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Unformatted text preview: MATHEMATICS 234—CALCULU'S III—MIDTERM I
Tuesday, October 10, 2006 Instructor: Paul Milewski NAME (print): Instructions: 1. Write your name on each page.
2. Circle the name of your TA from the list below. GUO PANTEA RAULT SOLOMOU
3. Closed Book. No calculators. 1 sheet notes allowed. 4. Answer your questions on the exam paper. There are some extra pages
in the back if you need them. 5. Show all your work. Partial credit is given only if your work is clear. 6. Time allowed: 75 minutes. GOOD LUCK! 1. ____(20)
2. _____(205
3. ____(20)
4. ___(20)
5. ____(20)
Total. ____(100) NAME (print): 1) Consider the function 1 2 ﬂay) = w2  1y (a) In the my plane sketch carefully the level curves of the function for f=mf=—Lf=—4 (b) Sketch the quadric surface :52 + ﬁyi‘ + z = O. NAME (print): (c) What is the relation between your sketch for (a) and the sketch for
(b)? ((1) Write the equation for the quadric surface of part (b) in spherical
coordinates in the form p = 9(6, ¢>). ' NAME (print): 2) If the velocity of a moving particle is given by v(t) = e‘ cos(t) i + etsin(t) j + x/get k. (a) Find the acceleration a(t). (b) Find the unit tangent T(t), the unit normal N(t) and the curvature Mt). NAME (print): (0) What is 3(t), the distance travelled along the curve as a function of
t? Assume that at t = O, s = 2. (d) Using your answers from part (b) and (c) verify the relation: dT_ jig—EN. NAME (print): 3) Consider the function {E f(w,y) = x2 _ yz'
(a) What is lim ?
(m,y)—*(1,1)f (b) Find V f . What is the direction of steepest decrease of f at (17 0)? (c) Find Dj f at the point (0, 1). Flom the result, what can you say about
the level curve of f at (0, 1)? NAME (print): 4) Find the following limits, or Show that they do not exist. (a)
hm rat/(w  y)
(awn—40,0) (x2 + 3/2)”2 . 3:2 + y2 ,
11m —'————
(z,y)—>(0,0) Sin(2at2 + 2y?) (c) Is the function f = xy differentiable at (O, 0)? Explain. NAME (print): 5) Consider the quadric surface :32 + 4y2 — 22 = 1
(a) What is the equation for the tangent plane to the surface at (1, 1/2, 1)? (b) What is the parametric equation of the tangent line to the surface (at
the same point) which is parallel to the yz plane? ...
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 Fall '08
 DICKEY
 Multivariable Calculus, Parametric equation, Conic section, quadric surface, GUO PANTEA RAULT SOLOMOU

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