DP Maths AA HL - Calculus - Limits and Continuity

DP Maths AA HL - Calculus - Limits and Continuity - Full notes

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DP Maths AA HL | Limits and Continuity

Limits and Continuity

This topic contains 2 notes:

5.12 Limits and Continuity
5.19 Maclaurin Series

DP MATHEMATICS AA · TOPIC 5: CALCULUS

5.12

Limits, Continuity and L'Hôpital's Rule

Full notes for HL · Syllabus AHL 5.12, 5.13

Contents

1	The concept of a limit
2	Evaluating limits algebraically
3	Limits at infinity
4	Continuity
5	Differentiability
6	Differentiation from first principles revisited
7	Indeterminate forms and L'Hôpital's rule
8	Repeated use and other indeterminate forms
9	Limits by factorizing: more practice
10	Limits by rationalizing
11	Limits at infinity: rational and exponential
12	Standard limits
13	L'Hôpital's rule: more practice
14	Indeterminate products and differences
15	Continuity of piecewise functions
16	Differentiability of piecewise functions
17	First principles for trigonometric and exponential functions
18	Mixed review: worked problems
19	Additional exam-style practice
20	Limits and series (preview)
21	Graphical interpretation of limits
22	Exam-style questions with markscheme
23	Common misconceptions and links
24	Check your understanding
25	Summary and topic map
26	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the Quick check.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
understand limits informally, including one-sided limits
evaluate limits by substitution, factorizing and rationalizing
find limits as x → ±∞ and relate them to asymptotes
determine whether a function is continuous at a point
determine whether a function is differentiable at a point
use first principles for general functions
evaluate limits of the forms 0/0 and ∞/∞ using L'Hôpital's rule, including repeated use

This booklet covers Additional Higher Level (AHL) content (AHL 5.12, 5.13). It is studied by HL students only.

Guiding questions

What does it mean for a function to approach a value? How can a ratio of two quantities that both tend to zero have a finite limit?

How this topic is examined

Limits and L'Hôpital's rule appear mainly on Paper 1, often as short questions ('find the limit') or within longer calculus questions. Continuity and differentiability of piecewise functions are typical 'find a and b' problems.

State the indeterminate form (0/0 or ∞/∞) before applying L'Hôpital's rule.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working
Solve	find the values of the unknowns, using algebra or technology as allowed
Show that	obtain the given result; every step must be shown and the result must not be assumed
Hence / Hence or otherwise	use the previous result (or another valid method)
Sketch	a diagram showing the general shape and key features, labelled
Prove	a formal, rigorous argument from accepted facts
Justify / Explain	give valid reasons or evidence

How to use these notes

Each section teaches one part of the syllabus. You'll see these boxes throughout; each has one job:

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or proof of a key result, so you understand where formulas come from
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently on Paper 2
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given); examples are graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet
Quick check	short questions to check the key ideas

1 The concept of a limit

Approaching, not necessarily reaching.

Think about it

What happens to (x2 − 1)/(x − 1) as x gets closer and closer to 1? The expression is undefined at x = 1, but does it approach a particular number?

We write limx→a f(x) = L if f(x) can be made as close to L as we like by taking x sufficiently close to a (but not equal to a). The value at x = a itself is irrelevant; f(a) may not even exist.

One-sided limits: limx→a⁻ f(x) uses x approaching from below, limx→a⁺ f(x) from above. The limit exists only if both one-sided limits exist and are equal.

x	0.9	0.99	0.999	1.001	1.01	1.1
(x2 − 1)/(x − 1)	1.9	1.99	1.999	2.001	2.01	2.1

2 Evaluating limits algebraically

Substitute, factorize, or rationalize.

If f is continuous at a, simply substitute: limx→2 (x2 + 3x) = 10. If substitution gives 0/0, simplify first: factorize and cancel, or rationalize a surd.

WORKED EXAMPLE 1 Factorizing (basic)

Find limx→3 (x2 − 9)/(x − 3).

Solution

= lim (x − 3)(x + 3)/(x − 3)M1

= lim (x + 3) = 6A1

WORKED EXAMPLE 2 Rationalizing (standard)

Find limx→0 (√(x + 4) − 2)/x.

Solution

Multiply by (√(x + 4) + 2): (x + 4 − 4)/(x(√(x + 4) + 2))M1

= 1/(√(x + 4) + 2) → 1/4A1

Practice exercise 1

1. Find limx→−2 (x2 + 5x + 6)/(x + 2).

2. Find limh→0 ((3 + h)2 − 9)/h.

3. Find limx→9 (x − 9)/(√x − 3).

3 Limits at infinity

Behaviour for large |x|.

For rational functions, divide numerator and denominator by the highest power of x in the denominator: limx→∞ (3x2 + 1)/(2x2 − x) = 3/2. A finite limit as x → ±∞ gives a horizontal asymptote. Exponentials dominate polynomials: lim xne−x = 0 as x → ∞; logarithms grow more slowly than any positive power: lim (ln x)/x = 0.

Practice exercise 2

1. Find limx→∞ (5x3 − x)/(2x3 + 4).

2. Find limx→∞ (2x + 1)/(x2 + 3).

4 Continuity

No breaks in the graph.

f is continuous at x = a if (1) f(a) is defined, (2) limx→a f(x) exists, and (3) the limit equals f(a). Informally, the graph can be drawn through x = a without lifting the pen. Polynomials, exponentials, sine and cosine are continuous everywhere; rational functions are continuous except where the denominator is zero.

For a piecewise function, check that the left and right pieces meet at the join.

WORKED EXAMPLE 3 Making a piecewise function continuous (standard)

f(x) = x2 + k for x < 2 and 3x − 1 for x ≥ 2. Find k so that f is continuous.

Solution

Left limit 4 + k; right value 5M1

4 + k = 5: k = 1A1

5 Differentiability

A smooth join.

f is differentiable at a if f′(a) exists: the limit of (f(a + h) − f(a))/h exists from both sides. A function that is differentiable at a is continuous at a, but not conversely. Corners (such as |x| at 0), cusps, vertical tangents and discontinuities prevent differentiability.

For a piecewise function to be differentiable at the join, it must be continuous and the derivatives of both pieces must agree there.

Figure 1. y = |x| is continuous but not differentiable at x = 0.

WORKED EXAMPLE 4 Continuity and differentiability (exam level)

f(x) = ax2 + bx for x ≤ 1 and 3 − 1/x for x > 1. Find a and b so that f is differentiable at x = 1.

Solution

Continuity: a + b = 3 − 1 = 2M1 A1

Derivatives: 2a + b = 1/x2 at x = 1 = 1M1 A1

Solve: a = −1, b = 3A1

Practice exercise 3

1. Is f(x) = x|x| differentiable at 0? Justify.

2. f(x) = ex for x ≤ 0 and mx + c for x > 0. Find m and c for differentiability at 0.

6 Differentiation from first principles revisited

The limit definition in general.

WORKED EXAMPLE 5 A reciprocal function (exam level)

Differentiate f(x) = 1/x from first principles.

Solution

(1/(x + h) − 1/x)/h = (x − (x + h))/(hx(x + h))M1

= −1/(x(x + h))A1

→ −1/x2 as h → 0A1

Practice exercise 4

1. Differentiate f(x) = √x from first principles.

2. Differentiate f(x) = x4 from first principles.

7 Indeterminate forms and L'Hôpital's rule

Differentiate top and bottom separately.

If lim f(x)/g(x) gives 0/0 or ∞/∞, then (provided the new limit exists):

This also works for a = ±∞. Note: differentiate the numerator and denominator separately, not with the quotient rule.

Why it works

Near a, with f(a) = g(a) = 0, f(x) ≈ f′(a)(x − a) and g(x) ≈ g′(a)(x − a) (tangent line approximations). So f(x)/g(x) ≈ f′(a)/g′(a). The Maclaurin series (5.19) give an alternative justification.

WORKED EXAMPLE 6 A trigonometric limit (standard)

Find limx→0 sin(3x)/x.

Solution

0/0 formR1

= lim 3cos(3x)/1M1

= 3A1

WORKED EXAMPLE 7 At infinity (standard)

Find limx→∞ x2/ex.

Solution

∞/∞ form; L'Hôpital: 2x/ex, still ∞/∞M1

Again: 2/exM1

→ 0A1

Common mistake

Applying the rule when the limit is not indeterminate: limx→0 (x + 1)/(x + 2) = 1/2 by substitution; L'Hôpital would wrongly give 1.

Practice exercise 5

1. Find limx→0 (ex − 1)/x.

2. Find limx→1 ln x/(x − 1).

3. Find limx→0 (1 − cos x)/x2.

4. Find limx→∞ ln x/√x.

8 Repeated use and other indeterminate forms

Rewrite as a quotient first.

If the result after one application is still 0/0 or ∞/∞, apply the rule again. Forms such as 0 × ∞ must first be rewritten as a quotient: x ln x = ln x/(1/x), which is ∞/∞ as x → 0+.

WORKED EXAMPLE 8 Rewriting 0 × ∞ (exam level)

Find limx→0⁺ x ln x.

Solution

= lim ln x/(1/x): −∞/∞ formM1

= lim (1/x)/(−1/x2) = lim (−x)A1

= 0A1

WORKED EXAMPLE 9 Repeated L'Hôpital (exam level)

Find limx→0 (x − sin x)/x3.

Solution

0/0: (1 − cos x)/(3x2), still 0/0M1

sin x/(6x), still 0/0; then cos x/6M1

→ 1/6A1

Practice exercise 6

1. Find limx→0 (ex − 1 − x)/x2.

2. Find limx→∞ x e−x.

9 Limits by factorizing: more practice

Cancel the common factor.

If direct substitution gives 0/0, factorize numerator and denominator and cancel the common factor (x − a), then substitute.

WORKED EXAMPLE 10 Cubic (standard)

Find limx→2 (x3 − 8)/(x2 − 4).

Solution

(x − 2)(x2 + 2x + 4)/((x − 2)(x + 2))M1

= 12/4 = 3A1

Practice exercise 7

1. Find limx→1 (x4 − 1)/(x − 1).

2. Find limx→−3 (x2 + x − 6)/(x + 3).

10 Limits by rationalizing

Multiply by the conjugate.

For expressions with square roots giving 0/0, multiply numerator and denominator by the conjugate, simplify, and substitute.

WORKED EXAMPLE 11 Conjugate (standard)

Find limx→0 (√(9 + x) − 3)/x.

Solution

Multiply by (√(9 + x) + 3): x/(x(√(9 + x) + 3))M1

= 1/6A1

Practice exercise 8

1. Find limx→4 (x − 4)/(√x − 2).

2. Find limx→0 (√(1 + x) − √(1 − x))/x.

11 Limits at infinity: rational and exponential

Dominant terms.

Divide by the highest power of x in the denominator. Exponentials dominate powers: xne−x → 0; logarithms grow slower than powers: (ln x)/xk → 0 for k > 0.

WORKED EXAMPLE 12 Rational (standard)

Find limx→∞ (4x2 − x)/(2x2 + 5).

Solution

Divide by x2M1

= 2A1

WORKED EXAMPLE 13 Exponential (standard)

Find limx→∞ (3ex + x)/(ex − 2).

Solution

Divide by ex: (3 + xe−x)/(1 − 2e−x)M1

= 3A1

Practice exercise 9

1. Find limx→∞ (√(x2 + x) − x).

2. Find limx→−∞ (2x + 1)/√(x2 + 1).

12 Standard limits

sin x/x and (ex − 1)/x.

Two important limits: limx→0 sin x/x = 1 and limx→0 (ex − 1)/x = 1 (x in radians). Combined with algebra, they give many others, such as lim sin(3x)/x = 3.

WORKED EXAMPLE 14 Using standard limits (standard)

Find limx→0 sin(5x)/(2x).

Solution

(5/2) × sin(5x)/(5x)M1

= 5/2A1

Practice exercise 10

1. Find limx→0 tan x/x.

2. Find limx→0 (e3x − 1)/x.

13 L'Hôpital's rule: more practice

0/0 and ∞/∞.

If lim f/g is 0/0 or ∞/∞, it equals lim f′/g′ (if this exists). Differentiate numerator and denominator separately; repeat if still indeterminate.

WORKED EXAMPLE 15 Trig and exponential (exam level)

Find limx→0 (ex − 1 − x)/(1 − cos x).

Solution

0/0: (ex − 1)/sin x, still 0/0M1

ex/cos x → 1M1 A1

WORKED EXAMPLE 16 Logarithm (standard)

Find limx→1 ln x/(x2 − 1).

Solution

0/0: (1/x)/(2x)M1

→ 1/2A1

Practice exercise 11

1. Find limx→0 (x − sin x)/x3.

2. Find limx→∞ x3/e2x.

14 Indeterminate products and differences

Rewrite as a quotient.

0 × ∞: rewrite as a quotient, for example x ln x = ln x/(1/x). ∞ − ∞: combine into a single fraction. Then apply L'Hôpital's rule.

WORKED EXAMPLE 17 Difference (exam level)

Find limx→0 (1/x − 1/sin x).

Solution

(sin x − x)/(x sin x): 0/0M1

L'Hôpital twice: (cos x − 1)/(sin x + x cos x) → −sin x/(2 cos x − x sin x)M1

→ 0A1

Practice exercise 12

1. Find limx→0⁺ x2 ln x.

2. Find limx→∞ x(e1/x − 1).

15 Continuity of piecewise functions

Matching values.

A piecewise function is continuous at the join if both pieces give the same value there (the left and right limits equal the function value). Solve for unknown constants.

WORKED EXAMPLE 18 Continuity (standard)

f(x) = ax2 + 1 for x < 2 and 3x − a for x ≥ 2. Find a.

Solution

4a + 1 = 6 − aM1

a = 1A1

Practice exercise 13

1. f(x) = ex for x ≤ 0, x + k for x > 0. Find k for continuity.

16 Differentiability of piecewise functions

Matching gradients too.

For differentiability at the join, the function must be continuous and the left and right derivatives must be equal. This gives two equations for two unknowns.

WORKED EXAMPLE 19 Smooth join (exam level)

f(x) = x2 + bx for x ≤ 1 and a ln x + 2 for x > 1. Find a and b for differentiability.

Solution

Continuity: 1 + b = 2: b = 1M1 A1

Derivatives: 2 + b = a: a = 3M1 A1

Practice exercise 14

1. Is f(x) = |x − 2| differentiable at x = 2?

2. Find p, q so that f(x) = px3 for x ≤ 1 and qx + 2 for x > 1 is differentiable.

17 First principles for trigonometric and exponential functions

Using standard limits.

Using sin(x + h) = sin x cos h + cos x sin h and the limits sin h/h → 1, (cos h − 1)/h → 0, first principles gives d/dx(sin x) = cos x. Similarly (ex+h − ex)/h = ex(eh − 1)/h → ex.

WORKED EXAMPLE 20 Sine from first principles (exam level)

Show that d/dx(sin x) = cos x.

Solution

(sin x(cos h − 1) + cos x sin h)/hM1 A1

→ sin x × 0 + cos x × 1 = cos xR1 AG

Practice exercise 15

1. Show from first principles that d/dx(cos x) = −sin x.

18 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 21 Mixed problem (exam level)

Find limx→0 (sin 2x − 2x)/x3.

Solution

0/0; L'Hôpital: (2cos 2x − 2)/(3x2)M1

Again: (−4 sin 2x)/(6x), then (−8 cos 2x)/6M1

→ −4/3A1

Practice exercise 16

1. Find limx→0 (cos 3x − 1)/x2.

2. Find limx→∞ ln(x)/√x.

19 Additional exam-style practice

Paper 1.

WORKED EXAMPLE 22 Paper 1 question (exam level)

(a) Find limx→0 (e2x − 1)/tan x. (b) Find limx→π/2 (1 − sin x)/cos2x.

Solution

(a) 0/0: 2e2x/sec2x → 2M1 A1

(b) 0/0: −cos x/(−2 cos x sin x) = 1/(2 sin x) → 1/2M1 A1

Practice exercise 17

1. Find limx→0 x/(ex − e−x).

20 Limits and series (preview)

Infinite sums.

The sum of an infinite series is the limit of its partial sums. Geometric series with |r| < 1 converge; the harmonic series diverges even though its terms tend to zero. Maclaurin series (5.19) give another route to limits.

WORKED EXAMPLE 23 Partial sums (standard)

Find the limit of Sn = 1 − 1/2n.

Solution

1/2n → 0M1

Limit 1A1

21 Graphical interpretation of limits

Holes, jumps and asymptotes.

A hole (removable discontinuity) has a finite limit but no defined value; a jump has different one-sided limits; a vertical asymptote has infinite one-sided limits. Sketches make these distinctions clear.

Figure 2. A hole at x = 1 and an asymptote at x = 3.

WORKED EXAMPLE 24 Classifying (standard)

Classify the discontinuities in the figure.

Solution

x = 1: removable (limit 2)A1

x = 3: infinite (vertical asymptote)A1

22 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 1 style) (about 12 minutes)

(a) Find limx→0 (e2x − 1)/sin x. [3]

(b) Find limx→0 (cos 2x − 1)/x2. [4]

Markscheme

Part	Answer	Marks
(a)	0/0, apply L'Hôpital	M1
	2e2x/cos x	A1
	2	A1
(b)	0/0: −2 sin 2x/(2x)	M1 A1
	0/0 again: −4 cos 2x/2	M1
	−2	A1

Examiner's comments. The indeterminate form should be identified each time the rule is used.

Question 2 (Paper 1 style) (about 10 minutes)

f(x) = x3 + p for x < 1 and qx + 1 for x ≥ 1.

(a) Find p and q such that f is differentiable at x = 1. [4]

(b) Sketch y = f(x) near x = 1. [1]

Markscheme

Part	Answer	Marks
(a)	continuity: 1 + p = q + 1	M1
	derivatives: 3 = q	M1
	q = 3	A1
	p = 3	A1
(b)	smooth join at (1, 4)	A1

Examiner's comments. Both conditions, continuity and equal derivatives, are needed.

23 Common misconceptions and links

Students often think…	Actually…
the limit equals f(a) always	only if f is continuous at a
continuous implies differentiable	\|x\| is a counterexample
L'Hôpital uses the quotient rule	differentiate top and bottom separately
apply L'Hôpital to any quotient	only to 0/0 or ∞/∞

How 5.12 connects to the rest of the course

Topic	Connection
5.1 Differentiation	first principles
2.13 Rational functions	asymptotes as limits
1.1 Series	infinite sums as limits
5.19 Maclaurin series	series approach to limits

24 Check your understanding

Section A: multiple choice

1. limx→2 (x2 − 4)/(x − 2) =

A. 4 B. 0 C. undefined D. 2

2. L'Hôpital applies to

A. any limit B. only 0/0 C. 0/0 and ∞/∞ D. 1/0

3. |x| at x = 0 is

A. differentiable B. continuous but not differentiable C. discontinuous D. undefined

4. limx→0 sin x/x =

A. 0 B. ∞ C. undefined D. 1

5. limx→∞ (2x + 3)/(5x − 1) =

A. 0 B. 2/5 C. ∞ D. −3

Section B: structured

6. Find limx→0 tan x/x. [2]

7. Explain why f(x) = 1/x is not continuous at 0. [1]

Answers

1. A 2. C 3. B 4. D 5. B

6. L'Hôpital: sec2x → 1.

7. f(0) is undefined.

25 Summary and topic map

Idea	Remember
Limit	value approached as x → a
Algebraic methods	substitute, factorize, rationalize
At infinity	divide by highest power; asymptotes
Continuity	f(a) = lim f(x)
Differentiability	continuous and matching derivatives
L'Hôpital	0/0 or ∞/∞: lim f′/g′
Other forms	rewrite 0 × ∞ as a quotient

Before the exam, make sure you…

•state the indeterminate form
•differentiate separately
•check both conditions at a join
•simplify before applying the rule

Limits, Continuity and L'Hôpital on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 3. Topic map for Limits, Continuity and L'Hôpital.

26 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. 1

2. 6

3. 6

Practice exercise 2

1. 5/2

2. 0

Practice exercise 3

1. yes: derivative from both sides is 0

2. c = 1, m = 1

Practice exercise 4

1. rationalize: h/(h(√(x + h) + √x)) → 1/(2√x)

2. (4x3h + 6x2h2 + 4xh3 + h4)/h → 4x3

Practice exercise 5

1. 1

2. 1

3. 1/2

4. 0

Practice exercise 6

1. 1/2

2. 0

Practice exercise 7

1. 4

2. −5

Practice exercise 8

1. 4

2. 1

Practice exercise 9

1. 1/2

2. −2

Practice exercise 10

1. 1

2. 3

Practice exercise 11

1. 1/6

2. 0

Practice exercise 12

1. 0

2. 1

Practice exercise 13

1. k = 1

Practice exercise 14

1. no: left derivative −1, right +1

2. p = q + 2 and 3p = q: p = −1, q = −3

Practice exercise 15

1. (cos x(cos h − 1) − sin x sin h)/h → −sin x

Practice exercise 16

1. −9/2

2. 0

Practice exercise 17

1. 1/2

DP MATHEMATICS AA · TOPIC 5: CALCULUS

5.19

Maclaurin Series

Full notes for HL · Syllabus AHL 5.19

Contents

1	Approximating functions with polynomials
2	The Maclaurin series formula
3	Standard Maclaurin series
4	New series by substitution
5	Products, differentiation and integration of series
6	Approximations and limits using series
7	Maclaurin series from differential equations
8	Deriving series from derivatives: more practice
9	Substitution into standard series
10	Multiplying and dividing series
11	Differentiating and integrating series
12	Approximations and errors
13	Limits using series
14	Series solutions of differential equations
15	Interval of validity
16	Graphs of Maclaurin polynomials
17	Mixed review: worked problems
18	Additional exam-style practice
19	Euler's formula from series
20	Choosing a method for series
21	Exam-style questions with markscheme
22	Common misconceptions and links
23	Check your understanding
24	Summary and topic map
25	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the Quick check.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
understand a Maclaurin series as a polynomial approximation built from derivatives at 0
derive the Maclaurin series of ex, sin x, cos x, ln(1 + x) and (1 + x)p
obtain new series by substitution, multiplication, differentiation and integration
use series to approximate values and evaluate limits
find the Maclaurin series of the solution of a differential equation

This booklet covers Additional Higher Level (AHL) content (AHL 5.19). It is studied by HL students only.

Guiding questions

How can complicated functions be approximated by polynomials? How does a calculator find sin(0.3)?

How this topic is examined

Maclaurin series appear on Papers 1 and 2: derive the first few terms from derivatives, use standard series to build new ones, find limits, approximate integrals, and (often) find the series solution of a differential equation.

Show the derivatives evaluated at 0 when deriving a series, and state the range of validity when it is required.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working
Solve	find the values of the unknowns, using algebra or technology as allowed
Show that	obtain the given result; every step must be shown and the result must not be assumed
Hence / Hence or otherwise	use the previous result (or another valid method)
Sketch	a diagram showing the general shape and key features, labelled
Prove	a formal, rigorous argument from accepted facts
Justify / Explain	give valid reasons or evidence

How to use these notes

Each section teaches one part of the syllabus. You'll see these boxes throughout; each has one job:

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or proof of a key result, so you understand where formulas come from
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently on Paper 2
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given); examples are graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet
Quick check	short questions to check the key ideas

1 Approximating functions with polynomials

Match the value and the derivatives at 0.

Think about it

Near x = 0, the graph of sin x looks very like the line y = x. How could you improve the approximation by adding more terms?

The tangent line at x = 0 matches the function's value and gradient there. Adding an x2 term can also match the curvature (second derivative), an x3 term the third derivative, and so on. Each additional term makes the polynomial agree with the function over a wider interval.

Figure 1. Polynomials of increasing degree approximate sin x over a wider interval.

2 The Maclaurin series formula

Coefficients from derivatives.

Why it works

Suppose f(x) = a0 + a1x + a2x2 + a3x3 + .... Setting x = 0 gives a0 = f(0). Differentiating and setting x = 0 gives a1 = f′(0). Differentiating again, 2a2 = f″(0); again, 3!a3 = f‴(0). In general an = f(n)(0)/n!.

WORKED EXAMPLE 1 Deriving a series (standard)

Find the Maclaurin series of ex up to the x4 term.

Solution

Every derivative is ex, equal to 1 at x = 0M1

ex = 1 + x + x2/2! + x3/3! + x4/4! + ...A1

WORKED EXAMPLE 2 Deriving ln(1 + x) (standard)

Find the first four non-zero terms of ln(1 + x).

Solution

f(0) = 0; f′ = (1 + x)−1: 1; f″ = −(1 + x)−2: −1; f‴ = 2(1 + x)−3: 2; f(4) = −6(1 + x)−4: −6M1 A1

x − x2/2 + x3/3 − x4/4 + ..., valid for −1 < x ≤ 1A1

Practice exercise 1

1. Derive the Maclaurin series of cos x up to x4.

2. Find the first three terms of the Maclaurin series of e2x by differentiation.

3 Standard Maclaurin series

Learn these (they are in the formula booklet).

Function	Series	Valid for
ex	1 + x + x2/2! + x3/3! + ...	all x
ln(1 + x)	x − x2/2 + x3/3 − ...	−1 < x ≤ 1
sin x	x − x3/3! + x5/5! − ...	all x
cos x	1 − x2/2! + x4/4! − ...	all x
arctan x	x − x3/3 + x5/5 − ...	\|x\| ≤ 1
(1 + x)p	1 + px + p(p − 1)x2/2! + ...	\|x\| < 1

sin x has only odd powers (it is odd) and cos x only even powers (it is even). The binomial series is the extended binomial theorem of 1.10.

4 New series by substitution

Replace x with an expression.

Substitute directly into a standard series: e−x² = 1 − x2 + x4/2 − ...; sin 2x = 2x − (2x)3/6 + .... Validity conditions change accordingly: ln(1 + 3x) is valid for −1 < 3x ≤ 1.

WORKED EXAMPLE 3 Substitution (standard)

Find the first three non-zero terms of ln(1 − 2x) and state its validity.

Solution

Replace x by −2x: (−2x) − (−2x)2/2 + (−2x)3/3M1

= −2x − 2x2 − (8/3)x3A1

Valid for −1 < −2x ≤ 1, i.e. −½ ≤ x < ½A1

Practice exercise 2

1. Find the series of cos(3x) up to x4.

2. Find the series of 1/(1 − x2) up to x6.

3. Find the series of ln((1 + x)/(1 − x)) up to x5.

5 Products, differentiation and integration of series

Combine known series.

Products: multiply the series and collect terms up to the required power. Differentiation/integration: differentiate or integrate term by term: d/dx of the sin x series gives the cos x series; integrating 1/(1 + x2) = 1 − x2 + x4 − ... gives the arctan x series.

WORKED EXAMPLE 4 Product of series (exam level)

Find the series of exsin x up to x3.

Solution

(1 + x + x2/2 + ...)(x − x3/6 + ...)M1

x + x2 + x3/2 − x3/6A1

= x + x2 + x3/3A1

WORKED EXAMPLE 5 Integrating a series (exam level)

Use the series of e−x² to estimate ∫ from 0 to 0.5 of e−x² dx.

Solution

e−x² ≈ 1 − x2 + x4/2 − x6/6A1

∫ = [x − x3/3 + x5/10 − x7/42] from 0 to 0.5M1

≈ 0.5 − 0.041 667 + 0.003 125 − 0.000 186 = 0.4613A1

Practice exercise 3

1. Find the series of x cos 2x up to x5.

2. Find the series of exln(1 + x) up to x3.

3. Differentiate the series of ln(1 + x) to obtain the series of 1/(1 + x).

6 Approximations and limits using series

Leading terms dominate near 0.

Truncated series give numerical approximations: e0.1 ≈ 1 + 0.1 + 0.005 + 0.000 167 = 1.105 167. For limits as x → 0, replace each function by its series; the lowest-power terms dominate, giving the limit directly (an alternative to L'Hôpital's rule).

WORKED EXAMPLE 6 A limit using series (exam level)

Find limx→0 (ex − 1 − x)/(1 − cos x).

Solution

Numerator: x2/2 + x3/6 + ...; denominator: x2/2 − x4/24 + ...M1 A1

Divide by x2: (1/2 + x/6 + ...)/(1/2 − x2/24 + ...)M1

→ 1A1

Practice exercise 4

1. Use series to find limx→0 (sin x − x)/x3.

2. Estimate ln 1.2 using four terms of the series.

3. Use series to find limx→0 (arctan x − x)/x3.

7 Maclaurin series from differential equations

Differentiate the equation repeatedly.

If y satisfies a DE with y(0) given, find y′(0) from the DE, then differentiate the DE (implicitly) to find y″(0), y‴(0), ..., and substitute into the Maclaurin formula. This gives an approximate solution when an exact one is hard to find.

WORKED EXAMPLE 7 Series solution (exam level)

y satisfies dy/dx = x + y2, y(0) = 1. Find the Maclaurin series of y up to x3.

Solution

y′(0) = 0 + 1 = 1A1

y″ = 1 + 2yy′: y″(0) = 1 + 2 = 3M1 A1

y‴ = 2(y′)2 + 2yy″: y‴(0) = 2 + 6 = 8A1

y ≈ 1 + x + (3/2)x2 + (4/3)x3A1

Practice exercise 5

1. y′ = y + x, y(0) = 2. Find the series of y up to x3.

2. y″ + y = 0, y(0) = 0, y′(0) = 1. Find the first three non-zero terms and identify the function.

8 Deriving series from derivatives: more practice

Evaluate derivatives at 0.

Compute f(0), f′(0), f″(0), ... and use f(x) = Σ f(n)(0)xn/n!. Organize the derivatives in a table to avoid errors.

n	f(n)(x) for f = ln(1 + 2x)	f(n)(0)
0	ln(1 + 2x)	0
1	2/(1 + 2x)	2
2	−4/(1 + 2x)2	−4
3	16/(1 + 2x)3	16

WORKED EXAMPLE 8 Series from derivatives (standard)

Find the Maclaurin series of ln(1 + 2x) up to x3.

Solution

0 + 2x − 4x2/2 + 16x3/6M1

= 2x − 2x2 + (8/3)x3A1

Practice exercise 6

1. Find the first three non-zero terms of excos x by differentiation.

2. Find the series of tan x up to x3.

9 Substitution into standard series

Replace x by an expression.

Replace x in a standard series by kx, x2, −x, etc. Adjust the interval of validity accordingly.

WORKED EXAMPLE 9 Substitution (standard)

Find the series of e−2x and of sin(x2) up to x6.

Solution

e−2x = 1 − 2x + 2x2 − (4/3)x3 + ...A1

sin(x2) = x2 − x6/6 + ...A1

Practice exercise 7

1. Find the series of 1/(1 + x2) up to x6.

2. Find the series of cos(3x) up to x4.

10 Multiplying and dividing series

Collect terms up to the required power.

Multiply series term by term, keeping only terms up to the required power. For a quotient, write 1/g(x) as a series (using 1/(1 + u) = 1 − u + u2 − ...) and multiply.

WORKED EXAMPLE 10 Product (exam level)

Find the series of ex ln(1 + x) up to x3.

Solution

(1 + x + x2/2)(x − x2/2 + x3/3)M1

x + x2/2 + x3(1/3 − 1/2 + 1/2)A1

= x + x2/2 + x3/3A1

WORKED EXAMPLE 11 Quotient (exam level)

Find the series of tan x = sin x/cos x up to x3.

Solution

1/cos x = 1/(1 − x2/2) = 1 + x2/2 + ...M1

(x − x3/6)(1 + x2/2) = x + x3/3A1

Practice exercise 8

1. Find the series of (1 + x)e−x up to x3.

2. Find the series of sin x/(1 − x) up to x3.

11 Differentiating and integrating series

Term by term.

Differentiating or integrating a Maclaurin series term by term gives the series of the derivative or integral (within the interval of convergence). This gives series for arctan x and ln(1 + x), and estimates of integrals with no elementary antiderivative.

WORKED EXAMPLE 12 Integrating (exam level)

Use the series of e−x² to estimate ∫ from 0 to 0.5 of e−x² dx.

Solution

1 − x2 + x4/2 − x6/6A1

[x − x3/3 + x5/10 − x7/42] from 0 to 0.5M1

≈ 0.4613A1

Practice exercise 9

1. Differentiate the series of sin x to obtain that of cos x.

2. Integrate 1/(1 + x2) = 1 − x2 + x4 − ... to get the arctan series.

12 Approximations and errors

How accurate is a truncated series?

For small x, a few terms give accurate values. For an alternating series with decreasing terms, the error is less than the first omitted term.

WORKED EXAMPLE 13 Estimate with error (standard)

Estimate sin(0.2) using two terms and bound the error.

Solution

0.2 − 0.008/6 = 0.198667A1

Next term 0.25/120 = 2.7 × 10−6: error less than thisR1

Practice exercise 10

1. Estimate e0.1 using four terms.

2. Estimate ln 1.1 using three terms.

13 Limits using series

Lowest powers dominate.

Replace each function by its series; cancel powers of x; let x → 0. This is often quicker than repeated L'Hôpital.

WORKED EXAMPLE 14 Limit (exam level)

Find limx→0 (x − sin x)/(x(1 − cos x)).

Solution

Numerator x3/6 + ...; denominator x × x2/2 = x3/2M1 A1

→ 1/3A1

Practice exercise 11

1. Find limx→0 (ex − 1 − x − x2/2)/x3.

2. Find limx→0 (ln(1 + x) − x)/x2.

14 Series solutions of differential equations

Derivatives from the equation.

Given y′ = f(x, y) and y(0), find y′(0) from the equation, differentiate the equation to find y″(0), y‴(0), and substitute into the Maclaurin formula.

WORKED EXAMPLE 15 Series solution (exam level)

y′ = 1 + xy, y(0) = 1. Find the series of y up to x3.

Solution

y′(0) = 1A1

y″ = y + xy′: y″(0) = 1A1

y‴ = 2y′ + xy″: y‴(0) = 2: y ≈ 1 + x + x2/2 + x3/3A1

Practice exercise 12

1. y′ = y2 + x, y(0) = 1. Find the series up to x2.

15 Interval of validity

Where the series converges.

ex, sin x and cos x converge for all x. ln(1 + x) converges for −1 < x ≤ 1, and (1 + x)n (non-integer n) for |x| < 1. After substitution, transform the interval: ln(1 − 3x) is valid for −1 < −3x ≤ 1, i.e. −1/3 ≤ x < 1/3.

WORKED EXAMPLE 16 Validity (standard)

State the interval of validity for the series of 1/(2 − x).

Solution

= ½(1 − x/2)−1: |x/2| < 1M1

|x| < 2A1

Practice exercise 13

1. State the validity of the series of ln(4 + x).

2. State the validity of √(1 + 9x2).

16 Graphs of Maclaurin polynomials

Approximations near zero.

The Maclaurin polynomial of degree n agrees with the function and its first n derivatives at x = 0. Graphs show the approximation is good near 0 and improves with more terms.

Figure 2. Polynomial approximations to ex.

WORKED EXAMPLE 17 Comparing (standard)

Compare the degree-3 approximation with ex at x = 1 and x = 2.

Solution

x = 1: 2.667 vs 2.718A1

x = 2: 6.333 vs 7.389: worse further from 0A1 R1

17 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 18 Mixed problem (exam level)

(a) Find the series of ln(cos x) up to x4 using ln(1 + u) with u = cos x − 1. (b) Hence find limx→0 ln(cos x)/x2.

Solution

(a) u = −x2/2 + x4/24: ln(1 + u) = u − u2/2 = −x2/2 + x4/24 − x4/8M1 A1

= −x2/2 − x4/12A1

(b) −1/2A1

Practice exercise 14

1. Find the coefficient of x4 in the series of ex²cos x.

18 Additional exam-style practice

Paper 1.

WORKED EXAMPLE 19 Paper 1 question (exam level)

Let f(x) = √(1 + x). (a) Find the Maclaurin series up to x2. (b) Hence estimate √1.1. (c) Use the series to find limx→0 (√(1 + x) − 1 − x/2)/x2.

Solution

(a) 1 + x/2 − x2/8A1 A1

(b) 1 + 0.05 − 0.00125 = 1.04875A1

Practice exercise 15

1. Use series to show that sin x ≈ x − x3/6 gives sin(0.5) to 3 d.p.

19 Euler's formula from series

eiθ = cos θ + i sin θ.

Substituting iθ into the series for ex and separating real and imaginary parts gives the series for cos θ and sin θ. This explains Euler's formula, used throughout the complex numbers topic (1.12).

WORKED EXAMPLE 20 Euler (standard)

Use the series to show that eiθ = cos θ + i sin θ.

Solution

1 + iθ − θ2/2! − iθ3/3! + θ4/4! + ...M1

Real part = cos θ series; imaginary part = sin θ seriesR1 AG

20 Choosing a method for series

Derivatives or known series?

Derive from derivatives when the function is new or the question asks you to. Otherwise, build from standard series by substitution, multiplication or integration, which is usually quicker and less error-prone.

WORKED EXAMPLE 21 Choosing (standard)

How would you find the series of x e3x?

Solution

Substitute 3x into ex, then multiply by x: x + 3x2 + 9x3/2 + ...R1 A1

21 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 1 style) (about 15 minutes)

Let f(x) = ln(1 + sin x).

(a) Show that f′(x) = cos x/(1 + sin x) and find f′(0) and f″(0). [4]

(b) Hence write down the Maclaurin series of f(x) up to the x2 term. [2]

Markscheme

Part	Answer	Marks
(a)	chain rule	M1
	f′(0) = 1	A1
	f″(x) = (−sin x(1 + sin x) − cos2x)/(1 + sin x)2	M1
	f″(0) = −1	A1
(b)	f(x) ≈ x − x2/2	A2
(c)	0.095	A1
	exact 0.0952: good agreement	R1

Examiner's comments. Simplifying f″(x) using sin2x + cos2x = 1 gives f″(x) = −1/(1 + sin x).

Question 2 (Paper 1 style) (about 10 minutes)

(a) Write down the series of e3x and cos 2x up to the x3 term. [2]

(b) Hence find limx→0 (e3x − 1 − 3x)/(1 − cos 2x). [4]

Markscheme

Part	Answer	Marks
(a)	1 + 3x + 9x2/2 + 9x3/2	A1
	1 − 2x2	A1
(b)	(9x2/2 + 9x3/2)/(2x2)	M1 A1
	→ 9/4	A2

Examiner's comments. Only the lowest-order terms matter for the limit.

22 Common misconceptions and links

Students often think…	Actually…
the series of sin x includes even powers	only odd powers
all series are valid for all x	ln(1 + x) needs −1 < x ≤ 1
divide f(n)(0) by n	divide by n!
substituting 2x into ex gives 2ex	replace every x with 2x

How 5.19 connects to the rest of the course

Topic	Connection
1.10 Extended binomial	(1 + x)p
5.12 L'Hôpital	limits
5.18 Differential equations	series solutions
1.1 Series	infinite sums

23 Check your understanding

Section A: multiple choice

1. The coefficient of x3 in sin x is

A. −1/6 B. 1/6 C. 1/3 D. 0

2. The series of ex is valid for

A. |x| < 1 B. x > 0 C. all x D. −1 < x ≤ 1

3. The coefficient an in a Maclaurin series is

A. f(n)(0) B. f(n)(0)/n! C. f(n)/n D. n!f(0)

4. The x2 coefficient of cos 3x is

A. −3/2 B. 9/2 C. −9 D. −9/2

Section B: structured

5. Find the series of excos x up to x3. [3]

6. Use the series of arctan x to show that π/4 = 1 − 1/3 + 1/5 − .... [2]

Answers

1. A 2. C 3. B 4. D

5. 1 + x − x3/3.

6. Substitute x = 1: arctan 1 = π/4.

24 Summary and topic map

Idea	Remember
Formula	Σf(n)(0)xn/n!
Standard series	ex, sin x, cos x, ln(1 + x), arctan x, (1 + x)p
Substitution	replace x; adjust validity
Products	multiply and collect
Calculus	differentiate or integrate term by term
Limits	lowest powers dominate
DEs	find y′(0), y″(0), ... from the equation

Before the exam, make sure you…

•divide by n!
•state validity
•collect terms carefully in products
•use unrounded coefficients

Maclaurin Series on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 3. Topic map for Maclaurin Series.

25 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. 1 − x2/2 + x4/24

2. 1 + 2x + 2x2

Practice exercise 2

1. 1 − 9x2/2 + 27x4/8

2. 1 + x2 + x4 + x6

3. 2x + 2x3/3 + 2x5/5

Practice exercise 3

1. x − 2x3 + (2/3)x5

2. x + x2/2 + x3/3

3. 1 − x + x2 − x3 + ...

Practice exercise 4

1. −1/6

2. 0.2 − 0.02 + 0.002 667 − 0.0004 = 0.182 27

3. −1/3

Practice exercise 5

1. 2 + 2x + (3/2)x2 + x3/2

2. x − x3/6 + x5/120: sin x

Practice exercise 6

1. 1 + x − x3/3

2. x + x3/3

Practice exercise 7

1. 1 − x2 + x4 − x6

2. 1 − 9x2/2 + 27x4/8

Practice exercise 8

1. 1 − x2/2 + x3/3

2. x + x2 + 5x3/6

Practice exercise 9

1. 1 − x2/2 + x4/24

2. x − x3/3 + x5/5

Practice exercise 10

1. 1.105167

2. 0.1 − 0.005 + 0.000333 = 0.095333

Practice exercise 11

1. 1/6

2. −1/2

Practice exercise 12

1. y′(0) = 1, y″ = 2yy′ + 1 = 3: 1 + x + 3x2/2

Practice exercise 13

1. ln 4 + ln(1 + x/4): −4 < x ≤ 4

2. |x| < 1/3

Practice exercise 14

1. (1 + x2 + x4/2)(1 − x2/2 + x4/24): 1/24 − 1/2 + 1/2 = 1/24

Practice exercise 15

1. 0.5 − 0.0208 = 0.4792 vs 0.4794

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