1. Sketch the two vectors listed after the formula for r (t).
2. Sketch, on the same plane, the curve C dterminated by r (t), and indicate the orientation for the given values of t.
   
   r (t)=t*i +4*cos (t)j +9*sin (t)k,
   
   r (0), r(Pi/4),  t>=0
   
   

How to convert an  algebraic equation to a parametrized one ?

How to sketch the curve of a circular cylinder , x^2+y^3^=a^2 with the parametric equations lying on it ?

x=a*cos't) , y=a*sin(t) , z=b*t , if t varies from 0 to 2*Pi, the point P starts at(a,0,0) and moves upward.

How to show that the parametric equations are indeed lying on the cylinder ?

How to plot the whole things ?

A concho-spiral is a curve C that has a parametrization :

y=

where a, b, , are constants.

1. Show that C lies on the cone a^2*z^2=b^2*(x^2+y^2).
2. Sketch C for a = b = 4 and =-1.
3. Find the length of C corresponding to the t-interval [0,].

 1) find  the direction in which f(x,y) increases most rapidly at the point P(1,2), and find the maximum rate of increase
of f at P.

Interpret 1) using the graph of f.

Let f(x,y) = 2+x^2 +1/4*(y^2)

Find a series solution of the differental equation

y'=2*x*y