What to memorize in a Calculus class?

Question

When enrolled in a calculus class, what is good to have memorized before and while taking it? EX: table of derivatives, table of antiderivatives, half and dbl. angle formulas. ETC. Answer as though you assumed no prior knowledge of such things. Just, what's good to just KNOW before taking it? Thanks

Answer 1

Things you should know before taking a calculus class:

Algebra (duh)
Definitions of trigonometric functions (all six of them)
Pythagorean identity (sin² x + cos² x = 1, and the variant 1+tan² x = sec² x)
Double angle formulas
Power-reduction formulas for sin² x and cos² x
Properties of exponential and logarithmic functions

Things you should know during a calculus class:

Limits:

Intuitive definition of a limit
ε-δ definition of a limit (yeah, you won't use it much until analysis, but you should know it and understand it before then)
Properties of the extended real line (including ∞ and -∞)
Method of rationalizing a denominator and/or numerator
L'hopital's rule

Differential calculus:

Definition of the derivative
Linearity of the derivative (sometimes broken into two observations, called the sum rule and constant multiple rule)
Power rule
Chain rule
Product rule
Quotient rule
Derivatives of e^x and ln x
Derivatives of sin x, cos x, and tan x (yes, you should memorize the derivative of tangent separately, even though you can convert it to an expression of sines and cosines, it comes up often enough to make doing that a pain in the ASCII)

Integral calculus:

Definition of the integral
Fundamental theorem of calculus
Linearity of the integral (sometimes broken into two observations, called the sum rule and constant multiple rule)
Inverse power rule
Integral of e^x and 1/x (yes, you did memorize this from the other direction during the differential section)
Integral of sin x and cos x (ditto)
u-substitution method
Inverse chain rule (actually, u-substitution is just a formalization of this)
Integration by parts
Inverse product rule (from which integration by parts is derived)
Derivatives of arcsin x and arctan x
Integrals of 1/√(1-x²) and 1/(1+x²) (Yes, I know this is redundant with the previous line, but you should memorize this from both perspectives)
Expressions of sin x and cos x in terms of complex exponentials (very helpful when integrating products of sines and cosines).

Series:

Sigma notation (duh)
∑1, ∑x, ∑x², ∑a^x (where x is the index of summation)
Method of summation by parts (actually, you won't use this very often. But I'm convinced that if the class is taught properly, you _should_ use it very often)
Convergence tests for infinite series, including:
- comparison test
- root test
- ratio test
- integral test
- alternating series test
Taylor's theorem (which for some reason gets cited without proof or justification _way_ more often than is required)
Arc-length formula

You'll also do some work with solids of revolution and surfaces of revolution in calc 2. I recommend that (with the exception of the arc-length formula) you do NOT try to memorize forumlae for dealing with these, instead using a conceptual understanding to derive the correct integral for the region being described. This helps you enormously in calculus 3, where there is no general formula and you have to be able to visualize the regions you're working with.

Vector calculus:

Definition of differentiability for functions of several variables.
Definition of partial derivatives
Definition of gradient
Definition of directional derivative
Polar, cylindrical, and spherical coordinates
How to change the order of integration for double and triple integrals
Formula for change of variables in multivariate calculus, in particular for switching from rectangular to polar, spherical, or cylindrical coordinates
Dot and cross products
Line integrals over vector and scalar fields
How to find a potential function for conservative vector fields
Surface integrals
Green's theorem
Stokes's theorem
Divergence theorem

If you manage to memorize all that stuff, you'll have no trouble passing your classes at all.

Answer 2

Half and double angles will serve you well, as will the definition of limits, definition of derivatives, and definition of integrals.

You will also be well-served by the law of sines and cosines, and series representations of both, as well as for e.

There are plenty of laws in calculus that you'll need to memorize, the division rule and integration by parts are particularly valuable.

Answer 3

Same as above also:

when function are continuous
familiarize with concept of + and - infinity
trig and algebra functions
differentiation and integration properties

Fundamental theorem of calc found here: http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus