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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
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
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 >.
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
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.
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
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
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
(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.
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
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 >
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
(b) L1 : r1 (t) = (1 − t)i + (3 + 4t)j;
L2 : r2 (s) = (−2 + 8s)i + (4 + 2s)j
14. Given the graph of f , find the following:
(a) lim f (x)
x→1
6
(b)
(c)
lim f (x)
4
x→−2+
2
lim f (x)
x→−4−
−6
−4
2
−2
(d) lim f (x)
x→3
−2
(e) lim f (x)
−4
x→5
−6
(f) The equations of any vertical asymptotes
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4
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
15. Find all vertical asymptotes of the function f (x) =
the graph at each asymptote.
16. Calculate the following limits.
(a)
lim
x→−3+
(b) lim
x→2
(c) lim
x→0
x2 (x − 2)
x+3
x−1
(x − 2)2
x−2
x(x − 1)3
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x2 + 4x + 3
and describe the behavior of
(x + 3)(x2 − 5x + 4)