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    AAHL Sample Paper 3 Agnesi

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    annaageeva118
    This document contains a multi-part question about exploring properties of the "witch of Agnesi" curve. The curve is derived from a circle centered at (0,a) with radius a. It is shown that the equation of the curve is y = 8a^3/(x^2 + 4a^2) and that this is an even function. The points of inflection and asymptotes are found. The area under the curve from x = -∞ to ∞ is calculated and it is explained how doubling a would affect this area.

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    © All Rights Reserved
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    AAHL Sample Paper 3 Agnesi

    Uploaded by

    annaageeva118
    84 views2 pages
    This document contains a multi-part question about exploring properties of the "witch of Agnesi" curve. The curve is derived from a circle centered at (0,a) with radius a. It is shown that the equation of the curve is y = 8a^3/(x^2 + 4a^2) and that this is an even function. The points of inflection and asymptotes are found. The area under the curve from x = -∞ to ∞ is calculated and it is explained how doubling a would affect this area.

    Original Description:

    Paper 3 IB examinations Math AA HL Practice

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    This document contains a multi-part question about exploring properties of the "witch of Agnesi" curve. The curve is derived from a circle centered at (0,a) with radius a. It is shown that the equation of the curve is y = 8a^3/(x^2 + 4a^2) and that this is an even function. The points of inflection and asymptotes are found. The area under the curve from x = -∞ to ∞ is calculated and it is explained how doubling a would affect this area.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    84 views2 pages

    AAHL Sample Paper 3 Agnesi

    Uploaded by

    annaageeva118
    This document contains a multi-part question about exploring properties of the "witch of Agnesi" curve. The curve is derived from a circle centered at (0,a) with radius a. It is shown that the equation of the curve is y = 8a^3/(x^2 + 4a^2) and that this is an even function. The points of inflection and asymptotes are found. The area under the curve from x = -∞ to ∞ is calculated and it is explained how doubling a would affect this area.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    −2−

    Answer all questions in the answer booklet provided. Please start each question on a new page. Full
    marks are not necessarily awarded for a correct answer with no working. Answers must be supported
    by working and/or explanations. Solutions found from a graphic display calculator should be supported
    by suitable working. For example, if graphs are used to find a solution, you should sketch these as part
    of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided
    this is shown by written working. You are therefore advised to show all working.

    1. [Maximum mark: 29]

    This question asks you to explore the derivation and properties of a curve known
    as the “witch of Agnesi.”

    In this question, let a be constant such that a ∈ ℤ+.


    A circle centred at the point C(0, a) with radius a units is shown in the diagram
    below. The point P(x, y) is positioned such that it is directly below point C and has

    N
    the same y-coordinate as point A. Both points A and C lie on a line passing

    IO
    through the origin at an angle of inclination θ, where 0 < θ < π.

    AT
    AM LE
    IN
    P
    EX M
    K A
    S
    C
    O

    x
    (a) (i) Show that cos θ = . [2]
    x 2 + 4a 2
    M

    (ii) Show that the equation of the line passing through OC is y = x tan θ. [2]

    (iii) Using the result of part (ii), show that x = 2a cot θ. [2]

    (This question continues on the following page)


    −3−

    (Question 1 continued)

    The y-coordinate of P is 2a sin2 θ . The point P moves as the value of θ varies.

    (b) By eliminating θ, show that the equation of the path of point P is given by:
    8a 3
    y= [5]
    x 2 + 4a 2

    8a 3
    (c) Show that y = is an even function. [1]
    x 2 + 4a 2

    (d) Find the coordinates of the points of inflexion. Give your answer in terms of a. [7]

    N
    8a 3
    (e) Sketch the graph of y= , labelling the stationary point, points of
    x 2 + 4a 2

    IO
    inflexion and any asymptotes. [3]

    AT
    AM LE
    ∞ R

    ∫k R→∞ ∫k
    You may use the fact that f (x) d x = lim f (x) d x without proof.
    IN
    P
    EX M

    8a 3
    (f) (i) Find the area under the curve y = for x ∈ ℝ. [5]
    x 2 + 4a 2
    K A
    S

    (ii) Describe the effect on the area under the curve if the value of a is doubled.
    Justify your answer with an appropriate calculation. [2]
    C
    O
    M

    Turn over

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