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  • 2r

    y 2 -

    y 1x 2

    - x

    1

    m =

    r2

    Michael Keating, Derek Mulvany and James OLoughlin

    Special Advisors: Oliver Murphy, Colin Townsend and Jim McElroy

    Active Maths 4

    I SBN 9 7 8 - 1 - 7 8 0 9 0 - 5 3 1 - 0

    9 7 8 1 7 8 0 9 0 5 3 1 0folensonline.ie

    leaving certificate HigHer level

    Book 1

    2714

    Supplementary Material for 2015 exam and

    onwards

  • Chapter 13 Differential Calculus I

    13.10 Implicit Differentiation 1

    Answers ............................................................................................................................................................... 5

    Additional copies of this booklet are available to download from folensie

    Editor: Priscilla OConnorDesigner: Liz WhiteLayout: CompuscriptIllustrations: Compuscript

    ISBN: 978-1-78090-531-0

    Michael Keating, Derek Mulvany, James OLoughlin and Colin Townsend, 2014

    Folens Publishers, Hibernian Industrial Estate, Greenhills Road, Tallaght, Dublin 24, Ireland

    All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission from the publisher The publisher reserves the right to change, without notice, at any time, the specification of this product, whether by change of materials, colours, bindings, format, text revision or any other characteristic

    Contents

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    Learning Outcomes Use differentiation to find the slope of a tangent to a circle

    13.10 impliCit DifferentiationThe functions that we have met so far can be described by expressing one variable explicitly in terms of another variable, for example:

    y = ______

    x2 + 5 or y = 2x2 cos x

    Sometimes, we deal with equations in x and y in which the relation between x and y is implicit.

    Eq. (1) x 2 + y 2 = 36

    or

    Eq. (2) x 2 + y 2 + 2x + 2y 34 = 0

    The graph of Eq. 1 above is a circle with centre (0,0) and radius length 6, while the graph of Eq. 2 is a circle with centre (1,1) and radius length 6 (See Active Maths 4, Book 2, Chapter 11).

    On our course we need to be able to use differentiation to find the slope of a tangent to a circle. Therefore, we will need to know how to differentiate equations such as Eq. (1) and Eq. (2) above.

    Worked example 13.28

    Use differentiation to find the slope of the tangent to the circle x 2 + y 2 = 25 at the point (3,4).

    SolutionDifferentiate both sides of the equation x 2 + y 2 = 25 with respect to x.

    d(x2 + y2)

    _________ dx

    = d(25)

    _____ dx

    d(x2)

    ____ dx

    + d(y2)

    ____ dx

    = d(25)

    _____ dx

    d(x2)

    ____ dx

    = 2x

    d(25)

    _____ dx

    = 0 (as 25 is a constant)

    What about d(y2)

    ____ dx

    ?

    Use the Chain Rule here:

    Write d(y2)

    ____ dx

    = d(y2)

    ____ d()

    d()

    ___ dx

    Replace () with y

    d(y2)

    ____ dx

    = d(y2)

    ____ dy

    dy

    ___ dx

    = 2y dy

    ___ dx

    Therefore:

    2x + 2y d ___ dx

    = 0

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    Worked example 13.29

    (i) Use differentiation to find the slope of the tangent t to the circle x 2 + y 2 + 6x 2y 15 = 0 at the point (6,3).

    (ii) Hence, find the equation of t.

    Solve this equation for dy

    ___ dx

    :

    2y dy

    ___ dx

    = 2x

    dy

    ___ dx

    = 2x ___ 2y

    dy

    ___ dx

    = x __ y

    Now find the value of dy

    ___ dx

    at (3,4). This will be the required slope.

    dy

    ___ dx

    |(3,4)

    = 3 __ 4

    Alternative Method:

    Write y explicitly in terms of x:

    x 2 + y 2 = 25

    y 2 = 25 x 2

    y = _______

    25 x 2

    y = _______

    25 x 2 is the semicircle of radius 5 units and centre (0,0) on or above the x-axis.

    y = _______

    25 x 2 is the semicircle of radius 5 units and centre (0,0) on or below the x-axis.

    As we are looking for the slope of the tangent at the point (3,4), we will use y =

    _______ 25 x 2

    y = (25 x 2 ) 1 __ 2

    Use the Chain Rule:

    y = u 1 __ 2 u = 25 x 2

    dy

    ___ du

    = 1 __ 2 u 1 __ 2

    du ___

    dx = 2x

    dy

    ___ dx

    = dy

    ___ du

    du ___ dx

    = 1 __ 2 u 1 __ 2 (2x)

    = x ___ u

    1 __ 2

    = x ________

    _______ 25 x 2

    = x __ y

    The required slope is dy

    ___ dx

    |(3,4)

    = 3 __ 4

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    iSolution (i) x 2 + y 2 + 6x 2y 15 = 0

    2x + 2y dy

    ___ dx

    + 6 2 dy

    ___ dx

    + 0 = 0

    (2y 2) dy

    ___ dx

    = 2x 6

    (y 1) dy

    ___ dx

    = x 3

    dy

    ___ dx

    = x 3 _______ y 1

    dy

    ___ dx

    |(6,3) = (6) 3 _________ 3 1 = 6 3 _____ 4

    = 3 ___ 4

    Slope = 3 __ 4

    (ii) Point (6,3) Slope = 3 __ 4

    y (3) = 3 __ 4 (x (6))

    4(y + 3) = 3(x + 6)

    4y + 12 = 3x 18

    t: 3x + 4y + 30 = 0

    exercise 13.9

    1. For each of the following circles, express dy

    ___ dx

    in terms of x and y.

    (i) x 2 + y 2 = 16 (iv) 3 x 2 + 3 y 2 = 64

    (ii) x 2 + y 2 = 49 (v) a x 2 + a y 2 = b (where a, b are of equal sign)

    (iii) 2 x 2 + 2 y 2 = 81

    2. For each of the following circles, express

    dy

    ___ dx

    in terms of x and y.

    (i) x 2 + y 2 + 2x 2y + 1 = 0

    (ii) x 2 + y 2 + 4x 6y + 11 = 0

    (iii) 2 x 2 + 2 y 2 2x 8y 1 = 0

    (iv) 5 x 2 + 5 y 2 + 2x 2y 1 = 0

    (v) 7 x 2 + 7 y 2 + 2x 2y 3 = 0

    3. For each of the following circles, express

    dy

    ___ dx

    in terms of x and y.

    (i) (x 3 ) 2 + (y + 2 ) 2 = 14

    (ii) (x 8 ) 2 + (y 2 ) 2 = 36

    (iii) (x + 3 ) 2 + (y 9 ) 2 = 24

    (iv) (x 5 ) 2 + (y + 3 ) 2 = 25

    (v) (x 7 ) 2 + (y + 6 ) 2 = 49

    4. Use differentiation to find the slope of the tangent to the circle x 2 + y 2 = 100 at the point (6,8).

    5. Use differentiation to find the slope of the tangent to the circle x 2 + y 2 6x 2y 3 = 0 at the point (5,4).

    6. Use differentiation to find the slope of the tangent to the circle (x 1 ) 2 + (y + 2 ) 2 = 5 at the point (3,3). Hence, find the equation of the tangent to the circle containing the point (3,3).

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    i7. The graphs of the function f(x) = x 2 + 2x 3 and the circle (x 7 ) 2 + (y 1 ) 2 = 20 are shown.

    AC is a tangent to the circle at A(3,3) and a tangent to f(x) at C. B is the centre of the circle.

    (i) Use differentiation to find the slope of the tangent AC.

    (ii) Find the equation of AC.

    (iii) Use two different methods to find the co-ordinates of the point C.

    (iv) Show that |AC| : |AB| = 3 : 2.

    (v) Show that the area of the circle is more than four times greater than the area of the triangle ABC.

    8. c is the circle x 2 + y 2 + 8x + 14 = 0, and d is the circle x 2 + y 2 4x 8y 30 = 0. a is a tangent to c at the point F(3,1).

    (i) Use differentiation to find the slope of a.

    (ii) Find the equation of a.

    (iii) Show that a contains the centre of d.

    (iv) Find the co-ordinates of E, if E is an endpoint of the diameter [EF].

    (v) A tangent f is drawn from E to the circle c. Show that the equation of f is 79x 119y + 518 = 0.

    (vi) Find to the nearest degree the measure of the acute angle between the tangents a and f.

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    irevision exercises extra Questions

    13A.

    The circle (x 3) 2 + (y 3) 2 = 5 has centre A and contains the points B and C.

    (i) Write the equation of this circle in the form x 2 + y 2 + ex + fy + g = 0, where e, f, g R.

    (ii) Express the slope of a tangent to this circle at a point on the circle in terms of x and y.

    (iii) Hence, calculate the slope of the tangent BD.

    (iv) The point C is vertically above the centre point A. Find the slope of the tangent CD.

    (v) Find the co-ordinates of the point D.

    (vi) Calculate the radius of the circle. Hence, calculate the area of the kite ABDC.

    13B. C is the circle x 2 + y 2 4x 2y + 1 = 0.

    (i) Verify that the point A(3, 1 + __

    3 ) lies on C.

    (ii) Use differentiation to find the slope of the tangent to C at the point A.

    (iii) Show that the equation of the tangent at A

    can be written as y = 1 ___

    __ 3 x + 2

    __ 3 + 1.

    (iv) Sketch the circle and tangent.

    (v) Find the area of the triangle enclosed between the tangent at A and the x and y axes.

    AnswersChapter 13

    Exercise 13.91. (i) x __ y (ii)

    x __ y (iii) x __ y (iv)

    x __ y (v) x __ y