Understanding Function Values and Derivatives

This page provides detailed instruction on calculating function values and derivatives through graphical interpretation and practical examples.

The content begins with determining funktionswert berechnen at specific points x₁ = 5 and x₂ = 3, where f(5) = 2.2 and f(3) = 1.7 are obtained through graph reading.

Example: The difference calculation f(5)-f(2)= 0.5 corresponds to the shorter side of the slope triangle shown in the figure.

Definition: The slope between two points P₁(3,1.7) and P₂(5,2.2) is calculated as f(5)-f(3)/(5-3) ≈ 0.25.

Highlight: The derivative f'(5) = 0.2 represents the slope of the tangent at point P₂(5,2.2).

The page also includes practical applications:

Example: A population statistics problem where f(5)=81.2 indicates 81.2 million inhabitants in 2015 (2010+5).

Vocabulary: The rate of change $f(5.5)-f(5)$/(5.5-5)=0.7 represents the average population change of 0.7 million inhabitants per year during the first half of 2015.

The material concludes with a water tank scenario demonstrating how derivative signs indicate different phases:

• Positive derivative during filling
• Zero derivative while bathing
• Negative derivative during emptying