Math 210, Final Exam – Fall 2010 Name:
Final Exam
(20 pts) 1. Let u=h1,−1,0iand v=h2,1,3i.
(a) Is the angle between uand vacute, obtuse, or right?
(b) Find an equation for the plane through the point (1,−1,2) containing uand v.
(25 pts) 2. The curve r(t) = h2 sin(t),2 cos(t),−tidescribes the movement of a particle in R3.
(a) Find the velocity and the acceleration of the particle as a function of t.
(b) Find the tangent line to the curve at time t=π/4.
(c) Find the distance travelled between time t= 0 and t=π.
(20 pts) 3. Use Green’s theorem to compute IC
y dx +x2y d y where Ctraces the triangle with vertices
(0,0), (2,0), (1,1) traversed in this order.
(25 pts) 4. Find the critical points of z=x3+x2+y2−2xy −12xand use the second derivative test
to classify them as local maxima, local minima or saddles.
(25 pts) 5. Consider the vector field F=hcx 2y2−ey,2x3y−x e yion R2where cis a constant.
(a) Find the value for cthat makes Fa conservative vector field.
(b) With cas in (a) find a function φ(x , y) so that F=∇φ.
(25 pts) 6. Compute the volume of the region in R3bounded by the paraboloid z=x2+y2, the cylinder
x2+y2= 9, and the plane z= 0.
(20 pts) 7. Given the function f(x , y) = x y 2+ycos(x) find:
(a) the gradient ∇fat the point P= (0,1),
(b) the directional derivative Dvf(0,1), where vis the unit vector from P= (0,1) towards
Q= (2,3).
(25 pts) 8. Use the method of Lagrange multipliers to find points where f(x , y) = x y attains its maximum
and minimum subject to the constraint: x2+ 4y2= 2.
(15 pts) 9. Given the function f(x , y) = x exy compute the partial derivatives:
∂f
∂x ,∂f
∂y ,∂2f
∂x ∂x ,∂2f
∂x ∂y ,∂2f
∂y ∂y .
Hand in this sheet along with your exam booklet!
