Tutored calculus for a couple of hours the other day, just before the student started Calculus II. Here are some things we went over.
We talked about the purpose of a derivative and how you can optimize things early on and how integrals can find the areas under various curves and on into three dimensions, etc.
Went over how logarithms can be useful, what the natural number e is, and the behavior of the natural logarithm.
How algebra can simplify calculus before doing derivatives or integrals.
How to deal with fractional exponents.
How the order of approach for fractional exponents can simplify things.
Thinking about the behavior of functions for limits.
How exponents like 1/4 and 1/6 work.
How the ln function increases and decreases very slowly.
How to divide by fractions by multiplying by the reciprocal.
Using common denominators.
Going over integration by parts.
Doing some derivatives and keeping track of the sign.
How squared trig functions look.
Factoring out algebraic terms.
Use the product rule when differentiating a product.
Picking u and v for integration by parts won’t always work out, if it seems to not be working, you may want to choose something else.
Doing an integral is taking an anti-derivative and finding something that when you take the derivative of it will give you what you started with between the integral sign and the differential.