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

 

         Infinite 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  a0 + a1x + a2x2 + a3x3 + … 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 a1 + 2a2x + 3a3x2 +

         (3)  f(x) has an indefinite integral which is given as the sum of the power series  c + a0x +  (a1/2)x2 + (a2/3)x3 + (a3/4)x4 + …

         (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]