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Math_20C_Review

Math_20C_Review - Math 20C Summer 2010 Review Example 1 The...

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(8/31/10) Math 20C. Summer 2010. Review. Example 1 The three forces, F 1 = < 2 , 1 , 2 >, F 2 = < 1 , 1 , 1 > , and F 3 = < - 2 , - 3 , - 1 > , measured in pounds, are applied at the same point on an object. What is the magnitude of the combined force? Answer: F 1 + F 2 + F 3 = < 1 , 1 , 2 > [Magnitude] = 1 2 + 1 2 + 2 2 = 6 pounds Example 2 Which angle in the triangle with vertices P = (1 , 1 , 1) , Q = (3 , 4 , 2), and R = ( - 1 , 2 , 2) is the right angle? Answer: The angle at P is the right angle because the dot product of −→ PQ = < 2 , 3 , 1 > and −→ PR = < 2 , 1 , 1 > is zero. Example 3 Is the smallest positive angle θ between the vectors A = i - j and B = j + k an acute angle, a right angle, or an obtuse angle? Answer: The angle is obtuse, because cos θ = A · B | A || B | = 1 2 2 = 1 2 is negative. Example 4 Give parametric equations of the line that is perpendicular to the surface x 2 + y 3 + z 4 = 3 at (1 , 1 , 1). Answer: ( x 2 + y 3 + z 4 ) = < 2 x, 3 y 2 , 4 z 3 > equals < 2 , 3 , 4 > at (1 , 1 , 1). Line: x = 1 + 2 t y = 1 + 3 t z = 1 + 4 t Example 5 Give an equation of the plane that contains the lines, L 1 : x = 2 + t y = 3 + t z = 4 - t and L 2 : x = 2 + 6 t y = 3 z = 4 - 7 t. Answer: < 1 , 1 , 1 > × < 6 , 0 , 7 > = < 7 , 1 , 6 > Plane: 7( x 2) + ( y 3) 6( z 4) = 0 Example 6 What is the area of the triangle with vertices P = (0 , 0 , 0) , Q = (3 , 1 , - 2), and R = (1 , 4 , 0)? Answer: −→ PQ × −→ PR = < 3 , 1 , 2 > × < 1 , 4 , 0 > = < 8 , 2 , 11 > [Area] = 1 2 | −→ PQ × −→ PR | = 1 2 8 2 + 2 2 + 11 2 = 189 Example 7 The curve C : x = t 2 + t - 2 , y = cos(2 t )+2 , - 2 . 75 t 1 . 75 is shown in Figure 1 Draw the velocity vector at t = - 1, using the scales on the axes to measure its components. (Use the values cos( - 2) . = - 0 . 42 and sin( - 2) . = - 0 . 91.) x - 3 - 2 - 1 1 2 3 y 3 FIGURE 1 1
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Math 20C Review p. 2 Answer: R ( t ) = < t 2 + t 2 , cos(2 t ) + 2 > v = R ( t ) = < 2 t + 1 , 2 sin(2 t ) > R ( 1) = < 2 , cos( 2) + 2 > . = < 2 , 1 . 58 > v ( 1) = < 1 , 2 sin( 2) > . = < 1 , 1 . 82 > Figure A7 x - 3 - 2 - 1 1 2 3 y 3 Figure A7 Example 8 Give a definite integral that equals the length of the curve in Example 7. Do not simplify the integrand or attempt to carry out the integration. Answer: [Length] = integraldisplay 1 . 75 2 , 75 radicalbig (2 t + 1) 2 + ( 2 sin(2 t )) 2 dt Example 9 Figure 2 shows the graph of g ( x, y ) = 10 cos( xy ) 1 + 2 y 2 . Find, without using derivatives, the global maximum of z = g ( x, y ) and the values of ( x, y ) where it occurs. y x z FIGURE 2 Answer: [Global maximum] = 10 along the x -axis (where y = 0)
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p. 3 Math 20C Review Example 10 (a) Draw the level curves of f ( x, y ) = radicalbig x 2 + y 2 where it has the values 1, 2, and 3. (b) Describe the graph of f . Answer: (a) Figure A10 (b) The graph is a circular cone with its vertex at the origin and the positive z -axis as axis. x 4 y 4 Figure A10 Example 11 Explain the shape of the graph of H ( x, y ) = 2 sin( y 2 ) + 8 1 + x 2 + y 2 in Figure 3. y x z FIGURE 3 Answer: z = 2 sin( y 2 ) is independent of x and oscillates increasingly rapidly between 1 and 1 as y moves away from 0, while z = 8 1 + x 2 + y 2 is a circularly symmetric bump with its highest point at x = 0 .y = 0.
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