MATHHX A

MATHHX A

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1.1 Stamfunktioner

  • Definition 1.1.1
    Lad \(f\) være en funktion. En stamfunktion til \(f\) er en funktion \(F\), som opfylder

    \[F'(x)=f(x)\]

En stamfunktion til \(f\) er altså en funktion som giver \(f\), når man differentiere den.

  • Eksempel 1.1.1
    Funktionen \(F(x)=x^2\) er en stamfunktion til funktionen \(f(x)=2x\), fordi

    \begin{align*} F'(x) & =2x \\ & =f(x) \end{align*}

Øvelse 1.1.1

Undersøg:

  • a) Er \(F(x)=4x\) en stamfunktion til \(f(x)=4\)?

  • b) Er \(F(x)=2x^6\) en stamfunktion til \(f(x)=4x^5\)?

Løsning 1.1.1

  • a) Ja

  • b) Nej

  • Eksempel 1.1.2
    Vi vil bestemme en stamfunktion til \(f(x)=3x^2+1\). Vi skal altså finde et udtryk, som giver \(3x^2+1\) når man differentiere det.

    Vi husker fra differentialregning, at \(x^3\) differentieret giver \(3x^2\), og at \(x\) differentieret giver \(1\). Derfor gætter vi på at svaret er \(F(x)=x^3+x\). Vi tester:

    \begin{align*} F'(x) & =3x^2+1 \\ & =f(x) \end{align*} Den var god nok.

Der er regler til at finde stamfunktioner (for simple funktioner), men ofte kan man tænke sig frem til stamfunktionen, ud fra den viden man har fra differentialregning.

  • Eksempel 1.1.3
    Vi vil bestemme en stamfunktion til \(f(x)=x^4\).

    Vi leder først efter noget som giver \(x^4\), når man differentiere det. Vi prøver \(x^5\), fordi vi kan huske at man trækker en fra eksponenten, når man differentiere en potensfunktion. Men \(x^5\) differentieret giver \(5x^4\), og det var ikke helt det vi skulle have. Vi kan dog reparere på det ved at gange med \(\frac {1}{5}\), så vi får

    \[F(x)=\frac {1}{5}x^5.\]

    Vi tester vores svar:

    \begin{align*} F'(x) & = \frac {1}{5} \cdot 5x^4 \\ & =x^4 \\ & = f(x) \\ \end{align*} Den var god nok.

Øvelse 1.1.2

Bestem en stamfunktion til følgende funktioner

  • a) \(f(x)=2x+1\)

  • b) \(f(x)=3x^2-x^5+2\)

Løsning 1.1.2

  • a) \(F(x)=x^2+x\)

  • b) \(F(x)=x^3-\frac {1}{6}x^6+2x\)

Metoden vi lige har brugt virker kun for simple funktioner, hvor man hurtigt kan gennemskue hvad stamfunktionen må være. Det viser sig, at det generelt er sværere at bestemme stamfunktioner, end det er at differentiere. Er funktionen simpel er det dog nemt nok. Man slår bare stamfunktionen op i en tabel.

  • Eksempel 1.1.4
    Vi vil bestemme en stamfunktion til \(f(x)=\ln (x)\), men vi kan ikke lige komme i tanke om noget som giver \(\ln (x)\) når man differentiere det. Vi åbner formelsamlingen, finder en tabel over stamfunktioner og ser at stamfunktionen er

    \[F(x)=x\ln (x)-x.\]

Øvelse 1.1.3

Lad \(f(x)=\sqrt {x}\).

  • a) Bestem en stamfunktion til \(f\).

Løsning 1.1.3

  • a) \(F(x)=\frac {2}{3}x^{\frac {3}{2}}\)

I eksempel 1.1.1 så vi at \(F(x)=x^2\) var en stamfunktion til \(f(x)=2x\), men det er ikke den eneste stamfunktion. Vi kunne i stedet vælge f.eks. \(F(x)=x^2+1\), eller hvad med \(F(x)=x^2+27\)? Hvis man differentiere dem, så får man også \(f\). Helt generelt gælder der, at hvis \(F\) er en stamfunktion til \(f\), så er \(F(x)+c\), hvor \(c\) er en konstant, også en stamfunktion til \(f\). Der findes dog ikke andre stamfunktioner til \(f\) end dem på formen \(F(x)+c\), og det betyder at vi kan fastlægge en stamfunktion ved at kræve, at dens graf skal gå igennem et punkt.

(-tikz- diagram)

Stamfunktioner for \(f(x)=x^2\). Vi har \(F(x)=\frac {1}{3}x^3+c\) med \(c=-2\), \(c=-1\), \(c=0\), \(c=1\), \(c=2\).
  • Eksempel 1.1.5
    Lad \(f(x)=2x\). Vi vil bestemme den stamfunktion til \(f\), som går igennem punktet \(P(4,5)\).

    Vi finder først en stamfunktion, men denne gang lægger vi et \(c\) til. Vi får så

    \[F(x)=x^2+c.\]

    Vi mangler nu bare at finde ud af hvad \(c\) skal være, for at grafen for stamfunktionen kommer til at gå igennem \(P\). Hvis funktionen skal gå igennem \(P\), så skal \(F(4)=5\). Det giver os ligningen:

    \[4^2+c=5\]

    Vi isolerer \(c\) og får \(c=-11\). Altså er forskriften

    \[F(x)=x^2-11.\]

Øvelse 1.1.4

Lad \(f(x)=4x-1\).

  • a) Bestem den stamfunktion til \(f\), hvis graf går igennem punktet \(P(1,3)\)

Løsning 1.1.4

  • a) \(F(x)=2x^2-x+2\)

Øvelse 1.1.5

Lad \(f(x)=e^x+2x\).

  • a) Bestem den stamfunktion \(F\), der opfylder \(F(0)=-7\)

Løsning 1.1.5

  • a) \(F(x)=e^x+x^2-8\)

Øvelse 1.1.6

Her ses grafen for en stamfunktion \(F\) til \(f(x)=-\frac {1}{x}+1\).

(-tikz- diagram)

  • a) Bestem først ud fra forskriften for \(f\) en forskrift for \(F\) på formen \(F(x)+c\).

  • b) Aflæs nu et pænt punkt på grafen og bestem ud fra det konstanten \(c\).

  • c) Opskriv den endelige forskrift for \(F\)

Løsning 1.1.6

  • a) \(F(x)=x-\ln (|x|)+c\)

  • b) \(c=1\)

  • c) \(F(x)=x-\ln (|x|)+1\)

Man kan undre sig over, om alle funktioner har stamfunktioner. Det viser sig at der findes funktioner, som ikke har stamfunktioner, men hvis en funktion er kontinuert, så har den også en stamfunktion. Derfor vil vi ofte kræve at funktioner er kontinuerte ,når vi opskriver sætninger – så er vi sikre på at funktionerne har stamfunktioner. Vi husker at en funktion, løst sagt, er kontinuert, hvis grafen er sammenhængende (den er uden huller eller hop). I praksis er det ikke noget som vi vil bekymre os om, når vi regner opgaver.