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IG: @musclemath_ 1 NAME: TOPIC: Maclaurin Series Website: www.musclemathtuition.com Instagram: musclemath_ Facebook: www.facebook.com/musclemathtuition Email: [email protected] Concepts Overview: - Maclaurin Series Equation - Standard Series - Binomial Expansion - Small Angle Approximation - Unique Concepts: Difference between ascending and descending order questions Equation of tangent at =0 Accuracy of Approximates

NAME: TOPIC: Maclaurin Series...Example 13 [NYJC Prelim/2014/P2/Q3] Given that =ln√1+ 1+ , where −1

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  • IG: @musclemath_ 1

    NAME:

    TOPIC: Maclaurin Series

    Website: www.musclemathtuition.com

    Instagram: musclemath_

    Facebook: www.facebook.com/musclemathtuition

    Email: [email protected]

    Concepts Overview: - Maclaurin Series Equation - Standard Series - Binomial Expansion - Small Angle Approximation

    - Unique Concepts:

    • Difference between ascending and descending order questions

    • Equation of tangent at 𝑥 = 0 • Accuracy of Approximates

    mailto:[email protected]

  • www.musclemathtuition.com 2

    Maclaurin’s Series

    𝑓(𝑥) = 𝑓(0) + 𝑥𝑓 ′(0) + 𝑥2

    2!𝑓 ′′(0) +

    𝑥3

    3!𝑓′′′(0) + ⋯

    • In every Maclaurin series questions, we have to find 𝑓(0), 𝑓′(0), 𝑓′′(0), 𝑓′′′(0), … • Always try to simplify the expression with every step. • Recall: sin(0) = 0, cos(0) = 1, tan(0) = 0

    Standard Series (from MF26)

    (1 + 𝑥)𝑛 = 1 + 𝑛𝑥 +𝑛(𝑛 − 1)

    2!𝑥2 + ⋯ +

    𝑛(𝑛 − 1) … (𝑛 − 𝑟 + 1)

    𝑟!𝑥𝑟 + ⋯, |𝑥| < 1

    (Also known as the Binomial Expansion)

    𝑒𝑥 = 1 + 𝑥 +𝑥2

    2!+

    𝑥3

    3!+ ⋯ +

    𝑥𝑟

    𝑟!+ ⋯, (all 𝑥)

    sin 𝑥 = 𝑥 −𝑥3

    3!+

    𝑥5

    5!− ⋯ +

    (−1)𝑟𝑥2𝑟

    (2𝑟+1)!+ ⋯, (all 𝑥)

    cos 𝑥 = 1 −𝑥2

    2!+

    𝑥4

    4!− ⋯ +

    (−1)𝑟𝑥2𝑟

    (2𝑟)!+ ⋯ (all 𝑥)

    ln(1 + 𝑥) = 𝑥 −𝑥2

    2+

    𝑥3

    3− ⋯ +

    (−1)𝑟+1𝑥𝑟

    𝑟+ ⋯, −1 < 𝑥 ≤ 1

    (For sin and cos, 𝑥 must be in radians)

    In this chapter, we will touch on a few key points: ➢ How to properly present your solution in a Maclaurin’s series question ➢ How to deal with questions that incorporate various concepts outside of this chapter ➢ Dealing with common types of questions, which include: approximation, deducing variations of

    the original series, integration, etc.

  • IG: @musclemath_ 3

    Example 1 [N2003/P2/Q5] It is given that 𝑦 = sin[ln(1 + 𝑥)]. Show that

    (i) (1 + 𝑥)d𝑦

    d𝑥= cos[ln(1 + 𝑥)]

    (ii) (1 + 𝑥)2d2𝑦

    d𝑥2+ (1 + 𝑥)

    d𝑦

    d𝑥+ 𝑦 = 0

    Find the Maclaurin series for y, up to and including the term in 𝑥3.

  • www.musclemathtuition.com 4

    Example 2 [NJC Prelim/P2/Q2]

    Given that 𝑦 = 1

    √1−𝑥2 , show that

    (i) (1 − 𝑥2)d𝑦

    d𝑥= 𝑥𝑦

    (ii) d3𝑦

    d𝑥3= 0 when 𝑥 = 0

    Obtain the Maclaurin’s expansion of y up to and including the term in 𝑥2. Hence find the Maclaurin’s expansion of 𝑦 = sin−1𝑥 up to and including the term in 𝑥3.

  • IG: @musclemath_ 5

    Example 3 [ACJC Prelim/P2/Q4]

    (i) Given that 𝑦 = tan 𝑥, show that d2𝑦

    d𝑥2= 2𝑦

    d𝑦

    d𝑥. Hence find Maclaurin’s series for y, up to and

    including the term in 𝑥3 . (ii) Using the standard series expansion for ln(1 + 𝑥) and Maclaurin’s series for y, find the series

    expansion of ln(1 + tan 𝑥), in ascending powers of 𝑥 up to and including the term in 𝑥3 .

    (iii) Hence, show that the first three non-zero terms in the expansion of sec22𝑥

    1+tan 2𝑥 are 1 − 2𝑥 + 8𝑥2.

  • www.musclemathtuition.com 6

    Example 4 [VJC Prelim/2013/P1/Q2]

    It is given that 𝑦 = √1 + ln(1 + 𝑥). (i) By differentiating successively, find the Maclaurin’s series for 𝑦, up to and including the term in

    𝑥2.

    (ii) By substituting 𝑥 =1

    4, approximate ln (

    4

    5) , leaving your answer in exact form.

  • IG: @musclemath_ 7

    CHECKPOINT

    Example 5 [TMJC Prelim/2019/P2/Q1]

    Given that 𝑦 = √5 − 𝑒2𝑥, show that

    𝑦d𝑦

    d𝑥= −𝑒2𝑥 .

    [1] By further differentiation of this result, find the Maclaurin series for 𝑦 up to and including the term in 𝑥2. [4]

  • www.musclemathtuition.com 8

    CHECKPOINT

    Example 6 [SRJC Prelim/2010/P1/Q4]

    Given that 𝑒𝑦 = √𝑒 + 𝑥 + sin 𝑥3 . Show that

    3𝑒3𝑦d2𝑦

    d𝑥2+ 9𝑒3𝑦 (

    d𝑦

    d𝑥)

    2

    + sin 𝑥 = 0.

    [3] Hence, find in terms of 𝑒, the Maclaurin’s series for 𝑦, up to and including the term in 𝑥2. [4]

  • IG: @musclemath_ 9

    Binomial Expansion The binomial expansion (1 + 𝑥)𝑛 is a sub section of the Maclaurin’s Series chapter as this concept is heavily tested in many questions. In the MF26, the formula for the binomial expansion is as follows:

    (𝑎 + 𝑏)𝑛 = 𝑎𝑛 + (𝑛1

    ) 𝑎𝑛−1𝑏 + (𝑛2

    ) 𝑎𝑛−2𝑏2 + (𝑛3

    ) 𝑎𝑛−3𝑏3 + ⋯ + 𝑏𝑛,

    where 𝑛 is a positive real integer. However, in the A level syllabus, 𝑛 is not always a positive real integer. (e.g. 𝑛 might be a fraction, a negative number or both). Hence, for the majority of the time in the A levels, we will be using the binomial expansion formula that had been derived from the Maclaurin’s expansion in the MF26:

    (1 + 𝑥)𝑛 = 1 + 𝑛𝑥 +𝑛(𝑛 − 1)

    2!𝑥2 +

    𝑛(𝑛 − 1)(𝑛 − 2)

    3!𝑥3 + ⋯ +

    𝑛(𝑛 − 1) … (𝑛 − 𝑟 + 1)

    𝑟!𝑥𝑟 + ⋯

    In this form, 𝑛 can be positive real integers, fractions and negative numbers as well. However, for the expansion to be valid,

    |𝑥| < 1 Note:

    (𝑛𝑟

    ) =𝑛!

    𝑟! (𝑛 − 𝑟)!

    Example 7 [NYJC/1999]

    Expand (1 − 2𝑥2)−1

    2 in ascending powers of 𝑥 as far as the term in 𝑥6, and state the range of values of

    𝑥 for which the expansion is valid. Hence, evaluate 1

    √0.98 to 7 decimal places.

  • www.musclemathtuition.com 10

    Example 8 [NJC/1999]

    Expand (1

    𝑥− 4)

    −1, where |𝑥| <

    1

    4, in ascending powers of 𝑥 up to and including the term in 𝑥4. Show

    that the coefficient of 𝑥𝑛 in this expansion is 2𝑘 where 𝑘 is a non-negative integer in terms of 𝑛. Example 9 [SRJC Prelim/2017/P1/Q7(a)] By considering the Maclaurin expansion for cos 𝑥, show that the expansion of sec 𝑥 up to and including

    the term in 𝑥4 is given by 1 +1

    2𝑥2 +

    5

    24𝑥4. Hence show that the expansion for ln(sec 𝑥) up to and

    including the term in 𝑥4 is given by [1

    2𝑥2 + 𝐴𝑥4] where 𝐴 is an unknown constant to be determined.

  • IG: @musclemath_ 11

    Example 10 [ACJC Prelim/2017/P1/Q4] (i) Expand (𝑘 + 𝑥)𝑛, in ascending powers of 𝑥, up to and including the term in 𝑥2, where 𝑘 is a

    non-zero real constant and 𝑛 is a negative integer. (ii) State the range of values of 𝑥 for which the expansion is valid. (iii) In the expansion of (𝑘 + 𝑦 + 3𝑦2)−3, the coefficient of 𝑦2 is 2. By using the expansion in (i), find

    the value of 𝑘.

  • www.musclemathtuition.com 12

    CHECKPOINT

    Example 11 [ACJC Prelim/2019/P1/Q5]

    Given that 𝑦 = tan(1 − 𝑒3𝑥), show that d𝑦

    d𝑥= 𝑘𝑒3𝑥(1 + 𝑦2), where 𝑘 is a constant to be determined. By

    further differentiation of this result, or otherwise, find the first three non-zero terms in the Maclaurin series for tan(1 − 𝑒3𝑥). [5] The first two terms in the Maclaurin series for tan(1 − 𝑒3𝑥) are equal to the first two non-zero terms

    in the series expansion of 𝑥

    𝑎+𝑏𝑥. Find the constants 𝑎 and 𝑏. [3]

  • IG: @musclemath_ 13

    CHECKPOINT

    Example 12 [YIJC Prelim/2019/P1/Q1]

    (i) Expand sin (𝜋

    4− 2𝑥) in ascending powers of 𝑥, up to and including the term in 𝑥3. [3]

    (ii) The first two non-zero terms found in part (i) are equal to the first two non-zero terms in the series expansion of (𝑎 + 𝑏𝑥)−1 in ascending powers of 𝑥. Find the exact values of the constants 𝑎 and 𝑏. Hence find the third exact non-zero term of the series expansion of (𝑎 + 𝑏𝑥)−1 for these values of 𝑎 and 𝑏. [3]

  • www.musclemathtuition.com 14

    Common Question Types Common question types after obtaining the Maclaurin’s Series of 𝑓(𝑥) are:

    (a) ‘Estimation’ Questions. - Use a suitable value of 𝑥 to estimate … - Is 𝑔(𝑥) a good estimate? Explain.

    Concept: A series becomes more accurate when: 1. More terms are used in calculations, 2. The value of 𝑥 used in the question is small, preferably near 0. (b) ‘State Equation of Tangent’ Questions

    - State the equation of the tangent to the curve at 𝑥 = 0. Concept: Terms up to power 1 is the equation of tangent to 𝑦 = 𝑓(𝑥) at 𝑥 = 0.

    A graphical look to the Maclaurin’s Series

    𝑦 = 𝑥 −𝑥3

    3!+

    𝑥5

    5!

    𝑦 = 𝑥 −𝑥3

    3!

    𝑦 = sin 𝑥

  • IG: @musclemath_ 15

    Example 13 [NYJC Prelim/2014/P2/Q3]

    Given that 𝑦 =ln √1+𝑥

    1+𝑥, where −1 < 𝑥 < 1, show that 2(1 + 𝑥)

    d𝑦

    d𝑥+ 2𝑦 =

    1

    1+𝑥.

    (i) By further differentiation, find the Maclaurin series for 𝑦 up to and including the term in 𝑥3. [5]

    (ii) Verify that the same result is obtained if the standard series expansions are used. [3]

    (iii) Deduce the approximate value of ∫ 𝑦 d𝑥.1

    4𝜋

    0 Explain why the approximation is not good. [2]

    (iv) State the equation of the tangent to the curve 𝑦 at 𝑥 = 0. [1]

  • www.musclemathtuition.com 16

    Example 14 [HCI Prelim/2019/P2/Q2]

    It is given that 𝑦 =𝑒sin 𝑥

    √1+2𝑥.

    (i) Show that 1

    𝑦

    d𝑦

    d𝑥+

    1

    1+2𝑥= cos 𝑥. [2]

    (ii) By further differentiation of the result in part (i), find the Maclaurin series for 𝑦 in ascending powers of 𝑥, up to and including the term in 𝑥3. [5]

  • IG: @musclemath_ 17

    (iii) Use your result from part (ii) to approximate the value of ∫𝑒sin 𝑥

    √1+2𝑥 d𝑥.

    1

    0 Explain why this

    approximation is not good. [2]

    (iv) Deduce the Maclaurin series for 1

    𝑒sin 𝑥√1−2𝑥 in ascending powers of 𝑥, up to and including the term

    in 𝑥3. [1]

  • www.musclemathtuition.com 18

    Small Angle Approximation Small angle approximation is a small sub-topic under Maclaurin’s series but questions on this sub-topic can be worth up to quite a few marks. Every angle in calculus is in radians. Besides extensive usage of the standard series in the MF26 for this particular sub-topic, one has to know the properties below. If 𝑥 is sufficiently small,

    sin 𝑥 ≈ 𝑥

    cos 𝑥 ≈ 1 − 1

    2𝑥2

    tan 𝑥 ≈ 𝑥 If a question does not mention (directly, indirectly), we assume 𝑥3 and higher powers of x to be neglected.

    Example 15 [AJC Prelim/2000/P1/Q2] Express sin22𝑥 + 2 cos2𝑥 in ascending powers of 𝑥, given that 𝑥 is sufficiently small for 𝑥3 and higher powers of 𝑥 to be neglected.

    Example 16 If 𝜃 (in radians) is small, show that 1−2 sin 𝜃

    5−3 cos 𝜃≈

    1

    2− 𝜃 −

    3

    8𝜃2.

  • IG: @musclemath_ 19

    Example 17 [TPJC Prelim/2013/P1/Q11 (edited)]

    Given that 𝑥 is sufficiently small such that sin(𝑒𝑥 − 1) ≈ 𝑥 +𝑥2

    2−

    5𝑥4

    24, show that

    1

    sin2(𝑒𝑥 − 1)≈

    1

    𝑥2−

    1

    𝑥+

    3

    4

  • www.musclemathtuition.com 20

    Example 18 In triangle ABC, ∠𝐵 =

    𝜋

    2+ 𝜃, ∠𝐶 =

    𝜋

    6− 𝜃. D is a point on BC such that AD = DC = 1. Show that

    𝐵𝐷 ≈1

    2(1 + 𝜃√3) if 𝜃 is sufficiently small for 𝜃3 and higher powers of 𝜃 to be neglected.

  • IG: @musclemath_ 21

    CHECKPOINT

    Example 19 [HCI Prelim/2019/P1/Q3]

    In the triangle 𝐴𝐵𝐶, 𝐴𝐵 = 1, 𝐴𝐶 = 2 and angle 𝐴𝐵𝐶 = (𝜋

    2− 𝑥) radians. Given that 𝑥 is sufficiently

    small for 𝑥3 and higher powers of 𝑥 to be ignored, show that 𝐵𝐶 ≈ 𝑝 + 𝑞𝑥 + 𝑟𝑥2, where 𝑝, 𝑞, 𝑟 are constants to be determined in exact form. [5]

  • www.musclemathtuition.com 22

    Example 20 [VJC Prelim/2014/P1/Q11(ii), (iii)]

    Let 𝑓(𝑥) =2 cos 2𝑥

    1+sin 2𝑥.

    (i) Given that 𝑥 is small and 𝑥 is in radians, find, without differentiation, the expansion of 𝑓(𝑥) in ascending powers of 𝑥, up to and including the term in 𝑥2. (a) State the equation of the tangent to the curve 𝑦 = 𝑓(𝑥) at the point (0, 2).

    (b) If the term in 𝑥𝑛 in the expansion of 𝑓(𝑥) has coefficient 𝑘, find the value of d𝑛𝑦

    d𝑥𝑛 at 𝑥 = 0,

    giving your answer in terms of 𝑛 and 𝑘. Denote the expansion of 𝑓(𝑥) in part (i) by 𝑔(𝑥).

    (ii) By substituting 𝑥 =𝜋

    12 in 𝑔(𝑥), find an approximation for √3, giving your answer in the form

    𝑎 + 𝑏𝜋 + 𝑐𝜋2, where 𝑎, 𝑏 and 𝑐 are constants to be found. Suggest how the error in the approximation can be reduced.

    END