Calculus Function Analysis Quiz

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Questions and Answers

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?

0

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?

<p>1 cm</p> Signup and view all the answers

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?

<p>True (A)</p> Signup and view all the answers

What is the limit of xln(x) as x approaches 0 from the right side?

<p>0</p> Signup and view all the answers

Is the function f(x) continuous from the right at 0?

<p>True (A)</p> Signup and view all the answers

What is the limit of the expression f(x)/x as x approaches positive infinity?

<p>+∞</p> Signup and view all the answers

What is the geometric interpretation of the limit as x approaches positive infinity of f(x)/x?

<p>The geometric interpretation of the limit as <em>x</em> approaches infinity of <em>f(x)/x</em> is the slope of the asymptote of the curve (C) as <em>x</em> approaches infinity.</p> Signup and view all the answers

What does the limit of f(x) as x approaches positive infinity tell us about the curve (C)?

<p>It tells us that the curve (C) has an asymptote as <em>x</em> approaches positive infinity. The limit of <em>f(x)</em> as <em>x</em> approaches infinity indicates the behavior of the curve as <em>x</em> becomes very large.</p> Signup and view all the answers

What type of branch does the curve (C) have near positive infinity?

<p>The curve (C) has a parabolic branch near positive infinity.</p> Signup and view all the answers

What is the derivative of f(x) in terms of ln(x)?

<p>f'(x) = ln^2(x) - 2</p> Signup and view all the answers

What interval does x belong to in the derivative equation of f(x)?

<p>The interval is from 0 to positive infinity.</p> Signup and view all the answers

What is the value of f(e^-√2)?

<p>1.2</p> Signup and view all the answers

Does the curve (C) have a single point of inflection located at (1 ; 0)?

<p>True (A)</p> Signup and view all the answers

What is the equation of the tangent line to the curve (C) at the point (1 ; 0)?

<p>(T)</p> Signup and view all the answers

Does the curve (C) intersect the x-axis twice within the interval from 0 to positive infinity?

<p>True (A)</p> Signup and view all the answers

What needs to be determined?

<p>We need to determine the x-coordinates where the curve (C) intersects the x-axis.</p> Signup and view all the answers

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?

<p>True (A)</p> Signup and view all the answers

What is the integral of xln(x) from 1 to e?

<p>(3e + 1)/4</p> Signup and view all the answers

Flashcards

What is the function f(x)?

A function defined on the interval [0, +∞) with the rule f(x) = xln(x)(-2 + ln(x)) for x>0, and f(0) = 0.

What is C?

The graph of the function f(x) in an orthonormal coordinate system.

What is continuity on the right at 0?

The limit of the function as x approaches 0 from the right. It exists if the left-hand limit and right-hand limit are equal.

What is an asymptotic parabola near +∞?

A function has an asymptotic parabola near +∞ if the difference between the function and the parabola approaches 0 as x approaches +∞.

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What is f'(x)?

The derivative of the function f(x) for x in the interval (0, +∞).

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What is an inflection point?

A point on the curve where the curvature changes from concave up to concave down or vice versa.

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What is a tangent line?

A line that touches the curve at a particular point and has the same slope as the curve at that point.

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Where does the curve intersect the x-axis?

The points where the graph of the function intersects the x-axis.

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What is a primitive of a function?

A function whose derivative is the original function.

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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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