Maturprüfung Mathematik 2024 – Aufgabe 5: f(x)=1/x² (Tangente, Fläche, Newton)

⬇️ DOWNLOAD MATERIAL (PDF): • Problem sheet (without solution): https://drive.google.com/file/d/1LZK3... • Solution sheet (final results): https://drive.google.com/file/d/1CdNg... This analysis problem from the Matura (Austrian school-leaving exam) in mathematics (basic subject, AME), May 2024, reveals a surprising amount of complexity hidden within the simple function f(x) = 1/x². We solve step by step: • a) sketch the graph (table of values, symmetry, asymptotes), • b) show that t(x) = −2x + 3 is the tangent at point P(1, 1), • c) calculate the area (including the improper integral) – result 3/4, • d) find the point Q with the shortest distance to the origin, • e) find the intersection point with h(x) = sin(2x) using Newton's method. Everything is explained clearly, with complete solutions – ideal for preparing for the Matura, Abitur, and Passerelle exams. Chapters: 0:00 Opening 0:11 Note on the voiceover 0:19 Problem statement – ​​Overview 0:42 a) Sketch the graph of f(x) = 1/x² 1:18 b) Tangent t(x) = -2x + 3 at point P(1, 1) 1:57 c) Area (improper integral) 2:39 d) Point Q with shortest distance to the origin 3:19 e) Intersection point with sin(2x) – Newton's method Subscribe for more step-by-step solutions – ideal for preparing for the Matura, Abitur, and Passerelle exams! More step-by-step solutions: www.mathefrosch.com Note on the voiceover: The voiceover in this video was partially created with an AI-generated version of my own voice. Therefore, some sentences may sound slightly off. Do you notice a sentence that sounds unusual? Feel free to write it in the comments so the clone can be improved. The Math Frog says thank you! #MaturaExam #Mathematics #Analysis #Derivative #Integral #NewtonMethod #Passerelle #Abitur #Matura #MathFrog