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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]
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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]
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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