Homer Euler

I went through a sample AP Calculus AB test and put the twenty questions into categories, to give you a rough idea what you'd need to know.

    derivatives (7)

      derivative and conic section
      derivative by double u sub
      derivative of compound function by chain rule
      derivative of added (subtracted) functions
      derivative of multiplied (divided) functions
      second derivatives
      derivatives for limit

    integrals (6)

      integrate with u sub (2)
      integrate differential equation
      integrals, split intervals (2)
      integral, FTC *, parametric equation in t

    limits (3)
    trig (3)
    polynomial division (2)
    simplify exponential equation, ln
    t parameterized x and y
    two equations, two variables (3)
    average rate of change over interval
    read graph (3)

* fundamental theorem of calculus

That didn't really work, did it? The counts came to way more than twenty.

Even without the varieties of derivative and integral, they come to thirty.

The categories overlap. For example, using derivatives to solve this limit problem.

` \lim _{h \rightarrow 0} \frac{e^{4} e^{h}-e^{4}}{h} `

Every question is two questions. The first question: What's the question? What do you want from me? That's the hard question.

Sometimes you know what's wanted but you don't have it. You can't do it, not completely.

More often you don't know what's wanted.

Back to that limit problem. You probably want to use l'Hopital's rule, and take the derivative of the numerator and denominator. But the problem doesn't tell you that. It doesn't tell you to use l'Hopital's rule.

In nearly every limit problem some term is zeroing the denominator, and you want to jigger away that term.

Often algebra is enough.

Sometimes there's more than one way.

Sometimes there's no way, and the answer is DNE (does not exist).

Sometimes the answer is uglier than question. It doesn't feel like an answer.

Just a pretty picture inspired by Euler, from 3Blue1Brown :

eulerian space

Thirteen of these twenty questions (65%) involve a derivative or integral.

The other third of the questions don't require calculus, strictly speaking. Algebra is enough.

You don't always know that though. You have to decide. That's the hard part.

Another Euler from 3Blue1Brown :

eulerian space

You can practice that.

Find a list of questions that could require anything. Like these twenty. Not a list of questions at the end of a chapter on u substitution, for example, with a heading that says "Solve these u substitution problems."

Sometimes you find the right strategy after your first strategy falls short. Expect that. Expect to abandon one strategy for another. That's not always a failure.

Trying strategies can be a winning strategy. Sometimes there's a hard way and an easy way to solve something. Take a moment to look at your choices. On a test of two or three hours you'll get that time back.

A pretty picture insipired by l'Hopital:

pick a path