Instructions: Good luck! Math 21a First Midterm Exam Spring, 2009

Transcription

1 Math 21a First Midterm Eam Spring, 2009 Your Name Your Signature Instructions: Please begin b printing and signing our name in the boes above and b checking our section in the bo to the right You are allowed 2 hours (120 minutes) for this eam Please pace ourself accordingl You ma not use an calculator, notes, or other assistance on this eam In order to receive full credit, ou must show our work and carefull justif our answers The correct answer without an work will receive little or no credit If ou need more room, use the backs of the pages and indicate to the grader that ou have done so Please write neatl Illegible answers will be assumed to be incorrect Raise our hand if ou have a question Good luck! MWF 9 MWF 10 MWF 11 MWF 12 TTh 10 John Hall Janet Chen Peter Garfield Peter Garfield Jun Yin Problem Total Points Score Problem Total Points Score Total 100

2 Math 21a First Midterm Eam Spring, (8 points) (a) (4 points) Find an equation for the plane containing the three points P (3, 3, 1), Q(2, 1, 0), and R( 1, 3, 1) (b) (4 points) Are the four points P, Q, R, and S(7, 4, 1) coplanar? (Here P, Q, and R are the points from part (a)) Justif our answer

3 Math 21a First Midterm Eam Spring, (12 points) (a) (4 points) Find an equation for the plane given b the parameteriation r(u, v) = 3 + 2u, 5 u + v, 2u + 3v (b) (3 points) Suppose the curve C is parameteried with respect to arc length b r(t) (that is, this parameteriation has r (t) = 1 for all t) What is the distance along C between r(3) and r(10)? (c) (2 points) Suppose the traces of a quadric surface are parabolas ( = k), parabolas ( = k), and hperbolas ( = k) What quadric surface is this? Eplain our reasoning (d) (3 points) An ant is standing on the surface = e at the point (1, 0, 2) If the ant walks East (that is, in the positive direction), is he moving up or down? Eplain our reasoning

4 Math 21a First Midterm Eam Spring, (12 points) Consider the curve C parameteried b r(t) = t cos t, t sin t, t This curve wraps counterclockwise around the cone 2 = 2 + 2, as shown in the pictures below (a) (2 points) Show that C is smooth everwhere (That is, show r (t) 0 for an value of t) (b) (3 points) Give an intuitive reason wh the curvature of C should go to ero as the curve winds up the cone (c) (4 points) Compute κ(0), the curvature of the curve C at t = 0 You ma assume an of the formulas for curvature: κ = dt ds = T (t) r (t) = r (t) r (t) r (t) 3 (d) (3 points) Find an equation for the osculating plane to C at the origin

5 Math 21a First Midterm Eam Spring, (8 points) Let C be the intersection of the surfaces = 2 and = 2, as shown in the pictures below (a) (5 points) Find a parameteriation of C (b) (3 points) Write down the integral that represents the distance along the curve C between the point (1, 1, 1) and the point ( 1, 1, 1) You do not need to evaluate this integral!

6 Math 21a First Midterm Eam Spring, (9 points) Consider the solid described b the inequalities 0 6 and The surface of this solid consists of three pieces: a clinder, and two disks (a) (5 points) Find a parameteriation of each piece of the surface Give bounds on each parameter (b) (4 points) Draw in the grid lines on the surfaces below corresponding to the parameteriations ou found in part (a) = 0 = 6

7 Math 21a First Midterm Eam Spring, (12 points) (a) (4 points) Let L be the line given parametricall b = 4 + t, = 1 2t, = 5 + t Find the point on the line L which is closest to ( 2, 2, 1) (b) (4 points) Find the point on the plane 2 3 = 7 which is closest to the point (7, 2, 1) (c) (4 points) Find the point on the sphere ( 7) 2 + ( + 2) 2 + ( + 1) 2 = 16 which is closest to the plane 2 3 = 7

8 First Midterm Eam Math 21a 7 Spring, 2009 (10 points) Pick the picture that each equation describes, and mark our answers in the space indicated below (a) = cos( ) (b) 2 2 = 0 (d) = 0 (e) = 1 (c) = 1 (A) (B) (C) (D) (E) (F) (G) (H) (I) Mark our answers here: (a) (b) (d) (e) (c)

9 Math 21a First Midterm Eam Spring, (9 points) (a) (2 points) Which one of the following is the same as φ = π 6 (i) = in Cartesian coordinates (ii) = 3r in clindrical coordinates (iii) = r in clindrical coordinates (iv) 2 = 3( ) in Cartesian coordinates (v) None of the above in spherical coordinates? (b) (2 points) Which one of the following is a picture of the surface defined in clindrical coordinates b = r and 0 r 1? (i) (ii) (iii) (iv) (c) (2 points) Let U be the solid bounded below b = and above b = 2 Which one of the following is a description of U? (i) r 2 2 r 2 in clindrical coordinates (ii) ρ 2, φ π 4 in spherical coordinates (iii) r 2 2 r 2 in clindrical coordinates (iv) sin φ ρ 2 in spherical coordinates (v) None of the above (d) (3 points) Parameterie the surface described in spherical coordinates b θ = φ

10 Math 21a First Midterm Eam Spring, (10 points) Let A = (0, 0, 1) and B = (0, 2, 3) Find the set of points P (,, ) such that AP is orthogonal to BP Give a geometric description

11 Math 21a First Midterm Eam Spring, (10 points) Suppose a and b are vectors about which we know: a = 3, b = 2, and a b = 1, 5, 1 Find the following quantities, if possible If ou cannot find a particular value because there is not enough information, indicate this (a) (2 points) a b (b) (2 points) a b (c) (2 points) The acute angle between a line in the direction of a and a line in the direction of b (d) (2 points) proj a b (e) (2 points) An equation of the plane through the origin parallel to both a and b