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Calc 3 โ Final Exam This is a take-home, open-book, open notes exam; you even may use Maple to assist you in calculations. You must, however, complete it entirely on your own. It is due on the last day of exams โ no exception. Please indicate clearly where each problem starts and do not forget to put your name on your exam. 1. Please state the following: a) an equation relating the dot product of two vectors with the angle between them b) the definition of the gradient of a function ๐(๐ฅ, ๐ฆ, ๐ง) and its properties c) the main difference between Greenโs and Stokeโs Theorem as far as the vector field ๐นโ is concerned d) the Divergence Theorem (also known as Gaussโ Theorem) 2. Match the following pictures with the algebraic expressions below. [A] [B] [D] [C] [E] (1) f ( x, y) ๏ฝ 6 ๏ญ x ๏ญ y 2 (4) [F] (2) f ( x, y ) ๏ฝ y ๏ญ x 2 2 r (t ) ๏ฝ cos(t ), t, sin( t ) (5) 2 F ( x, y) ๏ฝ x,1 (3) r (t ) ๏ฝ cos(t ), sin( t ), t (6) F ( x, y) ๏ฝ 1, y 3. Determine if the plane through the points ๐(1,0,0), ๐(0,2,0), and ๐ (0,0,3) is perpendicular to the plane given by the equation 2๐ฅ โ 2๐ฆ โ 3๐ง = 1 4. A baseball is hit 4 feet above ground at an initial velocity of < 80, 80 > feet per second. Find the maximum height reached by the baseball. Will it clear a 15-foot high fence located 350 feet from home base? 5. Determine the following limits, if possible, or explain why they donโt exist. lim ๐ฅ+2๐ฆ+3 (๐ฅ,๐ฆ)โ(0,0) ๐ฅ 2 +2๐ฆ 2 +3 lim ๐ฅ 2 โ๐ฆ 2 (๐ฅ,๐ฆ)โ(0,0) ๐ฅ 2 +๐ฆ 2 lim ๐ฅ๐ฆ 2 (๐ฅ,๐ฆ)โ(0,0) ๐ฅ 2 +๐ฆ 4 6. Find all critical points and test them for relative extrema for the function f ( x, y ) ๏ฝ ๏ญ x 3 ๏ซ 3 xy ๏ญ 3 2 y 2 (Hint: There are two critical points) 7. Evaluate the following integrals: 2 a) โฌ๐ ๐ ๐ฅ ๐๐ด, where R is the triangular region bounded by y = 0, y = x, and x = 1 b) โซ๐ถ ๐ฅ 2 + ๐ฆ^2 ๐๐ , where C is the curve given by ๐(๐ก) =< 2 cos(๐ก) , 2 sin(๐ก) > for 0 โค ๐ก โค ๐ c) โซ๐ถ ๐นโ ๐๐โ, where ๐นโ (๐ฅ, ๐ฆ) =< โ๐ฆ, ๐ฅ > and C is the line segment from ๐(โ1, โ1) to ๐(2,3) d) The flux of the vector field ๐นโ (๐ฅ, ๐ฆ, ๐ง) =< ๐ฅ, ๐ฆ, ๐ง >, where S is the portion of the surface ๐ง = 10 โ 2๐ฅ โ 2๐ฆ between the coordinate planes. 8. For the following integrals there are at least two ways to evaluate them. Use the most convenient method and quote the appropriate theorem. a) ๏ฒ F ๏dr where F ( x, y, z) ๏ฝ๏ผ 2 xy3 ๏ซ x cos( x),3x 2 y 2 ๏ญ sin( y)e y ๏พ and C is the closed curve given C by the boundary of the square with corner points (-1,-1), (-1,1), (1, -1), and (1,1). b) ๏ฒ ๏จ3x 2 ๏ฉ y ๏ญ y 2 dx ๏ซ x 3dy where C is the closed curve given by the boundary of the triangle with C corner points (0,0), (0,1), and (1, 0), oriented counter-clockwise. ๏ฎ ๏ฎ c) 2 2 2 ๏ฒ๏ฒ F ๏ n dS where F ( x, y, z) ๏ฝ๏ผ x,2 y,3z ๏พ and S is given by x ๏ซ y ๏ซ z ๏ฝ 9 S ๏ฎ d) ๏ฒ F dr where F ( x, y, z) ๏ฝ๏ผ z 2 , x 2 , y 2 ๏พ and C is the boundary of the surface S given by C z ๏ฝ 1 ๏ญ x ๏ญ y , oriented counter-clockwise.