sin y

Given y=Acos(Bx+C) + D. Throughout the day the depth of water at the end of a pier varies with the tides. High tide occurs at 4 am with a depth of 6 meters. Low tide occurs at 10 am with a depth of 2 meters. Model the problem by using the given trigonometric equation to show the depth of the water t hours after midnight showing all your work. Solve the problem by finding the depth of the water at noon, explaining, the reasoning.
#. A large boat needs at least 4 meters of water to secure it at the end of the pier. (a) determine what span of time after neen, including both a starting and ending time, the boat can first safely be secured, justying your answer.

integration of 1/(x^2-4x+5)

Trace the curve y= 2x(x+1)(x+1), stating clearly all the properties used for doing so?