Here's one of the most common student questions: 
‘What do I need to know?’

Answer: 
the ‘Index Card Method’

When I teach, every concept becomes an index card (4" x 6", blank both sides—it's easier to write
math without lines).
The cards guide class discussions.
Students handcopy the cards (adding additional information, as needed)—by course end, they have a stack
that embodies every key idea. Cards can be mixed up so learning isn't orderdependent; ones that are well understood are
taken out to focus on those that remain.
Students hold the whole course in the palm of their hand.
I used this method for decades of teaching.
Early on, the classroom blackboard became ‘one side of a card’: as I lectured,
the card emerged on the board, and students took notes.
Later on, students got photocopies of the cards, or they were posted on the web.
Manyastudent has told me (years later) that they still have their stack of cards!
Below are snapshots of the Calculus index cards. They're in my handwriting. You can purchase
the entire set of jpg images for \$2.00. It's a zip file; you'll need to be
able to unzip it on your computer.
The purchase of this file includes the rights for teachers to make
as many copies (electronic or printed) as needed for their students.
There are 91 card images (jpg) in a zipped folder. The cards are numbered 1–89, with two ‘extended’ cards: 43abcd and 53abcd.
Click on any card below to see the fullsize image.
Beneath the images is a numbered list of all the cards by title/concept.
You can see the cards used in the context of a college course here.
STUDENTS:
These are a great resource, even if your teacher isn't using the ‘index card method’.
Just be sure you write them up yourself. Yep, you could print, cut, tape onto cards—but
you'd reap only about 10% of the potential benefit.

\$2.00 
When you buy,
you'll receive an email
with a download link.
(So, check your email
after checking out!)

Index Card Titles/Concepts
In each entry, the first line is side (a), the second line is side (b).
Two cards (43 and 53) have additional sides (c) and (d).

average rate of change;
the tangent problem

position functions;
the derivative of a function

intuitive derivative (notation);
example

limit of a function;
3 cases when $\,\lim_{x\rightarrow a} f(x) = \ell\,$

onesided limits;
the squeeze (pinching) theorem

continuity at a point;
three ways a function can fail to be continuous at $\,c\,$

key idea (for continuity at a point);
example

an important type of continuity problem;
relationship between differentiability and continuity

limit rules;
techniques for evaluating limits (examples)

indeterminate forms;
deciding if you have an ‘indeterminate form’

the derivative of $\,f\,$ at $\,a\,$;
using the definition of derivative

an alternative approach (compare with 11b);
another example: using the definition of derivative

the linearization of $\,f\,$ at $\,a\,$;
intuition for the linearization

local max/min (extreme values);
global max/min

strictly increasing/decreasing functions;
getting increasing/decreasing behavior from the derivative

concavity;
getting concavity info from the second derivative

power rule for differentiation;
examples: using the power rule
 partial proof of the power rule (both sides)

derivative of a constant times a function;
finding $\,\frac{d}{dx} (kx^n)\,$

derivatives of sums and differences;
proof

differentiating composite functions: motivation;
differentiating composite functions: the idea

the chain rule (prime notation);
the chain rule (Leibnitz notation)

Why is it called the ‘chain rule’?
Using the chain rule to differentiate $\,\bigl( f(x) \bigr)^n\,$

pattern for generalizing all the basic differentiation formulas;
examples

the product rule for differentiation;
proof of the product rule

When is $\,\lim_{h\rightarrow 0} f(x+h) = f(x)\,$?
When is this true? As $\,x\rightarrow c\,$, $\,f(x)\rightarrow f(c)\,$

the quotient rule for differentiation;
proof of the quotient rule

the irrational number $\,\text{e}\,$;
equivalent limit statements defining $\,\text{e}\,$

evaluating a limit involving $\,\text{e}\,$;
Why is $\,\lim_{n\rightarrow\infty} \left[ (1 + \frac 1n)^n \right]^{2k} = \text{e}^{2k}\,$?

derivatives of logarithms;
proof that $\,\frac d{dx} (\ln x) = \frac 1x\,$

more notation for derivatives;
notation: evaluating a derivative at a point

differentiating $\,{\text{e}}^x\,$;
differentiating $\,a^x\,$

what is $\,\lim_{x\rightarrow 0} \frac{\sin x}{x}\,$?
example

derivatives of sine and cosine;
derivatives of tangent and cotangent

derivatives of secant and cosecant
a useful observation—derivatives of cofunctions

finding the derivative of an inverse function;
two methods: finding the derivative of an inverse function

What does $\,(f^{1})'(x) = \frac{1}{f'\bigl(f^{1}(x)\bigr)}\,$ really mean?
summary

derivatives of the inverse trigonometric functions;
a typical derivation

differentiating variable stuff to variable powers: the log trick;
example

l'Hopital's rule;
motivation, ‘$\,\frac 00\,$’ case

basic use of l'Hopital's rule;
more advanced use of l'Hopital's rule

Where can a function change its sign?
negating ‘and’ and ‘or’ sentences

(a) info given by $\,f\,$, $\,f'\,$, $\,f''\,$;
(b) a basic sign analysis of a function;
(c) implications (and equivalent sentences);
(d) the contrapositive of an implication

Where can a function have a local max/min?
If $\,f\,$ has a local max/min at $\,c\,$, then ...

critical points, critical numbers;
Careful!! A critical point does not have to be a max or min!!

Where can a function have an inflection point?
If $\,f\,$ has an inflection point at $\,c\,$, then $\,\bigl(f''(c) = 0\ \ \text{or}\ \ f''(c) \text{DNE}\,$.

an algorithm for thorough function analysis;
example—thorough function analysis

Are critical points actually max or min? Two tests: 1st and 2nd derivative tests;
the first derivative test

open/closed intervals;
bounded subsets of $\,\Bbb R\,$

review of absolute/global max/min;
the Extreme Value Theorem

optimization problems (finding max/min);
example

a first optimization problem;
You try it!

(a) an optimization problem;
(b) ... continued;
(c) a useful observation;
(d) using the observation

an optimization problem;
... continued

explicit (‘showing’) versus implicit (‘hidden’);
implicit differentiation

related rate problems;
falling ladder problem

Is the car speeding?
... continued

the Mean Value Theorem;
the Mean Value Theorem (MVT) intuition

a Mean Value Theorem example;
a Mean Value Theorem example

bounded functions;
the definite integral of a function

a definite integral may or may not exist;
notation for the definite integral

dummy variables in definite integrals;
simple definite integral examples

properties of the definite integral;
linearity of the integral

definite integral example;
definite integral example

How might we estimate areas beneath a curve?
general notation for the area problem

summation notation;
left and right estimates using summation notation

precise definition of a definite integral;
Riemann sums

estimating a definite integral;
finding a definition integral using the definition (for a very simple function)

the Fundamental Theorem, Part I;
proof

antiderivatives;
‘undoing’ differentiation; antidifferentiation

Every differentiation formula gives an antidifferentiation formula!
some antiderivatives you should know

examples: using the Fundamental Theorem;
more examples

A very useful formula!!
integrating a rate of change gives total change

connection between area and antiderivatives;
making it more precise...

making it more precise, continued (both sides)

summary: the Fundamental Theorem of Calculus, Existence of an Antiderivative;
the Evaluation Theorem

examples: derivatives involving integrals;
a more complicated derivative involving an integral

indefinite integrals (general antiderivatives);
more antiderivative formulas

finding a particular antiderivative;
example

generalizing the formula for the derivative of $\,\ln x\,$;
the antiderivatives of $\,\frac 1x\,$

substitution;
format for substitution problems

a substitution example;
same problem, more compact

substitution with definite integrals;
2nd approach: merely indicate that limits are different

integration by parts;
basic strategy for using the parts formula

format for integration by parts;
using parts with a definite integral

periodically differentiable functions;
using parts with products of polynomials and periodically differentiable functions

inverse functions with simpler derivatives;
using parts for inverse functions

using parts with products of periodically differentiable functions;
... but it doesn't always work

analyzing falling objects;
example: integrating to find velocity and position information