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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 Mia52% 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 ﬁrstorder 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 twodimensional 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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 Spring '10
 REMALEY
 Math

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