MATH 260

FINAL EXAM

1. Consider the matrix A

Find a basis for the solution space of the equation AX=0. What is the row rank of this matrix?

2. Find the eigenvalues for the following matrix

Is the matrix diagonalizable?

3. Find the inverse of the following matrix (if it exists).

4. Find and classify all local extrema of the function f(x,y)=x³+y²+6xy.

5. Find the plane passing through the points (0,1,2), (3-1,2), and (2,2,1).

6. Find the angle between the lines x=1+2t, y=3-t, z=2, and x=3t, y=-4+6t, z=-2+4t. Do these lines intersect? If so where?

7. Find the volume of the solid bounded by the plane z=0 and the paraboloid z=32-2x²-2y².

8. Let w=f(x,y,z), x=A(t), y=B(s,t,u), z=C(u), t=D(a,b), and s=F(c,d). Find w/u, w/b, and w/c.

9. Let f(x,y)=sin(x²+y²). Find the gradient of f, and the directional derivative of f at (Öp,0) in the direction of the vector <1,1> (are you going uphill or downhill in this direction?)

10. At what point on the graph of y=e^x is the curvature a maximum? What is the maximum curvature?