Calculus Answers

Find the area enclosed by the curve r = a(1- cos theta)

If x sin y =sin (p + y), p €R ,show that sin(p)dy/de sin^2y=0

prove that nCr+1=nCr (n-r/r+1)

Examine whether lim x to 0 (e^1/x)/(e^1/x +1) exist or not.

state whether the following statements are true or false. justify your answers.
1) lim x to 0 (1/x^2 -1/sin^2x)is in (0/0) form.
2) f(x,y)={sin(x^2y/x^3 +y^3)}/ln(x+y/x) is a homogeneous funtion of degree 2.
3) Domain of f(x,y)=xy/(x^4 +y^4) is R^2.
4) The funtion f(x,y)=(x^3y +1, x^2 +y^2) is locally invertible at (1,2).
5) The funtion f(x,y)=x^3 +y^3 is integrable on [1,2]*[1,3].

If you save three pennies on January 1, six pennies on January 2, nine pennies on January 3, and continue this pattern for one year (not a leap year), what will be the value of your entire savings, in dollars, at the end of that one year? Express your answer as a decimal.

Solve x^4 -8x^3+ 21 x^2 -20 x +5 =0 given that the sum of two of its roots is 4.

Differentiate tan ^(-1) (( sin x -cos x ) /(sin x + cos x )) with respect to (1/2 )x ?

For which value (s) of k , is the function f , defined as below , continuous at x= 2 ?
f(x) ={ 3- kx , 1<= x<2 . (x^2/4) -3 , x>=2 . Further , at which other points in [1, infinity [ is f continuous , and why ?

If f(x) = (4x^2-7x-2) /(x-2) , x!= 2 , find a del > 0 such that |f(x)-9 |<(1/100) for 0< |x-2|<del . Hence show that Lim tends to 2 f(x) = 9 ?