Revision sheet

L'Hôpital's rule

1. Check the form is $0/0$ or $\infty/\infty$.
2. Differentiate top and bottom separately.
3. Take the new limit (repeat if still $0/0$).

Watch out: no indeterminate form, no L'Hôpital.

First derivative test

1. Find the critical numbers.
2. Sign chart of $f'$.
3. $+ \to -$: max; $- \to +$: min; no change: neither.

Watch out: $f'(c) = 0$ alone does not make an extremum.

Second derivative test

• $f' = 0$, $f'' > 0$: min. $f' = 0$, $f'' < 0$: max. $f'' = 0$: test fails.
• Inflection: $f''$ changes sign.

Watch out: $f''(c) = 0$ is not enough for an inflection point.

Closed interval method

1. Critical numbers inside $(a, b)$.
2. Evaluate $f$ there and at $a$ and $b$.
3. Largest = maximum, smallest = minimum.

Watch out: the EVT needs continuity on a closed interval. Give the value, not just where it occurs, when asked.

Mean Value Theorem

1. Check: continuous on $[a, b]$, differentiable on $(a, b)$.
2. Average rate $\dfrac{f(b) - f(a)}{b - a}$.
3. Solve $f'(c) =$ that rate, with $c$ in $(a, b)$.

Watch out: $c$ is not always the midpoint.

Motion on a line

• $v = s'$, $a = v'$; at rest when $v = 0$.
• Same signs of $v$ and $a$: speeding up.
• Distance: add each leg between turns.

Watch out: displacement and distance differ when the particle turns back.

Optimization

1. Quantity and constraint.
2. One-variable function and its domain.
3. Critical numbers vs endpoints.
4. Answer the question: the value or the place, with units.

Watch out: not every critical number is the optimum.