math_273_(spring_2009)_practice_test_1

math_273_(spring_2009)_practice_test_1 - MATH 273 PRACTICE...

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Unformatted text preview: MATH 273 PRACTICE TEST 1 NAME: 1. (10 points). Suppose that z = z(x,y) is a function defined implicitly by the equation 23 + x2 — y2 = 1. Use implicit differentiation to find 3—: and 2—3. - .Bz__ z g_ 2 Answenm- Mia-52%- 2. (10 points). Find an equation of the tangent plane to the surface given by the graph of f(:r,,y) = 2:122 + 4y2 — 5 at the point P(1, 41,0). Answer: 4(m — 1) + 2(y — ‘11) — z = O. 3. (10 points). Compute all first-order partial derivatives of f(r,s,t) = (1 — r2 — 32 — t2)e'"t. 4. (10 points). Prove that - 2 515?! 5103+y3 does not exist. Hint: find two different direction of approach (m,y) —> (0,0) leading to different ”limits”. lim<w,y>~<o‘0) 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(3: — 3) — 17(y + 1) — 7(2 - 2) = 0 6.(10 points). Find the maximal rate of change of f(m,y) = sin(3m — 4y) at P(§, fi) and the direction in which it occurs. Answer: 5, gi — gj. 7.(10 points). The curve is given by r(t) = 2\/Ei + t2j + tk. Find the unit (length must be 1) tangent vector at the point P corresponding to the value t = 1. Then find the equation of the tangent line to the curve at P. Answer: 71§i+72gj+71€k,$=2+t,y=1+2t,z=1+t. 8. (10 points). In the two-dimensional space, the trajectory of a particle is given by 1 . r0?) = 392593 + Vot, where g 2: 10m/s2, v0 = 2\/§i + 2j. 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) g, b) 445. 9. (10 points). Find the length of the curve r(t) = ti+ lgtzj + %t3k, 0 S t S 1. Hint: to integrate, represent the quantity under the square root as a complete square. This should be easy. can.» Answer: ...
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