final_review.pdf - Review for Final Part I MATH 223 Fall...

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MATH 223 Review for Final Part I Fall 2017 Here are some review problems for Part I of the Final Exam, covering 14.8, 12.7, 15.1-15.4, 15.6, 16.2, 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. (We saw them in some integration problems in Chapter 15.) To study for Part II, see the file “Guidelines for Final Exam Part II” on Canvas. 1. 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.] 2. Convert the point ( - 2 2 , 2 2 , 2) into cylindrical and spherical coordinates. 3. Consider the set defined by x 2 + y 2 4 in rectangular coordinates. Describe the set in (a) cylindrical coordinates, and (b) spherical coordinates. 4. Convert the equation z = 2 r cos θ in cylindrical coordinates to rectangular coordinates. What type of surface does this equation define? 5. Evaluate Z 2 0 Z

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