Exam_1_20C

# Exam_1_20C - -plane with the parametric equations C x = x t...

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Math 20C (Shenk). Summer, 2010. Exam 1. Name Section Work alone and use no books, notes, or calculators. Show your work with your answers on 8.5” × 11” paper and staple the pages to the exam when you turn them in. Problem 1 (10%) Find the constant k such that the vectors A = a 2 , 1 , 3 A and B = a− 1 , 4 , k A are perpendicular. Problem 2 (10%) Find the angle θ (0 θ π ) between the vectors C = a 1 , 2 , 2 A and D = a 1 , 2 , 2 A . Problem 3 (10%) What is the vertex R opposite P in the parallelogram PQRS if P = (1 , 1 , 1) , Q = (3 , 3 , 3) and S = ( 1 , 4 , 2)? Problem 4 (10%) Give parametric equations of the line through (3 , 1 , 2) and perpendicular to the plane x y + 2 z = 5. Problem 5 (10%) Give parametric equations of the line through the points P = (1 , 4 , 2) and Q = ( 2 , 3 , 0). Problem 6 (10%) Give an equation of the plane through the points P (2 , 3 , 2) , Q (4 , 3 , 5), and R (2 , 6 , 4). Problem 7 (10%) Calculate the scalar triple product, a 1 , 2 , 3 A · [ a 0 , 1 , 5 A × a 4 , 0 , 1 A ]. Problem 8 (10%) Draw the curve in an xy
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Unformatted text preview: -plane with the parametric equations C : x = x ( t ) , y = y ( t ) , ≤ t ≤ 4, where the graphs of x = x ( t ) and y = y ( t ) are in Figures 1 and 2. Show the curve’s orientation. t 1 2 3 4 x 1 2 3 x = x ( t ) t 1 2 3 4 y 1 2 3 y = y ( t ) x − 3 y 3 FIGURE 1 FIGURE 2 FIGURE 3 Problem 9 An object is at x = t − t 2 (miles) , y = t 2 + t (miles) in an xy-plane at time t (hours) for − 2 . 1 ≤ t ≤ 2 . 1. Its path is in Figure 3. (a) (10%) Show that the object’s speed is √ 2 + 8 t 2 miles per hour at time t . (b) (10%) Find the object’s velocity vector at the point where its speed is a minimum and draw it with the curve. Use the scales on the coordinate axes to measure the components of the vector. Scores: 1 2 3 4 5 6 7 8 9 Total 39...
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