Calculus Answers

1. Find the values of the following derivatives. Use Use f^' (x)=lim┬(h→0)⁡〖(f(x+h)-f(x))/h〗

a) ├ ds/dt┤| 〖┬(t=-1)〗⁡ if s=1-3t²
b) ├ dy/dx┤| 〖┬(x=√3)〗⁡〖 〗 if y= -1/x
c) ├ dr/dθ┤| 〖┬(θ=0)〗⁡ if r= 2/√(4-θ)
d) ├ dw/dz┤| 〖┬(z=4)〗⁡〖if w=z+√z〗

Find the length of an arc of the curve given
by x = a(theta — sin theta), y = a(1— cos theta), and obtain
the ratio in which theta = π/3 divides it.

1. Evaluate the following integrals.

a) ∫_1^4▒√(5/x ) dx

b) ∫_1^9▒〖(3x-2)/√x dx〗

Let f(x,y)=x^2y/(x^4+y^2) if x^4+y^2not=0
=0. if x=y=0
examine whether
lim(x,y) tends to(0,0) f(x,y) exist or, not.

Let the function f:R^2-->R is defined as follow
f(x,y)=[xy(x^2-y^2)]/x^2+y^2 if(x,y )not=(0,0)
=0 if (x,y)=(0,0)
Then show that
1)fx(0,y)=y for all y

2)fx(x,0)=x for all x.
So that fxy(0,0)not=fyx(0,0)

Prove that
lim x tend to 0 xsin (2/x)=0

Find the point on the ellipse,(x^2/4)+y^2=1
that is nearest to the origin.

Find the massof an object which is in the form of a cuboid [0,1]×[2,4]×[1,3]Thedensity at any point(x,y,z)on the cuboid is given by del(x,y,z)= x^2+y^2+z^2.

Let x=e1+e2-2e3,y=2e1-e2+e3 where e1,e2,e3 are unit vector .find |x+y| and|x+2y|

if z=e^(xy^2),x=tcost,y=tsint then find dz/dt when t=π/2.