98. Cheatsheet
98.0.1. Limits
Epsilon-Delta
For every distance around , there’s a -range around that keeps within of .
The limit of as approaches equals if and only if, for every , there exists a such that, whenever , implies that
| Limit Type | Name | Quantifiers |
|---|---|---|
| Epsilon-Delta | ||
| M-Delta | ||
| epsilon-N | ||
| M-N |
Finite Finite (Epsilon-Delta)
Finite Infinity (M-Delta)
Infinity Finite (Epsilon-N)
Infinity Infinity (M-N)
98.0.2. Derivatives
Example
Let
a. Find
b. Prove a
c. Prove b
p.f.:
Let
Choose
Suppose
Check
98.0.3. Integrals
98.0.4. Differential
If you have a function:
Then the differential is defined as:
This means:
-
is the derivative
-
is a small change in
-
is the corresponding small change in
So:
Example
Let’s say“
Then:
| Operation | Notation | Input | Output | Meaning |
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| Derivative |
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Slope/Rate of change at each |
| Indefinite Integral |
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Function whose slope is |
| Definite Integral |
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Total signed area between and |
| Rule | Rule | Example | Rule | Example |
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| Constant |
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| Power |
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| Constant Multiple |
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| Sum |
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| Difference |
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| Product |
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Integration by Parts |
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| Quotient |
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Algebraic Manipulation / Substitution | |
| Chain |
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Integration by Substitution |
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| Exponential |
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| Logarithmic |
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| Sin |
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| Cos |
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| Tan |
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98.0.5. Product Rule Integration by Parts
Example
Given a function:
Integrate (by parts):
Step 1: Choose and
We choose:
-
(easy to differentiate)
-
(easy to integrate)
Step 2: Compute and
Step 3: Plug into formula
Step 4: Compute the remaining integral:
Step 5: Finish the expression: