# final_review(1).pdf - Review for Final Part I MATH 223...

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MATH 223 Review for Final Part I Spring 2016 Here are some review problems for Part I of the Final Exam, covering Sections 14.7 Part II, 14.8, 15.1-15.4, 15.6, 16.1-16.3, and 17.1 . Note: You need to be able to recognize the equations of the Quadric Surfaces ( § 12.6) and know their basic shapes, as they may appear in some problems. (For example, we saw them a lot in integration problems in Chapter 15.) To study for Part II, see the file “Guidelines for Final Exam Part II” on Blackboard. 1. Find the global/absolute extreme values of f and the points at which they occur. (Note that these problems do not involve the Second Derivative Test.) (a) f ( x, y ) = x 3 + y 3 - 3 xy on the domain enclosed by the triangle with vertices (0 , 0) , (2 , 0) , and (0 , 2) (b) f ( x, y ) = 2 xy - x - y on the domain y 4 , y x 2 2. Use the Method of Lagrange Multipliers to solve the following problems. (a) Find the absolute maximum and minimum of f ( x, y ) = 3 x - 2 y on the circle x 2 + y 2 = 4 . (b) Find the absolute maximum and minimum of f ( x, y ) = x 2 y on the ellipse 4 x 2 + 9 y 2 = 36 . [Hint: Notice that for the first Lagrange Equation, 2 xy = 8 xλ, there are two possibilities: x = 0 or 2 y = 8 λ . You’ll need to follow both possibilities.] 3. Evaluate Z 2 0 Z 5 3 y ( x - y ) dx dy. 4. Evaluate Z 0 1 / 2 Z π/ 6 0 e 2 y sin 3 x dx dy. 5. Sketch the domain D and evaluate RR D f ( x, y ) dA. (a) f ( x, y ) = e