In algebra, and in most math subjects prior to calculus, there are recipes and formulas that give you the answer directly and exactly. When solving the quadratic equation $$ x^2 + x - 12 = 0,$$ you either factor $$x^2+x-12 = (x-3)(x+4)$$ or you apply the quadratic formula: $$ x = \frac{-1 \pm \sqrt{49}}{2} = 3 \hbox{ or } -4.$$ You probably don't think

There's got to be a solution somewhere between 2.9 and 3.1. Make that 2.99 and 3.01. No, make that 2.999 and 3.001.....

 

But that sort of reasoning is exactly what we do in calculus, as the following video explains:



This idea is the first of the six pillars of calculus:

  1. Close is good enough.
  2. Track the changes.
  3. What goes up has to stop before it comes down.
  4. The whole is the sum of the parts.
  5. The whole change is the sum of the partial changes.
  6. One variable at a time.

In order to solve problems in calculus, our modus operandi is to

  1. Find an approximate solution.
  2. Find a way to get a more accurate approximation.
  3. Keep going until we have as much accuracy as we need.
  4. If for some reason we need exact answers, take a limit.

In the next few slides, we'll go over some of the ways we do this.