Review Sheet Math H161

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

Theory that might be on Midterm  I  is marked by asterisks  ***.  Six items are marked.  One of them will be on the exam.

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]

Theorems--with proofs

Properties of logarithms [63 - 67]

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]

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

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]