Calculus Preliminaries

Calculus  takes basic mathematics and introduces the limit process.  In doing so it generalizes basic mathematical concepts, for example:

     We know how to find the slope of a line; calculus allows us to find the slope of a curve at each point along a curve.

     We can find the area a region bounded by line segments;  calculus allows to find the area of a region bounded by curves. 

     We can find the sum of a finite sequence of numbers;  calculus allows us to find the sum of an infinite series.

     We can find the center of a circle;  calculus allows us to find the centroid of a region.

     We can find the surfaces are and volumes of cylinders, prisms, etc.; calculus allows us to find the surface area and volume of generalized solids.

What follows is a basic review of elementary mathematics concepts required for studying calculus.

1)  set theory

  Set membership can be expressed this way:

         { x: 'some mathematical relationship involving x'}

example:   {x:  0 < x < 100 }

  "This would be the set of all numbers greater than 0 and

less than 100."   200 would not be a member of this set.

2) properties of real numbers

    Given the real numbers a:     

      i)   a must be positive, negative or equal to zero.

 Given the distinct real numbers a, b, c, and d:     

     ii)  Only one applies: a < b  or  b < a  or  a = b

 On the number line for the Real Numbers,  ℜ,   '<' can be thought of as: to the left of, and  '>' can be thought of as: to the right of.

     iii)  If  a < b   and b < c   then  a < c

    iv)  If  a < b,  then  a + c  <  b  + c

     v) If  a < b  and   c > 0,  then  ac < bc

We'll look at this one algebraically.  First if a < b  then

b - a > 0.   c > 0 means that  c(b - a) must also be greater than

zero.  c(b - a ) > 0.  Simplifying we get  bc - ac > 0 which means that

bc > ac, i.e., ac < bc.

     vi)  If a < b  and c < 0,     then  ac > bc

a < b  means  b - a > 0.   c < 0 means that c(b-a) < 0.

Simplifying, cb - ca < 0.   cb < ca, i.e., ac > bc.

   Now review inequalities: Inequalities and their Graphs

3) absolute value

A graphical look at absolute value:

For  the first graph, recall the first definition of absolute value, |a| = +a if a is positive or -a if a is negative.  Directly we have

+x < d  and  -(x) < d,   Using the rule vi. above on the second inequality we get:

    x > - d   and now together we have  -d < x < d .

Now, the for the second graph, 

 Using rule vi. again  we write  |x - c| < d  as two equations,

+(x - c ) < d   and  -(x - c) < d;   The first gives x < d + c   and the

second gives -x + c < d   so that c - d < x  and together we get

         c - d < x  <  c + d      which is the interval (c - d,  c + d)

Concept of an interval::

An interval is a range of real numbers.  It may or may not include the endpoints.

Special notation:    [ and ] imply the endpoint is included and ( and ) imply the endpoint is not included.

If the endpoints of an interval are not included, then we have an open interval.

A closed interval is an open interval together with the open interval's endpoints.

examples:

 ( 5, 12 )    all real numbers (reals) between 5 and 12 but not including both 5 and 12.  5.00000000000001 is a member as is 11.99999999999999999999999999991 for example.

 [-2, 0)      all reals between -2 and zero including -2 but not 0.         

  -0.0000000000000000000000000001  is a member for example.

 [ 4, 7 ]     all reals between 4 and seven including both 4 and 7

( and )  names an "open interval" 

[ and ]  name a "closed interval."

(5, 12) is an open interval,   5 and 12 are not part of this interval.

[4, 7] is a closed interval,  4 and 7 are part of this interval.

[-2, 0) is neither open nor closed.

Most of the time we concern ourselves with open intervals.

Interval notation together with set notation:

The interval ( 5 , 12 )  represents  { x: 5 < x < 12 }

"The set of all real x such that x is greater than 5 and x is less than 12."

The interval [4, 7]  represents { x: 4 ≤ x ≤ 7 }

"The set of all real x such that x is greater than or equal to 4 and x is less that or equal to 7."

Note how the interval and inequality signs match above.

The interval [-2, 0) represents { x: -2 ≤ x < 0 }

"The set of all real x such that x is greater than or equal to -2 and x is less than 0."

The symbol   ∞  is used to mean "with no bound" or "with no limit."

  -∞  means " with no negative bound/limit "

   ∞  means " with no positive bound/limit "

The interval  [ 4, ∞ )  represents { x: x ≥ 4 }

The interval  ( -∞ , 0) represents { x: x < 0 }    all negative reals.

The symbol  ∞  can be troublesome.  It is NOT a number.  Numbers can be precisely defined in terms of limits, but by definition  ∞  means with no bound/limit, yet we use it as if it were a number in our equations and even call it 'infinity' as we call any other number by it's value, 10, for example.  But there is NO number to assign to the symbol ∞.   And thinking of  ∞ as a number can lead to serious conceptual errors.

So for example:

   The interval [-∞, ∞ ]  is meaningless; x cannot be equal to either

   -∞ or  ∞ since neither represent a number.

 Also [ and ] name a closed interval, recall, one with end points, yet  ∞  has no bound and therefore no endpoint

However (-∞ , ∞ )  is the set of all real numbers. 

   How does this work?  this  is { x:   -∞ < x < ∞ }

   x is greater than any possible negative value and less than any possible positive value,  unlimited possibilities of such negative and positive values exist, since any x whatsoever can satisfy this relationship, so x takes on all values.

  Yet, (-∞ , ∞ ) , admittedly is a strange concept to wrap your head around.  Think of it as notation alone until later when it becomes more clear.

Bounded Intervals: