Day 8, Mat128: Calculus A

• Roll
• We'll have a quiz today over section 1.2: I posted solutions to the homework problems yesterday evening, but you should definitely try them yourself...., of course....
• Our test next week will cover everything through section 1.3.
• Please read Section 1.4 for Monday. We'll transition from derivative at a point to the derivative function.
• Because of the logistical nightmare of securing graphing calculators anymore, you may use a phone with a graphical calculator app for quizzes and tests.

  What you may not do is use them for anything other than that. If you haven't signed my sheet (and I'm missing a couple of names), please do.

• We continued working on limits, and focused on our worksheet.
• Then we began derivative at a point, which is just an example of a limit.
  • Why are we concerned with "limits"? Here's why:

    The most important definition in calculus is the definition of the derivative (here is the derivative of $f$ at $a$): \[ f'(a)=\lim_{h\to 0}{\frac{f(a+h)-f(a)}{h}} \]

    And that's why we're so concerned about limits! Memorize this definition. Be able to write it at a moment's notice. Paint it on your bedroom ceiling, like Marcus.

    Alternate definition: \[ f'(a)=\lim_{x\to a}{\frac{f(x)-f(a)}{x-a}} \]

    We think of this as the slope of the tangent line to a curve (the graph of \(f\) at \(a\)), provided the tangent line exists.

  • The tangent ("touching") line osculates ("kisses") the curve at this point.

    Once we have the derivative at $x=a$, we can write the equation of the tangent line, graph of linear function $y=T(x)$, using "point-slope" form: \[ y - f(a) = f'(a)(x-a), \] or \[ y = f'(a)(x-a) + f(a). \] It's that simple!

• tangent lines (and their equations)
• slopes (and their estimates)
• algebraic resolution of the indeterminacy of the limit definition of the derivative.

I'll use Mathematica to go over the solutions: I hope that you've already given these problems a shot!

• Desmos page for secant lines (here's a copy appropriate to today's quadratic).
• My freshman physics professor at Miami University, Prof. Don C. Kelly wrote The Little Book of Calculus. Self-Help Guide to Calculus. It's a neat "little" (50 some pages) resource produced by him and his kids/grandkids. Check out page 12, limits, in particular.