Calculus Exam Notes

Continuity Refresher

• Continuity was learned before differentiability.
• Need a quick refresher on the two key features:
  • The point exists at that point.
    • f(a) exists at x = a.
  • The limit as x approaches that value of the function exists.
    • Want to show that \lim_{x \to a} f(x) exists.
  • These must be equal.
    • Show that \lim_{x \to a} f(x) = f(a).

Proving Continuity

• Argument will be lengthy; assume the reader knows nothing.
• Step 1: Show that the point exists, that I can actually plug in a and get something out.
  • Let a = 3.
  • The question is about t = 3 because that's where we switch from one function to the other.
  • f(3) can be found using the table because you can be equal to 3 there.
  • g(3) = 105.
• Establishing if the limit exists.
  • Look at two individual limits:
    • \lim_{x \to 3^-} f(t)
    • \lim_{x \to 3^+} f(t)
  • Hope that these numbers are equal.
  • When going from 3 to the left, trying to get as close to 3 as possible from the numbers to the left of 3.
  • Numbers to the left of 3 fit in the cubic equation.
  • 3^3 = 27, which reduces with 3 to make 9. So, 4 \times 9 = 36.
  • 3^2 = 9. What's 6 \times 9 = 54.
  • 5 \times 3 = 15.
  • 36 + 54 + 15 = 105.
  • Coming from 3 to the right, we're decreasing towards t = 3, so we'll be decreasing towards 105.
• Prove that the limit exists:
  • Since \lim{x \to 3^-} f(x) = \lim{x \to 3^+} f(x), then \lim_{x \to 3} f(t) = 105.
• Proving Continuity:
  • Is f(t) continuous at t = 3? Yes.
  • f(t) is continuous at x = 3 because the \lim_{x \to 3} f(x) = f(3).
• This is a sure-fire argument laying out evidence to prove continuity.
• On the FRQ in part b, this problem was worth two or three points. Knowing how to do continuity could significantly improve the score.

Concavity

• Concavity was a major skill in precalculus, and it's coming back again in calculus in a slightly different format.
• Determining concavity using calculus is related to a double derivative or second derivative, f''(x).
• How to find a derivative in a piecewise.
• Concavity at one, so which part of this function will I even care about? The cubic.
• Can find the first derivative of the cubic and then the second derivative.
• The second derivative is 8t + 12.
• If the double derivative is greater than zero, in other words, it's positive, then it's concave up.
• If the second derivative is less than zero, meaning we're negative, then it's concave down.
• Not enough to just find the second derivative; you have to know what to do with it.