Calculus II Midterm 2 Thursday, May 15, 2006 No calculator is allowed. No credit will be given for an answer without reasoning
1. Given that
r(t) =< e2t, t2−t,cos 2t >, calculate
• (2 pts)
tlim→0r(t) =?
• (2 pts)
d
dtr(t) =?
• (4 pts) Z s
1
r(t) dt =?
2. Given the position function
r(t) =< sin 2t, cos 2t, t >,
• (2 pts) find the velocity,v(t)
• (2 pts) find the acceleration,a(t)
• (6 pts) find the unit tangent vector T(t) and the principal unit normal vector N(t)
• (3 pts) find the binormal vector B(t) = T(t) × N(t)
• (3 pts) find the curvature κ = ||T||r00(t)||(t)||
• (4 pts) find the tangential and normal component of acceleration a(t), i.e. find aT and aN of a(t) = aTT(t) + aNN(t)
3. (8 pts) Show that the limit does not exist.
(x,y)→(0,0)lim 6x3y x6+ y2 4. (10 pts) Show that the limit exists. (You can use polar coordinates)
(x,y)→(0,0)lim x2y x2+ y2 5. (8 pts) Find the indicated partial derivatives.
f(x, y) = x3−3xy + y3; fx, fy, fxy, fxx
6. (8 pts) Find the equation of the tangent plane to the surface at the given point.
z= x2−y2+ 1 at (2, 1, 2)
7. (8 pts) For a differentiable function g(u, v) = f (x(u, v), y(u, v)) with x(u, v) = u cos v and y(u, v) = u sin v and where fxy and fyx are continuous, compute fu, fv and fuu.
f(x, y) = 4x2y3
8. (8 pts) Given that f (x, y) = x2−y2, find the gradient of f (x, y) at (2, 1) and compute the directional derivative of f in the direction of u =< 12,√23 >
9. (10 pts) For f (x, y) = x3−3x +13xy2, locate all critical points and classify them.
10. (12 pts) Find the maximum and minimum of the function f (x, y) subject to the constraint g(x, y) ≤ c.
f(x, y) = x3−3x +1
3xy2, subject to x2+ y2≤3
