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Final Math Exam
December 20, 2006
Read each problem carefully. Please show all your work for each problem! Use only those methods discussed thus far in class. Always simplify when possible. No calculators! 1. (10 points) Find a parametric equation of the line of intersection of the two planes: x 2y + z = 1, 3x 2y + z = 5.
2. (10 points) A projectile is launched at t = 0 seconds with the initial velocity v(0) = 100i + 100j + 100k meters/second from the point r(0) = 500i 200j meters. Assuming that the projectile moves under constant acceleration a(t) = 10 k meters/second2 due to gravity, determine: (a) The velocity v(t) of the projectile as a function of time. (b) The position r(t) of the projectile as a function of time. (c) The position and time at which the projectile will hit the ground (i.e. when z(t) = 0). 3. (10 points) (a) Find an equation of the tangent plane to the surface xezy + 2y 2 z = 0 at point (2, 1, 0). (b) Find a parametric equation of the normal to this surface at point (2, 1, 0). 4. (10 points) Let f (x, y) = x cos y + y cos x. (a) Find the directional derivative of f at point (/6, /3) in the direction v = i + (b) Find the rate maximum of change of f at point (/6, /3). 5. (10 points) Find and classify all the critical points of the function f (x, y) = 2x2 + 5xy + 4y 2 3x + 5y + 10. 6. (10 points) Use Lagrange Multipliers to nd the minimum value of the function f (x, y) = 3x 4y + 5 subject to the constraint x2 + y 2 = 4. 7. (10 points) Evaluate the double integral xe2y dA,
D
3 j.
where D is bounded by y = 0, y = x , x = 2. 8. (10 points) Use triple integral to compute the volume of a solid enclosed by the surfaces x2 + y 2 = 1, z = 0, z = 3 x + y. (Hint: use cylindrical coordinates). 9. (10 points) Use the Fundamental Theorem of line integrals to compute C F dr, where F = ex cos y i (ex sin y + 3) j, and C is any path that starts at (0, ) and ends at (, 0): (a) First determine that the vector eld F is conservative. (b) Then nd the potential function. (c) Use the result of part (b) to compute the integral. 10. (10 points) Use Greens Theorem to evaluate the integral C y dx x dy, if C consists of the straight line segment connecting (0, 0) to (0, 2), followed by the segment of the parabola y = 2 x2 , and then the straight line segment from (1, 1) to (0, 0).
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