10c_04f_fe

10c_04f_fe - the answer x y | x 2 y 2< 1(b Find ∇ g(2 1...

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Math 10C Final Exam 64 points Fall 2004 Print Name, ID number and Section on your blue book. BOOKS and CALCULATORS are NOT allowed. One page of notes (both sides) is allowed. You must show your work to receive credit. 1. (4 pts.) A street vendor sells a hot dogs and b soft drinks on a given day. He charges $2 for a hot dog and $1.50 for a soft drink, including taxes. If A = a a, b A and P = a 200 , 150 A , what is the meaning of the dot product A · P ? 2. (6 pts.) Find the equation of the plane containing the three points (1,0,0), (0,2,0) and (1,0,1). Express your answer in the form ax + by + cz = d for some constants a, b, c, d . 3. (6 pts.) Compute the angle between the two vectors a 1 , 2 , 3 A and a 3 , 2 , 1 A . You may leave a trig function in your answer. 4. (6 pts.) The equation x 3 x 2 y + y 3 = 5 implicitly de±nes y as a function of x . Compute dy/dx at the point x = 2, y = 1. Your answer should be a number. 5. (12 pts.) Let g ( x, y, z ) = ln( x + 2 y + z ). (a) Find the domain of g . Give your answer in set notation; for example (but not
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Unformatted text preview: the answer), { ( x, y ) | x 2 + y 2 < 1 } . (b) Find ∇ g (2 , 1 , − 3). (c) Find D u g (2 , 1 , − 3) where the unit vector u points in the direction of the origin from the point (2 , 1 , − 3). 6. (6 pts.) Use the chain rule to ±nd ∂z/∂s at ( s, t ) = (1 , 2), given that z = x 2 + xy + y 2 , x = s + t and y = st . 7. (6 pts.) Find the tangent plane to the surface z = 4 x 2 − y 2 +2 y at the point ( − 1 , 2 , 4). 8. (10 pts.) Find the critical points of the function f ( x, y ) = x 3 y − 3 xy + y 2 + 7. You do NOT need to classify them as max/min/saddle. (There are ±ve of them.) 9. (8 pts.) Find the absolute maximum and minimum values of f ( x, y ) = 2 x 4 + y 2 subject to the constraint x 2 + y 2 = 1. Also ±nd the points ( x, y ) at which these maximum and minimum values occur. END OF EXAM...
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