Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval). integral scaling rule Fundamental Theorem of Calculus FTC Rolle's Theorem, Graf funkcije gre od točke (a,f(a))(a, f(a))(a,f(a)) do točke (b,f(b))(b, f(b))(b,f(b)), ki sta na isti višini.Če je funkcija gladka (zvezna in odvedljiva), potem mora nekje vmes "obrniti smer" — točka, kjer je odvod enak 0 (vodoravna tangenta).

Rolle's Theorem, Graf funkcije gre od točke (a,f(a))(a, f(a))(a,f(a)) do točke (b,f(b))(b, f(b))(b,f(b)), ki sta na isti višini.Če je funkcija gladka (zvezna in odvedljiva), potem mora nekje vmes "obrniti smer" — točka, kjer je odvod enak 0 (vodoravna tangenta).

DIFFERENTIABILITY, INTEGRATION

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Calculus - Geometry

user_hodr Created by 6 mon ago

Cards in this deck(48)

the ... of f is the ... of the tangent line.

Let f:I(open interval) -> R be a function. f is ... at a€I if the function (f(x)-f(a))/x-a has a limit as x->a, in wich cas we call that L, limit the ... of x at a.

f is ... if it is ... at every a€I

odvajati

A number, a variable, or a product of a number and one or more variables

differentiaion is ...

f'(x)g(x)+f(x)g'(x)

g(x)f'(x)-f(x)g'(x)/g(x)^2

d/dx f(g(x)) = f'(g(x)) g'(x)

Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).

if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b) f '(c) = [f(b) - f(a)]/(b - a)

x^2 * sin (1/x) is an example of a function that is differentaible, but its derivative f' is not ..., so also not ...

Function with continuous first derivative.

Function with continuous first and second derivatives.

if it is n-times differentiable for all n

C^n, C^inf

the pure mathematics of continuously varying functions, defining continuity, differentiability, integrability ... and proving the theorems that explain the fundamental properties of these concepts

The particular tools and methods for calculating with continuously varying functions: derivates and their rules, integrales and their rules ...

Function where for x1 <= x2 -> f(x1) <= f(x2) <-> Vx f'(x)

Function where for x1 < x2 -> f(x1) <= f(x2) <- >Vx f'(x) >=0

Function where for x1 < x2 -> f(x1) < f(x2) <- Vx f'(x)>0 (ni ekvivalence, ker je odvod lahko tudi 0, za f(x) = x^3

Function where for x1 > x2 -> f(x1) > f(x2) <- Vx f'(x)<0

f is ... if f is either increasing or decreasing or constant

f is ... if f is either increasing or decreasing

noun for monotonic

a is a ... of f a = max{f(x) | x € I}

a is a ... of f if there exist eps>0 s.t. a = max{f(x) | x€I, |x-a| f'(a) = 0 (if f is differentiable) <- f'(a) = 0 and f''(a) < 0 (if f is twice differentiable)

a is a ... of f if a = min{f(x) | x€ I}

a is a ... if there exist eps>0 s.t. a = min{f(x) | x€I, |x-a| f'(a) = 0 (if f is differentiable) <- f'(a) = 0 and f''(a) > 0 (if f is twice differentiable)

ekstremi izrazi

if f'(0) == 0

the point where the graph changes concavity, the second derivative changes sign at 0

A ... of d:I->R is some a € I such that f(a) = 0

Geometrijska razlaga: Začetno točko x0x_0x0 projiciramo na graf funkcije. Nato potegnemo tangento na fff v tej točki. Tam, kjer tangenta seka os xxx, dobimo novi približek x1x_1x1. Postopek ponavljamo, dokler se vrednosti dovolj ne približajo ničli.

a value or quantity that is nearly but not exactly correct.

Calculus is built on ... of ... and ....

Two basic properties of integration are .... The fact that these two coincide is called ...

... is informally defined to be the area under the graph of f between a and b above the x axis.

If f is continuous at c than F is differentiable at c and F'(c) == f(c), ∫ f(x) dx on interval a to b = F(b) - F(a)

a function G such that G'(x) == f(x) is called an ... of f.

no limits, find antiderivative + C, use inital value to find C

integrabilnost je linearna

∫cacbf(x)dx=∫abf(cu)⋅cdu =c∫abf(cu) du= c \int_a^b f(cu)\,du=c∫abf(cu)du

glavni del funkcije intgerala

Arbitrary constant added to indefinite integrals.

fundamental theorem of calculuss

Integrals with infinite limits or discontinuous integrands.

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