Math_20C_Review

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

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Unformatted text preview: (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 , , − 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) ,Q = (3 , 1 ,- 2), and R = (1 , 4 , 0)? Answer: −→ PQ × −→ PR = < 3 , 1 , − 2 > × < 1 , 4 , > = < 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) . =- . 42 and sin(- 2) . =- . 91.) x- 3- 2- 1 1 2 3 y 3 FIGURE 1 1 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....
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Math_20C_Review - Math 20C Summer 2010 Review Example 1 The...

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