Math 151 Week in Review
Monday Sept. 13, 2010
Instructor: Jenn Whitfield
Thanks to Amy Austin for contributing some problems.
All prolbems in this set are copywrited
Section 1.3
• Given a line l, let r~0 be a position vector of
any point on line l and ~v be a vector parallel
to line l. The the vector equation for line l is
given by ~r(t) = ~r0 + t~v .
• The parametric equations of the line that
passes through the point P0 (x0 , y0 ) and is
parallel to the vector ha, bi are given by
x(t) = x0 + at y(t) = y0 + bt
1. Sketch the curve represented by the parametric equations and then eliminate the parameter to find the Cartesian equation of
the curve:
(d) Find an equation in x and y whose
graph is the path of the object.
Section 2.1
6. The point P (1, 2) lies on the curve ~r(t) =
(t2 )~i + (2t)~j for t = 1.
(a) If Q is the point (t2 , 2t), find the se1 −→
cant vector
P Q for each of the
t−1
following values of t:
(i) 1.5, 1.1, 1.01, 1.001
(ii) 0.5, 0.9, 0.99, 0.999
(b) Using the results from part (a), guess
the resulting tangent vector to the
curve at P (1, 2).
(c) Using the vector in part (b), write a
vector equation for the tangent line to
the curve at P (1, 2).
a.) x = 2t − 1, y = 2 − t, −3 ≤ t ≤ 3
b.) x = t2 − 6, y = 2t − 4, −4 ≤ t ≤ 2
c.) r(t) = h2 sin t, 3 cos ti, 0 ≤ t ≤ 2π
2. Find parametric equations and the vector
equation for the lines described below:
Section 2.2
The lim f (x) = L if and only if
x→a
lim f (x) = L and lim f (x) = L
x→a+
x→a−
f(x)
6
(a) The line passes thru the point (−1, 5)
and is parallel to the vector h2, −3i.
4
(b) The line passes thru the points (0, 3)
and (−3, 5).
2
−6
−4
3. Determine whether
L1: r(t) = (−4 + 2t)i + (5 + t)j
L2: r(t) = (2 + 3t)i + (4 − 6t)j
are parallel or perpendicular. If they are not
parallel, find the point of intersection.
4. Find a unit vector perpendicular to the line
described by the parametric equations x =
2t + 1, y = 3t + 5.
5. An object is moving in the xy-plane and its
position after t seconds is:
−→
r(t) = (t2 + 3t)~i + (2t − 4)~j.
−2
2
4
6
8
x
−2
−4
−6
7. Use the graph of f (x) above to compute the
following limits:
a) lim− f (x)
x→0
b) lim+ f (x)
x→0
c) lim f (x)
x→0
(a) Find the position of the object at time
t = 3.
d) lim f (x)
(b) At what time is the object at the point
(40, 6)?
e) lim f (x)
(c) Does the object pass through the point
(20, 2)?
f.) lim f (x)
x→5
x→7+
x→2
8. Sketch agraph of the function

2 − x x < −1


 x
−1 ≤ x < 1
f (x) =

4
x
=1



4−x x>1
Use the graph to find lim f (x) and
x→−1
lim f (x).
x→1
x−1
at x =
x2 − 4
1.7, 1.8, 1.9 1.99, 1.999, 2.3, 2.2, 2.1, 2.01, 2.001, 2.0001
(correct to six decimal places). Use these
results to find lim− f (x), lim+ f (x) and
9. Evaluate
f (x)
=
x→2
x→2
lim f (x).
x→2
10. Find lim
x→5−
6
.
x−5
11. Given f (x) =
x2 + x
x2 + 4x + 3
(a) find the vertical asymptotes for the
function.
(b) find lim f (x).
x→−1
(c) find lim f (x)
x→−3