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c Math 151 WIR, Spring 2014, 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. How much work is done? 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. A triangle has vertices A(−3, 2), B(4, 5), and C(2, −4). Using vectors, find the angle at vertex A. 6. Find a unit vector orthogonal to the vector < −6, −5 >. 7. Find the value(s) of x so that the vectors 2xi + 4j and 6xi − 6j are orthogonal. 8. Find the distance from the point (1, 3) 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 of motion as the parameter increases. √ (a) x = t + 1, y = t − 3 (b) x = −2 sin θ, y = 5 cos θ (c) x = tan t, y = cot2 t, π 6 ≤t≤ π 3 10. The motion of a particle after t seconds, where t ≥ 0, 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, if ever, does the particle pass through the point (1, 1)? (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 >. 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 >; L2 : r2 (s) =< 1 − 2s, −2 − 6s > (b) L1 : r1 (t) = (7 − t)i + (−15 + 2t)j; L2 : r2 (s) = (6 + 3s)i + (−11 − 4s)j 1 c Math 151 WIR, Spring 2014, Benjamin Aurispa 14. Given the graph of f , find the following: (a) lim f (x) 6 x→1 (b) (c) lim f (x) 4 x→−2+ lim f (x) 2 x→−4− (d) lim− f (x) x→3 −6 (e) lim f (x) x→3+ −4 2 −2 −2 (f) lim f (x) x→3 −4 (g) lim f (x) x→5 (h) The equations of any vertical asymptotes: −6 15. (a) Find all vertical asymptotes of the function f (x) = (b) Calculate lim f (x) x→4− 16. Calculate the following limits. x2 (x − 2) x+3 x→−3− x2 + 5 (b) lim x→5+ (x − 7)(x − 5) 2−x (c) lim x→0 x(x + 1)3 x−3 (d) lim x→−4 (x + 4)2 (a) lim 2 x2 + 4x + 3 . (x + 3)(x2 − 5x + 4) 4 6