Fall 2010 Math 152
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Section 11.1
Week in Review XII
Key Concepts:
courtesy: David J. Manuel
(covering 10.9, 11.1 and 11.2)
1. Regions in three-dimensional space
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2. Distance
p formula:
d = (∆x)2 + (∆y)2 + (∆z)2
Section 10.9
3. Equation of a sphere: (x − x0 )2 + (y − y0 )2 +
(z − z0 )2 = r2
Key Concepts:
4. Midpoint formula:
x2 + x1 y2 + y1 z2 + z1
,
,
2
2
2
1. The N th degree Taylor Polynomial of
f at x = a is given by TN (x) =
N
X
f (n) (a)
(x − a)n (Basically, it is the N th
n!
Examples:
n=0
partial sum of the Taylor Series!)
2. If RN (x) = |f (x) − TN (x)|, we can estimate
RN (x) on a given interval one of three ways:
1. The points (2, −1, 0) and (−3, 4, 2) are opposite vertices of the diagonal of a prism.
Find the coordinates of the other 6 vertices
and find the volume of the prism.
(a) Graphical (Matlab)
(b) If Alternating, use |s − sN | ≤ |aN +1 |
2. Find the center and radius of the sphere
whose equation is x2 + y 2 + z 2 − 8x + 4y +
10z + 9 = 0.
(c) Otherwise,
use Taylor’s Inequality
M
|x − a|n+1
|Rn (x)| ≤
(n + 1)!
3. Classify ABC as either a triangle (right,
isosceles, both, or neither) or collinear.
Examples:
(a) A(5, 0, −3), B(2, −1, 4), C(6, 4, −2)
1. Find the fourth-degree Taylor Polynomial of
1
f (x) =
centered at x = 0. Estimate
4−x
the error in using this approximation on the
interval [−2, 2].
(b) A(1, −1, 5), B(−2, 4, 3), C(4, −6, 7)
4. Describe the region of R3 (three-dimensional
space) represented by the following equations:
2. Find the
√ third-degree Taylor Polynomial of
f (x) = x centered at x = 1.
(a) y = 3x + 1
3. Use a seventh-degree
ˆ x Taylor Polynomial to
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determine f (x) =
e−t /2 dt. What degree
(b) x2 + z 2 = 9
0
polynomial is required to estimate f (1) correct to within 0.001?
5. Find the equation of the set of points equidistant from the points (3, 2, −1) and (0, −2, 4).
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Section 11.2
Key Concepts:
1. Vector Operations (+, −, ·, scalar mult)
p
2. |a| = a21 + a22 + a23
3. ijk notation
4. cos θ =
a·b
|a||b|
5. orthogonal vectors
6. direction angles/direction cosines
a·b
a·b
7. compa b =
a
, proja b =
|a|
|a|2
Examples:
1. Find the magnitude and direction cosines of
the vector 4i + 2j − 3k.
2. The vector a = −i − 2j + 4k is parallel to
the vector from P to Q. If P is the point
(4, −5, 0), find the coordinates of Q.
3. Given the vectors a = h3, −5, 4i, b =
h−2, 1, 7i, and c = h1, −3, −6i, find the following:
(a) a − 4b + 2c
(b) |a + b + c|a
(c) The cosine of the angle between a and c.
(d) The scalar and vector projections of b
onto c.
4. Find the value(s) of x such that hx, x, 3i and
hx, −1, −2i are orthogonal.
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