Mathematical formulas

Differential Calculus

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1Notation, functions and domains
2Limits
3Continuity
4Derivative and geometric interpretation
5Differentiation rules
6Higher-order derivatives
7Mean value theorem
8Function analysis
9Optimization
10Linear approximations and differentials
11Taylor series (introduction)
12L'Hôpital and indeterminate limits
13Implicit and related functions
14Related rates
15Graphs and asymptotic behavior
16Guide to differentiate and analyze
AAppendix A: extended derivative table

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Differential Calculus
Criteria on closed intervals
Formula

9.2. Criteria on closed intervals

Formula

In 9.2. Criteria on closed intervals

Formula

A=xy,\quad 2x+2y=P \Rightarrow A(x)=x\left(\frac{P}{2}-x\right)

Tags

Differential calculus
Derivative
Optimization
Criteria on closed intervals

Detail

Typical example: maximum area with fixed perimeter (rectangle → square). Here

P

is the full perimeter (

P=2x+2y

); the semiperimeter is

C=P/2=x+y

.

Explore the idea

Explore the idea

Variables

$x$ Independent variable
$2x+2y=P \Rightarrow A(x)=x\left(\frac{P}{2}-x\right)$ Dependent variable

Conditions

$2x+2y=P \Rightarrow A(x)=x\left(\frac{P}{2}-x\right)$

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