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Integralrechnung Aufgaben mit Lösungen Klasse 12 PDF - Mathe Klausur Analysis

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Integralrechnung Aufgaben mit Lösungen Klasse 12 PDF - Mathe Klausur Analysis

Amelie

@ameliee.

4 Follower

The document covers key concepts in integral calculus, including solving general integrals, finding antiderivatives, and calculating areas between curves. It provides practice problems with solutions for Integralrechnung Aufgaben mit Lösung Klasse 12 pdf and Mathe Klausur Klasse 12 Analysis.

Key points:
• Solving various types of integrals
• Finding antiderivatives and particular solutions
• Calculating areas between curves and the x-axis
• Applying integrals to real-world problems
• Using integration techniques like substitution

22.11.2023

3186

Name:
Klausur Integralrechnung Nr. 1 (Q1 GK)
Lösungs- und Rechenwege immer vollständig und nachvollziehbar angeben.
Aufgabe 1
Aufgabe 1 Best

Question 7: Velocity and Distance

The final question presents a function f(t) = 5t²e^(-0.5t) representing the velocity profile of a car in a 30-zone over 20 seconds. Students are asked to:

a) Calculate the distance traveled by the car in the first 20 seconds b) Verify that a given function F(t) is an antiderivative of f(t) c) Interpret the meaning of a given integral in the context of the problem

Highlight: This question combines integral calculus with kinematics, demonstrating the application of calculus to physics problems.

The problem requires students to understand the relationship between velocity and distance, apply integration techniques, and interpret mathematical results in a real-world context.

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Question 5: Real-World Application

This question presents a real-world scenario involving a railway tunnel with a parabolic cross-section. Students are asked to:

a) Justify why the given function f(x) = -0.5x² + 3 represents the parabolic shape b) Calculate the volume of concrete needed to build a 5m long section of the tunnel

Highlight: This question demonstrates the application of integral calculus to practical engineering problems.

The problem requires students to interpret the given function in the context of the physical shape and use integration to calculate a volume.

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Question 3: Particular Antiderivative

This question asks students to find a particular antiderivative of the function f(x) = x³ + x that passes through the point (2, 14). This involves:

Finding the general antiderivative
Using the given point to determine the specific value of the constant of integration

Highlight: This question combines integration with the concept of initial value problems in differential equations.

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Question 6: Volume of Revolution

Question 6 involves calculating the volume of a water glass formed by rotating the area between two functions around the x-axis. The functions given are:

f(x) = √(x + 2) g(x) = √(x - 1)

Students are asked to:

a) Calculate the maximum fill volume of the water glass b) Calculate the mass of the empty glass, given the density of glass

Vocabulary: Volume of revolution - The volume of a solid formed by rotating a region bounded by curves around an axis.

This problem tests students' ability to set up and solve complex integrals involving volumes of revolution.

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Question 2: Specific Integrals

Question 2 requires students to calculate specific integrals and provide the corresponding antiderivatives. Three integrals are given:

a) ∫ -sin(2x) dx b) ∫ (5x² - 1) dx c) ∫ 2e^(0.25x) dx

Example: For part (a), the solution would involve recognizing that the antiderivative of -sin(2x) is (1/2)cos(2x) + C.

This question tests students' ability to apply basic integration rules and recognize common integral forms.

Öffnen

Integral Calculus Exam Overview

This document contains an integral calculus exam for advanced high school or early university level mathematics. The exam covers a range of topics within integral calculus and its applications.

Highlight: The exam is designed to test students' understanding of key integral calculus concepts and their ability to apply these concepts to solve problems.

The exam consists of 7 multi-part questions, each focusing on different aspects of integral calculus. Students are instructed to show all their work and reasoning clearly.

Vocabulary: Integral calculus - A branch of mathematics concerned with the theory and applications of integrals and integration.

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Question 4: Area Calculations

Question 4 deals with the function f(x) = x⁴ - 3x² and asks students to:

a) Calculate the area enclosed between the graph of f and the x-axis b) Calculate the area bounded by the graph of f and the line g(x) = 6 in the interval [0, 2] c) Determine how much the graph of g(x) needs to be shifted to enclose an area of exactly 10 units with f(x) in the interval [0, 2]

Example: For part (a), students would need to find the roots of f(x), set up a definite integral, and calculate the area.

This question tests students' ability to apply integrals to area calculations and their understanding of how changes in functions affect enclosed areas.

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Question 1: General Integrals

This section asks students to solve a general integral. While the specific integral is not provided in the transcript, this type of question typically requires students to apply integration techniques to find an antiderivative.

Definition: General integral - An indefinite integral that includes a constant of integration, representing a family of antiderivatives.

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Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

Knowunity wurde bei Apple als "Featured Story" ausgezeichnet und hat die App-Store-Charts in der Kategorie Bildung in Deutschland, Italien, Polen, der Schweiz und dem Vereinigten Königreich regelmäßig angeführt. Werde noch heute Mitglied bei Knowunity und hilf Millionen von Schüler:innen auf der ganzen Welt.

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Ich liebe diese App so sehr, ich benutze sie auch täglich. Ich empfehle Knowunity jedem!! Ich bin damit von einer 4 auf eine 1 gekommen :D

Philipp, iOS User

Die App ist sehr einfach und gut gestaltet. Bis jetzt habe ich immer alles gefunden, was ich gesucht habe :D

Lena, iOS Userin

Ich liebe diese App ❤️, ich benutze sie eigentlich immer, wenn ich lerne.

Fächer

Mathe

Stochastik

Diskrete wahrscheinlichkeitsverteilungen

Wahrscheinlichkeitsrechnung (binomialkoeffizient)

Integralrechnung Aufgaben mit Lösungen Klasse 12 PDF - Mathe Klausur Analysis

Amelie

@ameliee.

4 Follower

22.11.2023

3186

12/13

Mathe

111

Question 7: Velocity and Distance

The final question presents a function f(t) = 5t²e^(-0.5t) representing the velocity profile of a car in a 30-zone over 20 seconds. Students are asked to:

Highlight: This question combines integral calculus with kinematics, demonstrating the application of calculus to physics problems.

The problem requires students to understand the relationship between velocity and distance, apply integration techniques, and interpret mathematical results in a real-world context.

Question 5: Real-World Application

This question presents a real-world scenario involving a railway tunnel with a parabolic cross-section. Students are asked to:

a) Justify why the given function f(x) = -0.5x² + 3 represents the parabolic shape b) Calculate the volume of concrete needed to build a 5m long section of the tunnel

Highlight: This question demonstrates the application of integral calculus to practical engineering problems.

The problem requires students to interpret the given function in the context of the physical shape and use integration to calculate a volume.

Question 3: Particular Antiderivative

This question asks students to find a particular antiderivative of the function f(x) = x³ + x that passes through the point (2, 14). This involves:

Finding the general antiderivative
Using the given point to determine the specific value of the constant of integration

Highlight: This question combines integration with the concept of initial value problems in differential equations.

Question 6: Volume of Revolution

Question 6 involves calculating the volume of a water glass formed by rotating the area between two functions around the x-axis. The functions given are:

f(x) = √(x + 2) g(x) = √(x - 1)

Students are asked to:

a) Calculate the maximum fill volume of the water glass b) Calculate the mass of the empty glass, given the density of glass

Vocabulary: Volume of revolution - The volume of a solid formed by rotating a region bounded by curves around an axis.

This problem tests students' ability to set up and solve complex integrals involving volumes of revolution.

Question 2: Specific Integrals

Question 2 requires students to calculate specific integrals and provide the corresponding antiderivatives. Three integrals are given:

a) ∫ -sin(2x) dx b) ∫ (5x² - 1) dx c) ∫ 2e^(0.25x) dx

Example: For part (a), the solution would involve recognizing that the antiderivative of -sin(2x) is (1/2)cos(2x) + C.

This question tests students' ability to apply basic integration rules and recognize common integral forms.

Integral Calculus Exam Overview

This document contains an integral calculus exam for advanced high school or early university level mathematics. The exam covers a range of topics within integral calculus and its applications.

Highlight: The exam is designed to test students' understanding of key integral calculus concepts and their ability to apply these concepts to solve problems.

The exam consists of 7 multi-part questions, each focusing on different aspects of integral calculus. Students are instructed to show all their work and reasoning clearly.

Vocabulary: Integral calculus - A branch of mathematics concerned with the theory and applications of integrals and integration.

Question 4: Area Calculations

Question 4 deals with the function f(x) = x⁴ - 3x² and asks students to:

Example: For part (a), students would need to find the roots of f(x), set up a definite integral, and calculate the area.

This question tests students' ability to apply integrals to area calculations and their understanding of how changes in functions affect enclosed areas.

Question 1: General Integrals

Definition: General integral - An indefinite integral that includes a constant of integration, representing a family of antiderivatives.

Nichts passendes dabei? Erkunde andere Fachbereiche.

Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

Knowunity wurde bei Apple als "Featured Story" ausgezeichnet und hat die App-Store-Charts in der Kategorie Bildung in Deutschland, Italien, Polen, der Schweiz und dem Vereinigten Königreich regelmäßig angeführt. Werde noch heute Mitglied bei Knowunity und hilf Millionen von Schüler:innen auf der ganzen Welt.

Laden im

Google Play

Laden im

App Store

4.9+

Durchschnittliche App-Bewertung

13 M

Schüler:innen lieben Knowunity

#1

In Bildungs-App-Charts in 12 Ländern

950 K+

Schüler:innen haben Lernzettel hochgeladen

Immer noch nicht überzeugt? Schau dir an, was andere Schüler:innen sagen...

iOS User

Ich liebe diese App so sehr, ich benutze sie auch täglich. Ich empfehle Knowunity jedem!! Ich bin damit von einer 4 auf eine 1 gekommen :D

Philipp, iOS User

Die App ist sehr einfach und gut gestaltet. Bis jetzt habe ich immer alles gefunden, was ich gesucht habe :D

Lena, iOS Userin

Integralrechnung Aufgaben mit Lösungen Klasse 12 PDF - Mathe Klausur Analysis

Question 7: Velocity and Distance

Question 5: Real-World Application

Question 3: Particular Antiderivative

Question 6: Volume of Revolution

Question 2: Specific Integrals

Integral Calculus Exam Overview

Question 4: Area Calculations

Question 1: General Integrals

Ähnliche Inhalte

Nichts passendes dabei? Erkunde andere Fachbereiche.

Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

4.9+

13 M

#1

950 K+

Immer noch nicht überzeugt? Schau dir an, was andere Schüler:innen sagen...

Ich liebe diese App so sehr, ich benutze sie auch täglich. Ich empfehle Knowunity jedem!! Ich bin damit von einer 4 auf eine 1 gekommen :D

Die App ist sehr einfach und gut gestaltet. Bis jetzt habe ich immer alles gefunden, was ich gesucht habe :D

Ich liebe diese App ❤️, ich benutze sie eigentlich immer, wenn ich lerne.

Integralrechnung Aufgaben mit Lösungen Klasse 12 PDF - Mathe Klausur Analysis

Question 7: Velocity and Distance

Question 5: Real-World Application

Question 3: Particular Antiderivative

Question 6: Volume of Revolution

Question 2: Specific Integrals

Integral Calculus Exam Overview

Question 4: Area Calculations

Question 1: General Integrals

Ähnliche Inhalte

Nichts passendes dabei? Erkunde andere Fachbereiche.

Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

4.9+

13 M

#1

950 K+

Immer noch nicht überzeugt? Schau dir an, was andere Schüler:innen sagen...

Ich liebe diese App so sehr, ich benutze sie auch täglich. Ich empfehle Knowunity jedem!! Ich bin damit von einer 4 auf eine 1 gekommen :D

Die App ist sehr einfach und gut gestaltet. Bis jetzt habe ich immer alles gefunden, was ich gesucht habe :D

Ich liebe diese App ❤️, ich benutze sie eigentlich immer, wenn ich lerne.