Exponential Functions and Derivatives
This page delves into complex mathematical concepts related to exponential functions and their derivatives, essential for Mathe im Abitur. The content covers various aspects of function analysis, including solving equations, finding extrema, and calculating limits.
The page begins with a Geogebra task involving an exponential function f(x) = e^(x² + a). Students are guided through several steps to analyze this function:
Example: Solving the equation f(x,a) = 0 yields x = √-a, demonstrating the relationship between the parameter 'a' and the function's roots.
A significant portion of the page is dedicated to derivative calculations and their applications:
Highlight: The first and second derivatives of f(x,a) are calculated, providing insights into the function's behavior and critical points.
The concept of limits is introduced, emphasizing the function's behavior as x approaches infinity:
Definition: The limit of f(x) as x approaches infinity is explored, showing that lim(x→∞) f(x) = 0.
The page also covers more advanced topics such as factorization and solving complex equations:
Vocabulary: Factorization (Faktorisieren) is applied to simplify expressions like e^(x² - 4x + a + 2).
Practical applications are demonstrated through a specific example of a function f₀.₆₅(x) = e^(-x+0.5) · (x² + 0.65), which is analyzed in detail:
Example: The maximum radius of a vase described by this function is calculated to be 1.07 dm at x₁ = 0 and x₂ = 1.59.
