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Unformatted text preview: Math 32: Calculus III Stewart ET Exam 1: Take Home Lucero: F09 Name:__________________________ Take Home Exam I Vectors, Vector Functions and the Geometry of Space For this take home exam you may use your books, notes, classmates, and me as a reference. You may NOT use tutors, other teachers or friends outside of our course. The in-class exam will be similar and I suggest you attempt ALL of these problems before the in class exam! I will assign “turn in” problems next week due with your in class exam. Please complete the “turn in” problems on a separate sheet of paper. The final draft of this take home exam should be neat, orderly and SHOW ALL YOUR WORK. The final draft should be yours and yours alone. You should identify each problem, show all your work and clearly label your answers. NO LATE PAPERS ACCEPTED. (If you will not be in class
send it in early or with someone else.) 1. Find the equation of a sphere if one of its diameters has endpoints ( 2, −3,5 ) and ( −1, 2,1) . 2. Describe the region in R3 and draw a sketch of the region: a. 0 ≤ z ≤ 3 c. z 2 + x 2 ≤ 25, 0 ≤ y ≤ 10 b. 4 ≤ ( x − 1) + ( y + 2 ) + z 2 ≤ 25
2 2 3. An observation cart holding two people is suspended above a river as shown. If the cart with the two people weighs 250 kg, find the components and the magnitude of the tension in each of the cables shown. 4. Given A ( 0, −5, 2 ) , B ( 3,1, −2 ) and C ( −2,1, 4 ) a. a = AB, b = BC , c = CA b. Find a , a - 2b, b + c c. Find a ⋅ b and the angle between a and b. d. Find b × c and a unit vector that is orthogonal to both a and b 5. A tow truck pulls a stalled car for 12 kilometers, how much work is done by the tow truck if the tension in the cable holding the car is 1250N and the cable is at a 35 with the road. 6. Find two vectors in opposite directions that are orthogonal to the plane through the points A ( 2,1,5) , B ( −1,3, 4 ) , C ( 3, 0, 6 ) . Then find the area of the triangle ΔABC . Revised on 9/17/2009 Math 32: Calculus III Stewart ET Exam 1: Take Home Lucero: F09 7. A bolt is turned by applying a 30 N force to a 25 cm wrench as shown in the diagram; find the magnitude of the torque about the center of the bolt and determine if the torque is pointing into or out of the page. 8. Write the parametric and symmetric equations for the line through ( −3,1,5) that is perpendicular to both i + j and i + k . 9. Determine if the lines defined below are parallel, skew or intersecting. If they intersect find the point of intersection: x −1 y z − 2 L1 : == −4 7 1 x +1 y − 3 z − 7 L2 : = = 2 −3 −5 10. Find the equation of a plane that passes through the point (1, 2,3) and contains the line
x = 3t , y = 1 + t , z = 2 − t . 11. Determine if the following two planes are parallel, perpendicular or neither. If neither find the angle between the planes and the parametric equations of the line of intersection. 2 x − 3 y + z = 5 and x + y − 3 z = 10 For the problems 12—14: Go to the class website to match the equations to graphs of the surfaces on the link: DP Graphs from TH Exam 1 12. Describe and sketch the surfaces: a. y −
12 x =0 2 b. z = sin y c. z = ln x 13. Draw the traces of the following surfaces in each of the coordinate planes. Sketch the surface and identify the surface as one of the quadric surfaces. 1 b. 4 x 2 + 4 z 2 − 9 y 2 = 36 c. 4 x 2 + 16 y 2 + z 2 = 16 a. x = y 2 + z 2 2 14. Reduce the equation to one of its standard forms, classify the surface and sketch it. a. x 2 + y 2 − z 2 − 2 x + 4 y + 6 z = 4 b. x 2 − 4 y 2 − z 2 − 4 x + 8 y − 10 z = 29 15. For the vector function r ( t ) = tan −1 ( t 2 − t 4 ) , et + t a. ( ) 1 t , 3+ t 1 + 9t 2 find the following: lim r ( t )
t →∞ b. r′ ( t ) Revised on 9/17/2009 Math 32: Calculus III Stewart ET Exam 1: Take Home Lucero: F09 16. Sketch the curve with the vector equation r ( t ) = sin 2t , t , cos 2t 17. If r ( t ) = t , e 2t , t 3 find r′ ( t ) , r′′ ( t ) , T ( t ) , T ( 0 ) , r′ ( t ) ⋅ r′′ ( t ) , r′ ( t ) × r′′ ( t ) ⎛ ⎞ et t2 i + t 3e 2 t j + k ⎟dt . 18. Find ∫ ⎜ 2t t 16 − t 2 ⎠ ⎝ e − 2e − 3
19. Find the arc length of the helix defined by r ( t ) = ( sin 2t ) i + ( t ) j + ( cos 2t ) k from t = 0 to t = 2π . 20. Find the curvature of 15t , et , e −t at ( 0,1,1) 21. A bug moves with position function t , 2 cos t ,sin t find velocity, speed and acceleration of the particle. Sketch the path of the bug and draw the velocity and acceleration vectors for t=0. 22. A batter hits a baseball 3 ft above the ground toward the center field fence, which is 10 ft high and 400 ft from home plate. The ball leaves the bat with speed 115 ft/s at an angle 50 above the horizontal. Is it a home run? (Does the ball clear the fence?) Revised on 9/17/2009 ...
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