Review Sheet 2    Math H162

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

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

No proofs on midterm 2.

 

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]

 

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]

 

           

 

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]
 

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]

 

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]

         Acceleration vectors  [115]

         Angle between two vectors  [116-117]
 

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]