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1. The diameter of a right circular cylinder is equal to its altitude. Letting V and S be the respective volume and surface area of the cylinder and 2r the diameter, use differentials to estimate the maximum possible error, relative error, and percentage error in computing
(a) the volume of this cylinder and
(b) the surface area of the cylinder
when the measured diameter is 20 cm and the maximum possible error in the measurement is 0 1 cm.
Integrate (cos(2√x)/√x)
Integrate (3-x^2)/(x^2+3)^2
The graph of the parametric equations is called a cycloid.
x=θ-sinθ and y=1-cosθ for 0≤θ≤2π
(a) find dy/dx
(b) find an equation of the tangent to the cycloid at the point where θ=π/3
(c) at what point is the tangent horizontal?
(d) graph the cycloid and the tangent lines in parts (b) and (c)
At a certain instant the dimensions of the rectangle are 8 and 12 feet, and they are increasing at the rates 3 and 2 feet per second, respectively. How fast is the area changing?
The diameter of a right circular cylinder is equal to its altitude. Letting V and S be the respective volume and surface area of the cylinder and 2r the diameter, use differentials to estimate the maximum possible error, relative error, and percentage error in computing
(a) the volume of this cylinder and
(b) the surface area of the cylinder
when the measured diameter is 20 cm and the maximum possible error in the measurement is 0.1 cm.
What is Integration? Discuss types of Integration with Example.
A tent of a given volume has a square base of side,2a four side of vertical length b of and is surmounted by a rectangular pyramid of height h. Use Lagrange’s method of undetermined multipliers to find the values of and in terms of such that the canvas required for its construction is minimum
use the definition of the derivative to evaluate v=4/2pir^3
The temperature T at any point (x, y, z) in space is T = 400xyz2
. Find the
highest temperature on the surface of the unit sphere x
2 + y
2 + z
2 = 1.
10. If u =
x + y
x − y
, v =
x + y
x
, w = x + y + z, determine whether there is a
functional relationship between u, v, w and if so, find it.
11. If u = xey
sin z, v = xey
cos z, w = x
2e
2y
, determine whether there is a
functional relationship between u, v, w and if so, find it.
