Final Review Sheet Math H161

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

 

Boldface for items beyond Midterm I that could be on Midterm II.

Italic for items beyond Midterm I and II that could be on Midterm III.(Nov. 25)

Larger Bold Italic for items beyond the Midterms that could be on the Final Exam.  (Dec. 10, 11:30 am – 1:18 pm in our classroom)

 

There are four items of theory marked *** and there will be a question about one of them on the final exam.

The only material  that you are newly responsible for on the final exam is centroids and springs (with Hooke's Law)

         Centroids  [231 – 234]   

Springs and Hooke's Law  [143 - 146, 148.0 – 149, 216 – 217]]

 

 

Tools

Number line  [1, 3]

                        To illustrate individual numbers  [2]

                        To illustrate sets  [2]

Calculator

                        Use in guessing limits [21.8.1]

Factoring out  | x – a|  in doing an epsilon  (e ) delta  (d )  calculation for a limit  [22.2]

           


 

Graphs

            Of continuous and noncontinuous functions  [23.2]

 

Axioms  (know them and be able to write them correctly)

            Least upper bound axiom  [7]

 

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

Upper bound and least upper bound for a nonempty set  [4, 5,6]

Riemann sum for a continuous function on a closed interval and its motivation from finding areas [10 – 11, 13.1 - 14]

            Bounded function on a closed interval  [16]

Geometric series  [20.1]]

            The epsilon  (e ) delta  (d )  definition of limit  [21.6, 21.8.0]

Motivation for the epsilon  (e ) delta  (d )  definition of limit [20.3 -  20.9, 21.4 – 21.5]

Continuous function at  x = a  {21.6 – 21.7, 23.3]

            Definition of definite integral as the limit of Riemann sums  [38]

            Antiderivative [42]

 

            Indefinite integral [42]

 

            ***Natural logarithm [62.1 – 62.3]

 

            ***The number  e  [64]

 

            Functions

 

                  Domain [96]

 

                  Range [96]

 

                  Exponential function [99]

 

                  Inverse functions [102]

 

                  Inverse sine function [130 - 133]

 

                  Inverse tangent function [139 - 140]

 

                  Hyperbolic functions [150]

 

                  Inverse hyperbolic functions [154, 155, 156]

 

         Definition of db  [107]

 

 

Theorems--with proofs

Properties of logarithms [63 - 67]

Fundamental Theorem of Calculus [93.0, 93.1, 95]

 
 

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

***Archimedean property of real numbers  [8, 9]

Existence of the definite integral for continuous functions on a closed interval  [15]

If  f(x)  is a continuous function on a closed interval, then  f(x)  is bounded on that interval  [16 – 20.0]

If the limit of  f(x)  as  x  approaches  a  is  L  and the limit of  g(x)  as  x approaches  a  is  M, then the limit of  f(x) + g(x)  as  x approaches  a  is  L + M  [21.8.3]

For a function continuous on a closed interval, Riemann sums approach a unique limit as the numbers of subdivision points on the interval increases and the length on every subinterval approaches zero.[38]

            Fundamental theorem  of calculus [43]

            Elementary properties of definite integrals [45.1 -  45.4]

***If  F(x)  is an antiderivative of  f(x), then all of the other antiderivatives of  f(x)  are the functions  F(x) + C, where  C  is a constant  [54 – 56.3]

Mean Value Theorem (for derivatives)  [56.0 – 56.2]

 

Calculations

 

            Sum of a finite geometric series  [20.2 – 20.3]

Epsilons and deltas for limits of linear functions  [20.4 – 20.9]

Calculation of indefinite integrals and definite integrals through the use of antiderivatives  [39.0,44, 45.0  ]

Calculation of derivatives and antiderivatives  [39.1 - 39.2, 41, 57]

Calculation of areas by using definite integrals [44 – 45.0, 46, 47, 49 ]

Substitution in integration [50 – 53, 58]

Techniques of integration [59 – 61, 68, and Oct.13, 14, 15]

Applications of integration [Oct. 13, 14, 15]

Arc length  [80 - 82]

Derivatives and integrals involving ex  [109.0 – 109.2]

Derivatives and integrals involving inverse sine function  [134 - 138]

Derivatives and integrals involving inverse tangent function  [139 – 140, 141.0 – 141.2]

Population growth and radioactive decay [110 - 117]

Completing the square [137 – 138, 141.0 – 141.2]

Derivatives and integrals involving hyperbolic and inverse hyperbolic functions  [151, 155, 157]

Motion along a straight line [88 - 89]

Motion under the force of gravity [90 – 92.1]

Surface area  [83.0 – 87]

Simple harmonic motion [143 – 149]

Techniques of integration [171 – 181.0, 182 – 209]

Work and energy  [214 - 225]

Hydrostatic force [226 – 230]

Centroids, center of mass [231 – 234]


 

Notation

Notation used in writing down Riemann sums, including the summation notation using S  [13.1 - 14]

Facts

            About upper bounds and least upper bounds  [4 - 6]

            More on the natural logarithm [97 - 98]

         About the exponential function [100 - 101]

         The derivative of an inverse function [103.0 – 105]

         Derivative of ex  [105]

         Properties of ex  [106 – 108]

         Properties of db  [107 – 108]

         Antiderivative of ex  [109.0]

 

Trigonometry [118 - 124]

         Hyperbolic functions [150, 151, 153]

         Inverse hyperbolic functions [154 - 157]