Finally. I am about to begin my 2009 efforts at learning a little calculus. I am selecting the book, "Calculus Know-It-All" (2009) by Stan Gibilisco. I began this book last year and my initial impression was very favorable. However I may well access other books as I proceed. And possibly the Web.

Part 1 is about differentiation in one variable. It consists of 10 chapters and 188 pages.

I have learned from previous efforts that much of this will involve hand writing, especially the exercises. But I will scan this work into these notes.

Part 1 Differentiation in One Variable

Chapter 1 Single Variable Functions

a) Mappings

• "Calculus is the mathematics of functions, which are relationships between sets consisting of objects called elements." [p. 3]

  I must keep a list of the key concepts and their definitions. Essentially I need a set of flash cards, and a table.

  
  

  Concept	Definition
  function	a relationship between 2 sets. It maps every element in the domain to one, and only one, element in the range
  set	a collection of objects
  element	an object in a set
  domain	the set of elements in the first set
  range	the set of elements in the second set
  mapping	a method for pairing the elements of two sets. It assigns values of the dependent variable to values of the independent variable
  variable	a collection of elements
  independent variable	a nonspecific element in the domain
  dependent variable	a nonspecific element in the range
  ordered pairs	one method of defining a mapping
  relation	a mapping represented by a set of ordered pairs
  inverse function	obtained by transposing the domain and range
  graph	a diagram showing all the ordered pairs of a relationship
  vertical line test	if all vertical lines intersect the graph only once, then it is a function
  horizontal line test	if all horizontal lines intersect the graph only once, then the inverse is a function

• Learning to write (and read) the notational conventions is an important component of mathematics. There is a system for writing about sets and ordered pairs and functions. The name of a function is usually given a letter like f or g, whereas the domain and range are given letters like x and y. Example: f(x) = x + 3 = y .

  No problem. This is all very familiar to me.

• An inverse function is obtained by transposing the domain and range. The notation for this is to write a superscript -1 after the name of the function.

  b) Linear Functions

  Concept	Definition
  linear function	the dependent variable changes in a constant proportion to the independent variable
  slope	the constant proportion
  intercept	the point where the graph cuts the dependent axis (the y-axis)
  standard form for a linear function	f(x) = ax + b, where a is the slope and b is the y-intercept.

We now have two different representational systems: an algebraic one and a geometric one. Each form has strengths and weaknesses. The geometic form often provides immediate insights into a situation, but it can occasionally be misleading. The algebraic form is less intuitive but more logically rigorous. One often begins a new situation by sketching a rough graph to obtain a sense of the situation and then proceed to see if one can be more precise by using algebraic/functional notation.

• "To calculate the slope of a line, we must know the coordinates of two points on the line." [p. 9]

  Here is a copy of a Challenge activity that I completed.

 

c) Nonlinear Functions

• Examples of quadratic, cubic, :broken", reciprocal and tangent functions are given.

  This is all basic review for me. But it is good to go slow and refresh my mind about the terms and how to use them. There are 10 Practice Exercises which I will look at next time.

   

• Tags: mathematics, calculus