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
Lagrange remainder
Formula

11.3. Lagrange remainder

Formula

In 11.3. Lagrange remainder

Formula

R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1},\qquad c\text{ between }a\text{ and }x,\ f^{(n+1)}\text{ exists and is continuous on the open interval between }a\text{ and }x

Tags

Differential calculus
Continuity
Derivative
Derivative formula
Lagrange remainder
Taylor

Detail

Requires that

f^{(n+1)}

exists and is continuous on the open interval between

a

and

x

.

Variables

$x$ Independent variable
$a$ Point or real constant
$c$ Intermediate point (MVT, Rolle)
$f$ Function
$n$ Derivative order or exponent
$c\text{ between }a\text{ and }x,\ f^{(n+1)}\text{ exists and is continuous on the open interval between }a\text{ and }x$ Dependent variable

Conditions

$c\text{ between }a\text{ and }x,\ f^{(n+1)}\text{ exists and is continuous on the open interval between }a\text{ and }x$

Related topics

Taylor series

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Related formulas

Related
Formula
$P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k$
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