final265sp2007sol

# final265sp2007sol - Math 265 Final Exam 2007/04/30 Name...

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Math 265 Name & Section: %PI e Final Exam 2007/04/30 Instructor: A 1113Jtj Answer each question completely. Show all work. No credit is allowed for mere answers with no work shown. Show the steps of calculations. State the reasons that justify conclusions. 1. (12 points) Show that the triangle with vertices P = (4,1,3), Q = (2,0,5) and R = (4,2,8) is a right triangle. Which vertex has the right angle? =.J[z-ry +(5-3)' = JJ* = 5-3 = ((q-~)-r+ ~14)5+(%-~\~ 3 Jrn =fi r- KP = S~Y-QI* ~1-2)'*~3-b); = jG? = JZL. 1% N ~p~~~+~ni'= @PP -Ih~~v.cq . 5u Q is he vrvh* dh thl: k~g~ ~MJ& dl 2. (12 points) Find the point(s) on the upper half of the ellipsoid 2x2 + y2 + 3a2 = 1 where the tangent plane is parallel to the plane 4x - 4y + 6z = 15.

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1 1 3. (14 points) Find all critical points of f (x, y) = - + xy + -. Indicate whether each such x Y point gives a local maximum or a local minimum, or whether it is a saddle point.
4. (12 points) Find the volume of the solid in the first octant bounded by the coordinate planes and the planes

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## final265sp2007sol - Math 265 Final Exam 2007/04/30 Name...

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