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Transcript
Fall 2013 Math 151
Exam I Review
courtesy: Amy Austin
(covering sections 1.1-3.1)
I will work a selection of these problems during class
Thursday Sept 26. Come to class prepared to ask questions over this set.
Section 1.1
1. For the following vectors, illustrate a + b, a − b and
b−a
9. A woman exerts a horizontal force of 25 pounds as
she pushes a crate up a ramp that is 10 feet long
and inclined at an angle of 20◦ above the horizontal.
Find the work done.
Section 1.3
10. Sketch the graph of the vector function
r(t) = 3 cos(2t)i + 5 sin(2t)j.
11. Find parametric equations of the line through the
points (2,-3) and (-4,5).
12. Find a cartesian
equation for the parametric equa√
tions x = t + 2, y = 2t − 1
13. Given the line (2 + 3t)i + (6 − 4t)j , find:
a.) A cartesian equation of the line.
b
a
b.) A vector perpendicular to the line.
14. Find the intersection of the lines
2. Given a = −3i − 5j and b = −4i + 2j , compute
| − 2a + 3b|.
~r(t) = h−4 + 2t, 5 + ti and
~s(w) = h2 + 3w, 4 − 6wi.
3. Given a and b above, find a unit vector orthogonal Section 2.2
to a − b.

2−x
if x < −1



4. An object on the ground is pulled by two forces F1
x + 4 if − 1 ≤ x < 1
15. f (x) =
and F2 . If |F1 | is 8 pounds with direction due east
4
if x = 1



and |F2 | is 20 pounds with direction N 60◦ E, find
4−x
if x > 1
the magnitude of the resultant force.
a.) Sketch the graph of f (x)
Section 1.2
5. Given the points A(1, 4), B(−1, 2) and C(3, 0), find
the three angles of ∆ABC.
6. Find the value of x such that the vector from
P (−4, 2) to Q(2, 1) is parallel to the vector from
R(9, x − 4) to S(6, 2 − x).
7. Find the vector and scalar projections of h3, 2i onto
h1, 4i. Sketch the vector projection.
8. A constant force with vector representation
F = 10i + 18j moves an object from the point (2, 3)
to the point (4, 9). Find the work done if the distance is measured in meters and the magnitude of
the force is measured in Newtons.
b.) Find the following limits:
lim f (x);
x→−1−
lim f (x);
x→1−
lim f (x);
x→−1+
lim f (x);
x→−1
lim f (x); lim f (x)
x→1+
x→1
c.) (question from section 3.1) Where is f (x) not
differentible?
16. Determine the limit or prove it does not exist:
x(4 − x)
lim
x→3 (x − 3)3
17. Find the vertical and horizontal asymptotes for
x3
6x
and
g(x)
=
f (x) =
3x − x2
x2 + 1
Sections 2.3 and 2.6
27. Given f (x) =
x2 − x − 2
x→−1 x2 + 13x + 12
1
4
19. lim
+ 2
x→−2 x + 2
x −4
lim r(t) where r(t) =
t→−∞
b.) What is the equation of the tangent line at the
point (3, 2)?
3t
t2 − 4
,√
2
3t − 6t + 3
t2 − 4t
|x|
21. Find lim 2
or prove it does not exist.
x→0− x − x
√
22. lim (x − x2 + 3x + 1)
x→∞
Section 2.5




x+1
a.) Using the limit definition, find the derivative of
f (x).
18. lim
20.
√
x2
if x < 0
−4
23. f (x) =
if 0 ≤ x < 2


 x3− 2
x −4
if x ≥ 2
x2
a.) Show f (x) is continuous at x = 2 or explain
why it is discontinuous.
b.) Show f (x) is continuous at x = 0 or explain
why it is discontinuous.
24. Use the Intermediate Value Theorem to prove there
is a solution to the equation
x3 + 2x + 1 = 0

 3x3 − 2x + 2a if x < 1
25. f (x) =
5
if x = 1

3ax − 1
if x > 1
(i) Find the value of a that makes lim exist.
x→1
(ii) Find the value of a that makes f (x) continuous
at x = 1, if possible. If it is not possible, be sure to
support your answer.
Sections 2.7 and 3.1
26. A ball is thrown into the air. The height of the ball
after t seconds is given by h(t) = 2t − 4t2 .
a.) Find the average velocity of the ball from t = 2
to t = 3.
b.) Using the limit definition, find the instantaneous velocity of the ball at t = 2 seconds.
28. Compute the derivative of f (x) =
definition of the derivative.
x
, using the
x+1
29. Where is f (x) = |2x − 3| not differentiable?