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Unformatted text preview: Math 1920 Prelim 1 — Solutions Sp2009 [l5ptsl 1. Consider the vectors 172 i+ 2j + ak and 13 : i+ 2j + k. (a) Find all values of the parameter a (if any) such that 77 is perpendicular to 16.
A: a = —5 (b) Find all values of the parameter a (if any) such that the area of a parallelogram determined
by 77 and IE is equal to v 20. A: a : —l, 3
(c) Find the projection of 16 onto the plane 23: + 1y + 22* = 4
—l 4 —l
A: 1 (me—’2 —>_>—
pm“ w <3 3 3> l l .
[lOpts] 2. Let 77(t) be the vector function deﬁned by 77(t) = i+ §t2j+ §t3k. Compute the arc length parameter 5(t) of ﬁt) starting from t:0. 1 3/2 1
A: = — 1 2 — —
3(t) 3 ( +t ) 3
4
[lOptsl 3. Find %, where f(w,a:, y, 2) = arctan(.r 2) + S 6939 + y ln + sin(y)
84f —1
A: — = my . my —
82$8y82 [e + rye ] + lOpts 4. True False — Problems are worth 2 points if correct, —l if incorrect. You do not need to ‘ustif
J Y
your answer, but guessing may cost you points. (a) For any vectors 17 and '17in V3, [27 X '17! : li7>< A: True.17><77:—27><27 (b) The cross product of two unit vectors is a unit vector. A: False. The magnitude also depends on the angle between the vectors.
(c) H1717: 0, then 17:0 or 17:0.
A: False. Another option would be if the vectors were perpendicular.
d d7 d7
(d) If 17(t) and 77(t) are differentiable vector functions, then a [17 X 77] = d—r: X 77 + d—: X 17.
d a a_d17 a( a d?7_d17 a £127 a
A. False. Order matters E [u X 22] _ dt X 7) . u x dt _ dt x 2} dt x u.
(e) The shortest distance between two lines is along a vector perpendicular to both lines.
A: True. Pythagorean theorem.
(f) If f$(a, b) and fy(a, 1)) both exist, then f(;r, is diﬁerentiable at (a, b).
. . _ 0 if my yé 0
A. False. Cons1der f(x,y) — { 1 iffy I 0 3 — 3
[10ptsl 5. Given f(:1:, y) = $2 + 1y and E = 0.01, show that there exists a 6 > 0 such that for all (133;),
at M < 5 implies that y) — f(070)l < 6' 1
A: If we set 6 = 6 e, we are guaranteed that for all (3:, y), «:02 +y2 < 6 implies that lf(:c,y) — f(0,0) < 6 333—33;
562+1 [ 9pts] 6. Sketch three equally spaced level curves (contours) for the function f(x,y) = A: [18pts] 7. A 1 kg rocket is ﬁred at a mountain slope with initial velocity :70 :< 27 27 4 > m / s and position 770 :< 0,070 > m. The rocket thrusters produce a force FR :< 272 — 5%,6 — 5x/t > N and
acceleration due to gravity is 10 772/52. (a) Write an equation for the acceleration of the rocket as a function of time.
A: at) = (27 2 — sﬁ, 6 — S'ﬂ — 10) m/s2 2 (b) Write the velocity and position of the rocket as a function of time. 10 10 l
A: 27(t) 2 <2: l 2,2: 3 t3” : 2, 4t 3 733/2 : 4> m/s
a a 2 2 4 5/2 2 4 5/2
A: r(t):fv(t)dt: t :2t,t 3t :2t, 2t 3t :4t m (c) If the mountain slope can be described by the equation x + y — 2 : 4. Where does the rocket
hit the slope? [18pts] 8. Given the following lines ,
F(t):<2+t,3+t,2+t> 17(t)=<1+2t,1+3t,1+4t> (a) Find the minimum distance between the lines. 2
A: The minimum distance is \/;. Hint: Consider two parallel planes. Plane 1 contains line 1 and plane 2 contains line 2.
What is the distance between the planes? There are at least 3 different ways to answer this
problem. (b) If the equations describe projectile paths7 express the distance between the projectiles as a
function of time. A: d(t) = \/(1 — 1?) + (2 — 21:)2 + (1 — 31:)? = «142:2 — 1615 — 6 (c) If the equations describe projectile paths7 at what time are the projectiles closest to each
other? A: t: NIH; ...
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 Spring '06
 PANTANO
 Derivative, Multivariable Calculus, Vectors, Velocity, Vector Motors, differentiable vector functions, arc length parameter

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