3.8
Consider a mug whose cross-sectional area varies with height—it is wider at the bottom and top and narrower in the middle.
When coffee is poured into this mug at a constant volumetric rate, the coffee level rises over time. The rate of this rise is inversely related to the cross-sectional area at that height.
The concavity of the curve depends on the sign of the second derivative of height with respect to time.
In the lower half of the mug, the cross-sectional area changes in a way that causes the height to accelerate. Because the height of the liquid accelerates, the second derivative is positive in this region, resulting in a concave-up curve.
On the other hand, cross-sectional area increases in the upper-half, and shows the opposite effect, height decelerates, meaning the second derivative is negative, and corresponds to a concave down region on the graph.
Inflection points mark where concavity changes.
In this example, the inflection point is located near the middle of the mug, where the cross-sectional area is minimum. So, the acceleration of height represented by its second derivative has decreased to zero following its passage from positive to negative values.
In de wiskundige analyse is het bepalen van de hoogste en laagste punten van een functie essentieel om inzicht te krijgen in het gedrag ervan. Deze punten, aangeduid als kritische punten, treden op waar de eerste afgeleide nul is of niet bestaat. Kritische punten vormen mogelijke plaatsen van lokale maxima en minima en kunnen worden geclassificeerd met behulp van de tweede-afgeleidetoets. Niet elk kritisch punt komt echter overeen met een lokaal maximum of minimum. Daarom wordt de tweede afgeleide geanalyseerd om deze punten te classificeren. De tweede-afgeleidetoets geeft informatie over de concaviteit van de functie:
Als f''(x) = 0, is de tweede-afgeleidetoets niet doorslaggevend en moeten aanvullende methoden, zoals de eerste-afgeleidetoets, worden toegepast. Beschouw de volgende functie:
\begin{equation*}f(x) = x^3 -3x^2 + 4\end{equation*}
1. Bepaal de eerste afgeleide:
\begin{equation*}f'(x) = 3x^2 -6x\end{equation*}
Stel f'(x) = 0 om de kritische punten te bepalen. Deze uitdrukking levert x = 0 en x = 2 op als kritische punten.
2. Bepaal de tweede afgeleide:
\begin{equation*}f''(x) = 6x -6\end{equation*}
3. Evalueer de tweede afgeleide in de kritische punten:
Een functie heeft een buigpunt waar de tweede afgeleide van teken verandert. Door f''(x) = 0 te stellen en op te lossen naar x, wordt x = 1 verkregen. Aangezien f''(x) bij x = 1 van teken verandert, is dit punt een buigpunt. Deze analyse illustreert hoe de tweede-afgeleidetoets helpt belangrijke kenmerken van de grafiek van een functie te identificeren.
Consider a mug whose cross-sectional area varies with height—it is wider at the bottom and top and narrower in the middle.
When coffee is poured into this mug at a constant volumetric rate, the coffee level rises over time. The rate of this rise is inversely related to the cross-sectional area at that height.
The concavity of the curve depends on the sign of the second derivative of height with respect to time.
In the lower half of the mug, the cross-sectional area changes in a way that causes the height to accelerate. Because the height of the liquid accelerates, the second derivative is positive in this region, resulting in a concave-up curve.
On the other hand, cross-sectional area increases in the upper-half, and shows the opposite effect, height decelerates, meaning the second derivative is negative, and corresponds to a concave down region on the graph.
Inflection points mark where concavity changes.
In this example, the inflection point is located near the middle of the mug, where the cross-sectional area is minimum. So, the acceleration of height represented by its second derivative has decreased to zero following its passage from positive to negative values.
From Chapter 3:
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