Calculus Answers

A curve is given x=a(cosθ + θsinθ ) , y=a(sinθ - θcosθ) , find the length of the arc from θ=1 to θ=α?

Find the particular integral of

y' - y = 2e^x

double integral of xdydx +2ydzdx+ 3zdxdy over the surface of the spherex^2 +y^2 +z^2 is

Find the equation of the sphere with the given conditions:

(a) Center (2,4,3), tangent plane 2x-3y+4z-5=0

(b) endpoints of the diameter are (-5,3,2) and (-1,-3,2)

Let f be the function given by f (x) = |x|. Which of the following statements about f is true?

I. f is continuous at x = 0

II. f is differentiable at x = 0

III. f has an absolute maximum at x = 0.

Integrate dx/(sqrt(4^x-4)

Integrate (cbrte^sin(1/x))/(x^2 sec(1/x))

Investigate convergence or divergence of the following sequence with justification:

(√n2 + 1)(√n).

Let {xn} n=1to ∞ and {yn}n=1to ∞ be sequences of positive numbers such that limn→∞ xn/yn = 0. Then show that if {yn}n=1to ∞ is bounded, then limn→∞ xn = 0

Let xn = ( 2 + (−1)^n/ n+2 ). Using the algebra of limits and standard results on limits to establish/justify if the sequence {xn} n=1 to infinity converges or diverges; find the limit in case the sequence is convergent.