# 273s09pt1 - P corresponding to the value t = 1 Then ﬁnd...

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MATH 273 PRACTICE TEST 1 NAME: 1. (10 points). Suppose that z = z ( x, y ) is a function deﬁned implicitly by the equation z 3 + xz - y 2 = 1. Use implicit diﬀerentiation to ﬁnd ∂z ∂x and ∂z ∂y . Answer: ∂z ∂x = - z 3 z 2 + x , ∂z ∂y = 2 y 3 z 2 + x . 2. (10 points). Find an equation of the tangent plane to the surface given by the graph of f ( x, y ) = 2 x 2 + 4 y 2 - 5 at the point P (1 , 1 4 , 0). Answer: 4( x - 1) + 2( y - 1 4 ) - z = 0. 1

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3. (10 points). Compute all ﬁrst-order partial derivatives of f ( r, s, t ) = (1 - r 2 - s 2 - t 2 ) e - rst . 4. (10 points). Prove that lim ( x,y ) (0 , 0) x 2 y x 3 + y 3 does not exist. Hint: ﬁnd two diﬀerent direction of approach ( x, y ) (0 , 0) leading to diﬀerent ”limits”. 2
5.(10 points). Find the equation of the plane passing through the points (3, -1, 2), (8, 2, 4) and (-1, -2, -3). Answer: 13( x - 3) - 17( y + 1) - 7( z - 2) = 0 3

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6.(10 points). Find the maximal rate of change of f ( x, y ) = sin(3 x - 4 y ) at P ( π 3 , π 4 ) and the direction in which it occurs. Answer: 5 , 3 5 i - 4 5 j . 4
7.(10 points). The curve is given by r ( t ) = 2 t i + t 2 j + t k . Find the unit (length must be 1) tangent vector at the point

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Unformatted text preview: P corresponding to the value t = 1. Then ﬁnd the equation of the tangent line to the curve at P . Answer: 1 √ 6 i + 2 √ 6 j + 1 √ 6 k , x = 2 + t, y = 1 + 2 t, z = 1 + t . 5 8. (10 points). In the two-dimensional space, the trajectory of a particle is given by r ( t ) =-1 2 gt 2 j + v t, where g ≈ 10 m/s 2 , v = 2 √ 3 i + 2 j . a) Find the time when the particle hits the ground; b) Find the distance d between the origin and the point of impact. (To do this, you would need to determine x-coordinate of the impact point). Answer: a) 3 5 , b) 4 √ 3 5 . 6 9. (10 points). Find the length of the curve r ( t ) = t i + √ 2 2 t 2 j + 1 3 t 3 k , ≤ t ≤ 1 . Hint: to integrate, represent the quantity under the square root as a complete square. This should be easy. Answer: 4 3 . 7...
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273s09pt1 - P corresponding to the value t = 1 Then ﬁnd...

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