Review Sheet 3 Math H162

[n]
means the topic is found on acetate n.

**Bold face for material beyond Midterm 1
that could be on Midterm 2**.

*Italics for material beyond Midterms 1 and
2 that could be on Midterm 3.*

*Bold italics for material beyond the
midterms that could be on the final exam.*

*Two items marked for proofs on midterm 3,*

*And two for the final exam.*

** **

**Tools**

Integration
techniques, including integration by parts

Least
Upper Bound axiom for the real number system

Triangle
inequality [9.1]

L'Hopital's
rule [25.0]

*Graphing
curves and surfaces in three-dimensional space [128, folders 0302 and 0303]*

*Cylindrical
and spherical coordinates [folder 0301]*

** **

**Definitions** (know them and be able to write them correctly)

Infinite series [2]

Infinite sequence [2]

Limit of a sequence [8.1 – 8.3]

Partial sums of a series [3, 6.1, 7]

Convergent series [6.1, 7]

Properties of sequences [9.2, 16 - 17]

Properties of series [19, 23.0, 24, 34 – 35, 53, 54.0, 55 - 59]

Divergent series [6.2, 7]

Absolutely convergent series [51]

Conditionally convergent series [51]

Sum of a series [3, 6.1, 7, 23.0]

**Power series [60 – 61]**

**Properties of power series [63.0 – 63.1]**

**Radius of convergence of a power series [64] **

**Taylor's formula [74 – 75]**

**Taylor polynomial [75]**

**Remainder in Taylor's formula
[75, 76, 79]**

**Polar coordinates [90.0-91]**

**Parametric equations [97-102.1]**

**Parameter
[97]**

**Vector [104]**

**Geometric representations of vectors [105]**

**Vector addition and subtraction [106.0-106.1]**

**Scalar multiplication [107]**

**Length of a vector [111.0]**

**Unit vector [111.0-111.1**

**Standard unit vectors [111.0]**

**Tangent or velocity vector [113-114]**

*Right-handed Cartesian coordinates in three
dimensions [125-126.1]*

*Vectors in three dimensions [129]*

*Standard unit vectors in three dimensions [131]*

*Cross-product of three-dimensional vectors [133]*

*Cones [158.0-158.1]*

*Conics [159-164.3]*

*General cylinders [165-167.2]*

*General cones [169.0-170]*

****Cylindrical and spherical coordinates
[folder 0301]** —theory for the
final exam, not midterm 3*

* *

** **

**Examples**

** **

** I**nfinite series
[2, 55 - 59]

Infinite
sequences [2]

Telescoping
series [4]

Power
series [bottom of 5]

Convergent
series [4, 45]

Divergent
series [6.2]

Harmonic
series [10]

Geometric
series [11]

p-series
[25.1, 31]

**power
series [5, 62.0]**

** **

** Taylor's
series [80-85.1]**

** **

**Theorems--with
proofs**

Adding
sequences [9.1 – 9.2]

Convergence and
divergence of geometric series, depending on the ratio [11, 40.8]

If an increasing
infinite sequence has an upper bound, then it converges [12.1 – 14]

If a series is
absolutely convergent then it is convergent. [52]

********Geometric proofs using vectors [144-146.3]—theory
for midterm 3
*

**Theorems--no
proofs** (know what they mean and be able to
state and use them correctly)

Theorems on
subtraction of sequences and multiplication and division of sequences, both by
a constant and by another sequence
[9.2]

Comparison test [20,
22, 25.1, 30]

p-series [25.1]

Limit comparison
test [26 - 30]

Integral test [37.0 – 37.2]

Ratio test [41 - 42]

Root test [46 – 47.2]

Alternating series
test [49]

**Radius of
convergence of a power series [64
- 67]**

**Taylor's
formula [74-79]**

** **

**Facts**

**If the power
series a**_{0}** + a**_{1}**x
+ a**_{2}**x**^{2}**
+ a**_{3}**x**^{3}**
+ … has a nonzero radius of convergence
R and f(x) is the function with domain all x such
that | x | < R, f(x) being defined as the sum of the power series for x
in that domain, then**

** (1) f(x) is continuous (at every x in its
domain)**

** (2) f(x) has a derivative f'(x), which is the sum of the power
series a**_{1}** + 2a**_{2}**x
+ 3a**_{3}**x**^{2}**
+**

** (3) f(x) has an indefinite integral which
is given as the sum of the power series
c + a**_{0}**x + (a**_{1}**/2)x**^{2}**
+ (a**_{2}**/3)x**^{3}**
+ (a**_{3}**/4)x**^{4}**
+ …**

** (4) The radius of convergence for the power
series in (2) and (3) is the same as that of the original power series. [70-71]**

**Connection
between parametric equations and vector-valued functions [112]**

**The tangent
vector at a point on a smooth curve is tangent to the curve at the point [113]**

**Laws of
arithmetic that apply to vectors [104, 107, 108]**

**Laws of
arithmetic that apply to the dot product
[118]**

**Triangle
inequality for vectors [111.0]**

*Parametric
equations for uniform circular motion
[120]*

**Calculations**

Sum of a finite geometric series

Partial fractions

Telescoping
series [4]

**Radius of
convergence of a power series [68
– 69]**

**Interval of
convergence of a power series [68 – 69]**

**Differentiating
and integrating power series
[86-87]**

**Multiplying and
dividing power series [88, 89.1, 89.2] **

**Changing between
polar and Cartesian coordinates [90.0, 92-93.0]**

**Graphing in polar
coordinates [93.0-94]**

**Arc length in
polar coordinates [95.0-95.3]**

**Area in polar
coordinates [96.0=96.2]**

** Positions
of points along a straight line
[109, 110]**

** Writing
a vector in terms of standard unit vectors [111.1]**

** Tangent
(velocity) vectors [114, ***120-121***]**

** Acceleration
vectors [115, ***120-121***]**

** Angle
between two vectors [116-117]**

*Nonzero vectors
are perpendicular if and only if their dot product is 0. [119]*

*Component of the
acceleration vector that is perpendicular to the velocity vector. [124-125]*

*Locating points
in three-dimensional Cartesian coordinates [127.0-127.1]*

*Vector operations
in three dimensions*

* Addition
[129]*

* Scalar
multiplication [129]*

* Dot
product [129]*

*Cross-product [133-135.1]*

* *

*3-by-3
determinants [135.1]
*

*Lines and planes
in three-dimensional space [136-144]*

*Curves in three
dimensions using parametric equations [147-148]*

*Arc-length for
curves in three dimensions [149-152]*

********More arc-length [153-154]—theory for midterm 3*

*Tangent and
velocity vectors in three dimensions [155]*

*Curves given by
two equations in three variables [156-157.2]*

*Normals to planes
[157.1-157.2]*

*Cones and conics
[158-164.3]*

******Parametric equation for general cylinders (two
parameters) [167.2] —theory for the final exam, not midterm 3*

*Right circular
cylinders [168] *

*Parametric
equation for general cones (two parameters) [169.4]*

*Right circular
cone [170]*

* *

**Notation**

Notation used in writing down limits and series [8.1]

**Summation notation using capital
sigma [throughout]**

** **

**Vectors [105]**

** **

**Tangent (velocity) vector
[115]**

** **

**Acceleration vectors [115]**

** **

**Dot product [116-118]**

** **

*Standard unit vectors i, j, k [131]*

** **