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c Math 151 WIR, Spring 2010, Benjamin Aurispa Math 151 Week in Review 2 Sections 1.2, 1.3, 2.2 1. Find a · b for the following. (a) |a| = 9, |b| = 8, and the angle between a and b is 120◦ (b) a = −3i − 7j, b = 2i − 8j 2. Given the vectors a =< 4, 1 > and b =< −2, 4 >, calculate the following. (a) The scalar projection of b onto a (b) The vector projection of a onto b 3. A wheelbarrow is pulled up a 15-meter ramp with a force of 10 N applied at an angle of 25◦ to the ramp. (a) How much work is done? (b) If the force is applied horizontally and the work done is 140 J, at what angle is the ramp inclined? 4. A force given by the vector F = −3i + 4j is used to move an object from the point (2, 3) to the point (−2, 9). How much work is done if the force is measured in pounds and the distance is measured in feet? 5. Find the angle between the vectors a =< −2, −7 > and b =< 3, −1 >. 6. Find a unit vector orthogonal to the vector < −6, −5 >. 7. Find the value(s) of x so that the vectors 2xi − xj and 6xi + j are orthogonal. 8. Find the distance from the point (1, 5) to the line y = −3x − 4. 9. Eliminate the parameter to create a Cartesian equation and sketch a graph for the following parametric equations. Indicate the direction traveled as the parameter increases. √ (a) x = t + 1, y = t − 3 (b) x = −2 sin θ, y = 5 cos θ (c) x = 3 cos t − 2, y = −3 sin t, 0 ≤ t ≤ π (d) x = sin t, y = − cos2 t 10. The motion of a particle after t seconds is given by the vector function r(t) =< t − 3, 2 − t3 >. (a) Find the position of the particle at time t = 2. (b) When does the particle pass through the point (1, −64)? (c) Eliminate the parameter to find a Cartesian equation for the path of the object and sketch a graph. 11. Find parametric equations of the line passing through the points (−5, 2) and (1, 3). 12. Find a vector equation of the line passing through the point (−2, 4) and perpendicular to the line with vector equation r(t) =< 1, 1 > +t < −3, 5 >. 1 c Math 151 WIR, Spring 2010, Benjamin Aurispa 13. Determine if the following pairs of lines are parallel, perpendicular, or neither. If not parallel, find the point of intersection. (a) L1 : r1 (t) =< 3 + t, −5 + 3t >; (b) L1 : r1 (t) = (1 − t)i + (3 + 4t)j; L2 : r2 (s) =< 1 − 2s, −2 − 6s > L2 : r2 (s) = (−2 + 8s)i + (4 + 2s)j 14. Given the graph of f , find the following: (a) lim f (x) 6 x→1 (b) (c) lim f (x) x→−2+ 4 lim f (x) x→−4− 2 (d) lim f (x) x→3 (e) lim f (x) −6 x→5 (f) The equations of any vertical asymptotes −4 2 −2 4 6 −2 −4 −6 15. Find all vertical asymptotes of the function f (x) = the graph at each asymptote. 16. Calculate the following limits. x2 (x − 2) x+3 x→−3+ x−1 (b) lim x→2 (x − 2)2 x−2 (c) lim x→0 x(x − 1)3 (a) lim 2 x2 + 4x + 3 and describe the behavior of (x + 3)(x2 − 5x + 4)