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Questions and Answers
What is the function f(x) defined for x greater than 0?
What is the function f(x) defined for x greater than 0?
f(x) = xln(x)(-2 + ln(x))
What is the value of f(x) when x equals 0?
What is the value of f(x) when x equals 0?
0
What represents (C) in the problem statement?
What represents (C) in the problem statement?
(C) is the curve representing the function f(x) in an orthonormal coordinate system.
What is the length of the unit in the orthonormal coordinate system?
What is the length of the unit in the orthonormal coordinate system?
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Is the expression xln(x^2) equivalent to 4(√x ln(√x))^2 for any value of x in the interval from 0 to positive infinity?
Is the expression xln(x^2) equivalent to 4(√x ln(√x))^2 for any value of x in the interval from 0 to positive infinity?
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What is the limit of xln(x) as x approaches 0 from the right side?
What is the limit of xln(x) as x approaches 0 from the right side?
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Is the function f(x) continuous from the right at 0?
Is the function f(x) continuous from the right at 0?
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What is the limit of the expression f(x)/x as x approaches positive infinity?
What is the limit of the expression f(x)/x as x approaches positive infinity?
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What is the geometric interpretation of the limit as x approaches positive infinity of f(x)/x?
What is the geometric interpretation of the limit as x approaches positive infinity of f(x)/x?
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What does the limit of f(x) as x approaches positive infinity tell us about the curve (C)?
What does the limit of f(x) as x approaches positive infinity tell us about the curve (C)?
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What type of branch does the curve (C) have near positive infinity?
What type of branch does the curve (C) have near positive infinity?
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What is the derivative of f(x) in terms of ln(x)?
What is the derivative of f(x) in terms of ln(x)?
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What interval does x belong to in the derivative equation of f(x)?
What interval does x belong to in the derivative equation of f(x)?
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What is the value of f(e^-√2)?
What is the value of f(e^-√2)?
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Does the curve (C) have a single point of inflection located at (1 ; 0)?
Does the curve (C) have a single point of inflection located at (1 ; 0)?
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What is the equation of the tangent line to the curve (C) at the point (1 ; 0)?
What is the equation of the tangent line to the curve (C) at the point (1 ; 0)?
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Does the curve (C) intersect the x-axis twice within the interval from 0 to positive infinity?
Does the curve (C) intersect the x-axis twice within the interval from 0 to positive infinity?
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What needs to be determined?
What needs to be determined?
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Is the function H(x) = 1/2(2x*ln(x) - x^2) a primitive of the function h(x) = xln(x) on the interval from 0 to positive infinity?
Is the function H(x) = 1/2(2x*ln(x) - x^2) a primitive of the function h(x) = xln(x) on the interval from 0 to positive infinity?
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What is the integral of xln(x) from 1 to e?
What is the integral of xln(x) from 1 to e?
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Study Notes
Problem Statement
- Function f(x) is defined on [0,+∞) with f(x) = xln(x)(-2+ln(x)) for x > 0 and f(0) = 0
- Curve (C) is the graph of the function in an orthonormal coordinate system (0, i, j) with 1cm = 1 unit
- Various analysis tasks are presented regarding the function and its graph.
Task 1
- Verify xln²(x) = (√x ln(√x))² for x ∈ [0,+∞) and derive the limit of xln²(x) as x approaches 0 from the right.
Task 2
- Prove that the function f(x) is continuous from the right at x = 0
Task 3
- Determine the limit of f(x) as x approaches +∞ and interpret the result geometrically.
Task 4
- Calculate the limit of f(x) as x approaches +∞ and demonstrate that the graph (C) has an asymptote of a parabolic branch near positive infinity and specify its direction
Task 5
- a) Prove that the derivative of f(x) is f'(x) = ln²(x) - 2 for x ∈ ]0,+∞[
- b) Analyze the sign of f'(x) on [0,+∞) and create a table of the function's variations.
Task 6
- Prove that curve (C) has only one inflection point at (1;0) and establish the equation of the tangent line (T) at this point
Task 7
- a) Show that (C) intersects the x-axis in two points in the interval [0,+∞) and find the x-values of these points.
- b) Construct the tangent line (T) and curve (C) on the same coordinate system.
Task 8
- a) Demonstrate that H(x) = ½(2x²ln(x) - x²) is an antiderivative of h(x) = xln(x) in the interval [0,+∞).
- b) Calculate the definite integral of xln(x) dx from 0 to positive infinity.
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Description
Test your understanding of the properties of the function f(x) = xln(x)(-2+ln(x)). This quiz covers continuity, limits, differentiability, and geometric interpretations of the graph. Dive into various tasks that challenge your grasp of calculus concepts and analysis.
