Calculus Answers

f(×)=6/ײ3×-10

a. X=-2

b. x= 0

c. x=5

It is required to inclose a rectangular field by a fence and then to divide it into two lots by a fence parallel to one of the sides. If the area of the field is given, find the ratio of the sides so that the total length of fence shall be a minimum.

1. Use double integration to find the volume of the solid bounded from above by the paraboloid z = 9x2 + y2, below by the plane z = 0, laterally by y=x^2 and y=x.

The hypotenuse of a right triangle is constantly 8 inches long. One leg is increasing at the

rate of 3in./min. At what rate is the area of the triangle changing when the increasing leg is

4in.

Is this statement true or false ? provide reason for your answer 

d/dx[ _x²∫(π/2) e^2cost dt ] = -2xe^{2(cosx^2)}

 if I_n = ₀∫∞ e^-xsinⁿ(x)dx , prove that (1+n²)I_n = n(n-1) I_n-2 for n >₌ 2

if f(x) = sin^-1x , show that ( 1-x²) f^'' (x) -x f^'(x) = 0 . Hence prove that , f⁽ⁿ⁺²⁾(0) = n² fⁿ (0)

Find the center of mass of the region bounded by y=(x-2)^2 and y=4.

Consider the decomposition of

(6x^3+x−3)/(x^2−2x+1)

Find the equation of the tangent and normal lines to the curve y3-xy+x2=1 at (1,0)