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Unformatted text preview: MATH 234 (A. Assadi) 1 Spring 2007 MlDTERM l Instructions. This test is CLOSED BOOK. You are allowed to have two note cards or one page of notebook paper with formulas only. Calculators & laptops are
allowed ONLY for computation and graphing. (1) All FINAL answers must be presented clearly and inside a box. TEN PERCENT
of the midterm grade is for writing in a coherent, logical and organized manner.
(2) The time is 75 minutes for wOrking on the test. ' (3) All functions and vector ﬁelds are assumed to be differentiable everywhere,
unless otherwise stated. (4) Problems 19 have total of 100 points. Problem 10 is extra credit and the
points will be added separately to improve Homework/Quiz final scores. (***) We’re] on honor s stem for the students to use onl their own work and
without askin from or offerin he] to other students. Please Observe the Honor em
Time: 75 minutes YOUR NAME: TA NAME:
SECTION: Problem 1. Suppose that an object P is moving so that its position vector at time t is
given by X(t) = (t+e',t——e",t2).
(i) Find the velocity V(t) and the acceleration vectors A(t) ofP at t = 1. (ii) Now considerthe curve described by the velocity V(t). Find the curvature of it; ‘ curve V(t) at t = 1. IO Problem 2. (a) Use the change of variable x = e“ sin 31 , y = es cos 3t , z = u and compute the function f (x, y, z) = x2 + y2 + z2 in terms of the variables (s, t, u) in its simplest form. Call this
function F (s, t, u) . (b) Find the gradient of F (s, t, u) with respect to (s,t, u) . Problem 3. (a) Find the equatibn of the tangent plane and thevnormal line to the surface S
given by x2y'7+22=.c2 at P = (2,2, 0) for any value of c. (b) Consider the parametric formula for the normal line L(t). For each value of c, we denote the normal by L(t, c) that could be regarded
as a function of two variables (I, c) . Find the general equation of the tangent plane to the
surface given by the parametric formula L( t, c). Problem 4. Consider the level surface S(u) given by the formula xzy2 — z = u for the function
F (x, y,z) = xzy2 — z . (1') Find the gradient vector of F(x, y, 2) at a general point P = (x, y,z) in
terms of the values of u. (ii) Write the parametric equation of the line L(t) that passes through
P and is normal to the surface S(u) at P. (iii) Find the unit vector X parallel to (2,2,2). (iv) Find the directional derivative DXF (x, y, 2) at all points P that results from the intersection of the z
axis and the level surface 5(4) . ' LII Problem 5. (i) Write down the parametric equation for the sphere S centered (3,4,0) and
radius 5. (ii) Use S for writing the spherical coordinates for the points on the surface of S that
are situated above the (x , y)plane, and describe this region given inequalities for the spherical
angles. V [[ Hint: The xcoordinate for the parametric equation for the sphere with center at (12, 14, 10)
and radius 17 is x = 17cos¢cosw+12. Use the two parameters (¢ , 1p) in your solution.]] Problem 6. For f(x, y) = x2 312 (i) Find the equation of its level curve that goes through the point (5,4) in its domain; (ii) Find the gradient vector Vf at (5,4). _ (iii) Find the directional derivative offat (5,4) in the direction of maximal
change. ' (iv) At (5,4), ﬁnd a unit vector X such that DXf(x, y) = 0. Problem 7. A solid V is the volume cut off from a spherical shell 8 by a cone K, such
that S is centered at the origin and it has the inner radius R1 = 9 and the outer radius .
R2 =10. The cone K is obtained from a 360 degree rotation of the triangle with
vertices (0,0,0) , (0, 0,10) , (0,10,10) about the zaxis. Describe the region of the space that belongs to the solid V by writing an appropriate set of inequalities (and equations,
when appropriate). Problem 8. The curve C in R3 given by the parameterization by arclength s in the interval [mg] via: X (s) = (0: cos 5,0: sin s,s2) where a is a suitably chosen constants. (a) Find the unit tangent vector T(s) and the unit normal vector N(s) as functions of the parameter s for the
curve C. (b) Does there exist a nonzero value of (1 such that at some point Q within the .717 , .
interval (0,—2—) the absolute values of the curvature and the tor51on of C are equal? If the answer is yes, ﬁnd a and use that to explicitly determine the coordinates of the point (or points) Q by the vector X(s) = (0: cos s,a sin s,s2). Remark: The binormal B(s) is deﬁned by the cross product of the tangent and the normal vectors. The torsion of
the curve is the rate of change of the binormal with respect to the arclength. Problem 9. A function F (t, s , u) of three variables (I, s , u) satisﬁes Laplace's Equation if:
62F 62F 62F _ ‘
aﬁ as2 au2 Such a function F (t, s , u) is called harmonic. Determine if the function deﬁned below is
harmonic or not by explicit calculation of the Laplace Equation: F(s,t,u) = s3t —1fs3 + loge(s2 + t2 + uz). O + + 10 Problem 10. (EXTRA CREDIT PROBLEM). Please do NOT attempt this problem before
you are DONE with regular problems 19. Find the indicated limits or demonstrate that it does not exist. Note: the origin (0,0) is shown by notation O, and the lirn it must be computed for (x,y)9(0,0). 2
x2+y lim ; (ii) lim
(mu>0 x+y (x,y)*0 xy ll ...
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 Spring '08
 DICKEY
 Multivariable Calculus

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