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# Test2 - Name(1 point 175 section 2 Test 2 3/1 You have the...

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Name: (1 point) 175 section 2, Test 2, 3/1 You have the full class period to complete the test. If anything seems unclear, please ask. 1. A particle moves along the xy -plane on the path given by x = 2 t , y = t 2 (assume distance is in m and time is in s ). (a) Find the velocity, speed and acceleration of the particle (as functions of t ). (4 points) (b) Say now that the function f ( x, y ) = x 2 + y 2 + 100 gives the temper- ature (in F ) at the point ( x, y ) on the plane. Find the directional derivative of f in the direction of h 1 , 1 i at the point ( - 1 , 2). (6 points) (c) What physical quantity does your answer to part (c) represent? (2 points) 1

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(d) Treating f as a function of t along the path of the particle, find df/dt at time t = 1. (5 points) (e) What physical quantity does your answer to part (e) represent? (2 points) 2. Match the following functions and graphs. (2 points each) f 1 ( x, y ) = cos( x ) f 2 ( x, y ) = sin( x ) f 3 ( x, y ) = x sin( x ) f 4 ( x, y ) = sin( x + y ) f 5 ( x, y ) = sin( x - y ) x 2 + y 2 . (a) 2
(b) (c) (d) 3

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(e) 3. Let f ( x, y ) = p 2 + x 2 + y . For which values of k R does f have a level curve at height k ? Find and sketch the level curves (if they exist) for k = - 1, k = 0, k = 1 and k = 2. (8 points) 4
4. The following picture is the graph of a differentiable function f ( x, y ). Find the sign (i.e. one of positive, negative or zero) of each of f x (2 , 2), f y (2 , 2), f xx (2 , 2), f xy (2 , 2). (5 points) 5. Let f ( x, y, z, t ) = cos( x 2 + y 2 ) z + t 2 . Find f x , f y , f xz and f tt . (6 points) 5

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6. Show that the limit of the function f ( x, y ) = x 2 - y 2 x + y as ( x, y ) (0 , 0) is 0. (7 points) 7. Show that the limit of the function f ( x, y ) = x 2 - y 2 x 2 + y 2 as ( x, y ) (0 , 0) does not exist. (6 points) 6
8. Say f is a differentiable function of x and y , and that x = r cos( θ ), y = r sin( θ ). Find ∂f ∂θ in terms of f x , f y and r and θ . (5 points) 9. Find the linear approximation to the function f ( x, y ) = x 2 cos( y ) + e x y 2 at the point (0 , π ). (5 points) 10. Say f ( x, y ) is a differentiable function such that f ( - 1 , 1) = 2 and (

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Test2 - Name(1 point 175 section 2 Test 2 3/1 You have the...

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