Limit of Functions
AL Pure Mathematics Informal Summary
Limit of Functions 1
Differentiation 2
Applications of Differentiation 4
Indefinite Integral 6
Definite Integral 10
Applications of Definite Integrals 13
Limit of Sequences 14
Binomial Theorem 17
Polynomials 18
Inequalities 20
Complex Numbers 21
System of Linear Equations 27
Coordinate Geometry 29
Limit of Functions
• Two Important Limits
1. [pic] (also, [pic])
2. [pic]
Sandwich Theorem:
If [pic] and [pic] then [pic].
• Continuity
[pic] is continuous at [pic] [pic] [pic]
Differentiation
• Definition (First Principle)
[pic]
• Formula of Differentiation
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
8. [pic]
9. [pic]
• Rules of Differentiation
1. Product Rule: [pic]
2. Quotient Rule: [pic]
3. Chain Rule: [pic]
e.g. [pic]
• Logarithmic Differentiation
Take ln before differentiate:
e.g. [pic] e.g. [pic]
[pic] [pic]
• Leibniz’s Theorem
[pic]
OR
[pic]
where [pic], etc
• Differentiability
[pic] is differentiable at [pic] if
[pic]
Note: differentiable [pic] continuous
not continuous [pic] not differentiable
Applications of Differentiation
• L’Hospital’s Rule
|For [pic] |For [pic], etc, take ln first |
|e.g. [pic] |e.g. [pic] |
|[pic] |[pic] |
|e.g. [pic] | |
|[pic] | |
• Proving Inequalities
To prove [pic], let [pic] and find [pic]
e.g. Prove [pic] for all [pic].
Let [pic]
[pic]
When [pic], [pic]
|[pic] | |1 | |
|[pic] |--------- |0 |++++++ |
[pic] [pic] is max. at [pic]
[pic] [pic] for all [pic]
[pic]
• Curve Sketching
Steps:
(1) Find [pic] and [pic]
─ fully simplify and factorize them
─ take ln may be useful
(2) Draw table
|e.g. |[pic] | |3 | |
| |[pic] |-------------- |0 |++++++++++ |
| |[pic] |+++++++++++++++++++++++ |
| |[pic] |[pic] |2 |[pic] |
| |[pic] |[pic] |
| |increasing |decreasing |
|[pic] |[pic] |[pic] |
|Concave upward | | |
|[pic] |[pic] |[pic] |
|Concave downward | | |
(3) identify vertical asymptote(s) [pic]
find oblique asymptote(s) [pic] by [pic]
(4) Draw and label asymptotes/ extreme points/point of inflexions/ intercepts
Then draw the curve
Indefinite Integral
• Definition:
If [pic] then [pic]
• Integration Formula
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
8. [pic]
9. [pic]
10. [pic]
Rules of Partial Fractions
e.g.
(1) [pic] (Linear factor)
(2) [pic] (Quadratic factor)
(3) [pic] (Repeated linear factor)
(4) [pic] (Repeated quadratic factor)
• Method of Substitution
e.g. [pic]
[pic] OR
• Integration of Rational Functions ([pic])
─ do long division if [pic]
─ then use partial fractions
• Integration of Rational Functions of [pic]
[pic] [pic]
• Integration of Irrational Functions
For[pic], try to put [pic]
For[pic], try to put [pic]
For[pic], try to put [pic]
• Integration by Parts
[pic]
“derivative transfer”
[pic] [pic] (by parts twice)
• Reduction Formula
─ usually generated by Integration by Parts
[pic]
Definite Integral
• Second Fundamental Theorem
If [pic]
• Method of Substitution
[pic]
• Integration by Parts
[pic]
• Reduction Formula
[pic]
[pic]
When n is odd, When n is even,
[pic] [pic]
• First Fundamental Theorem
[pic]
• Sum an Infinite Series by Definite Integrals
[pic]
e.g.
[pic]
[pic]
• Inequalities on Definite integrals
1. lf f(x)[pic]g(x) [pic][pic]
2. [pic]
Applications of Definite Integrals
• Plane Area
• Volume: Disc Method
• Volume: Shell Method
Limit of Sequences
• Sandwich Theorem
If [pic] for all [pic] , and [pic] then [pic].
e.g. Prove [pic].
Let [pic]
[pic]
[pic]
[pic] [pic]
[pic]
[pic]
[pic]
[pic]
• Monotone Convergence Theorem:
Monotonic increasing + bounded above [pic] Convergent
[pic] [pic]
Monotonic decreasing + bounded below [pic] Convergent
[pic] [pic]
usually proved by Method of Difference or M.I.
e.g. Let [pic] be a sequence of positive numbers, where
[pic] and [pic]
a) Prove that [pic] for all positive integers n.
Try Method of Difference:
[pic]
[pic]
[pic]
[pic] need to have [pic]
[pic] try M.I.
When n = 1, [pic]
[pic][pic] is true for n = 1
Assume [pic]
[pic]
=[pic]
=[pic]
[pic]
[pic][pic] is true for n = k+1
By M.I., [pic]
b) Prove that [pic] exists.
Need to prove monotonic increasing,
try method of difference:
[pic]
=[pic]
=[pic]
=[pic]
[pic] ([pic]
[pic][pic] is monotonic increasing and, by(a), bounded above by 3
[pic][pic] is convergent
c) Find [pic].
Let [pic]
[pic]
[pic]
[pic]
[pic] or [pic]
[pic]
[pic][pic]
[pic]
Binomial Theorem
[pic] total (n+1) terms
[pic] [pic]
[pic]
[pic]
Differentiation and Integration may help you find properties of Binomial coefficients.
e.g. [pic]
Put x = 1, [pic]
e.g. [pic]
Put x = 0, [pic]
Put x = 1, [pic]
[pic]
Polynomials
• Division Algorithm
[pic] Remainder
Quotient
deg [[pic]] < deg [[pic]]
or [pic] (that is, [pic] divisible by [pic])
• Remainder Theorem
When a polynomial [pic] is divided by [pic], the remainder is [pic].
• Factor Theorem
When [pic], [pic] is a factor of the polynomial [pic].
Generally, if [pic] are the roots of a polynomial [pic] of degree [pic],
then [pic], where [pic] is the leading coefficient.
• Euclidean Algorithm (輾轉相除法)
G.C.D. of [pic] and [pic] = G.C.D. of [pic] and [pic]
• Complex Roots Theorem
Let [pic] be a polynomial with real coefficients.
If [pic] is a root of [pic] ([pic]), [pic] is also a root of [pic].
Implications:
1. A polynomial of degree[pic] can be factorized into real linear and/or quadratic factors.
2. A polynomial of odd degree has at least one real root.
• Multiple Roots
[pic] has a multiple root [pic]
[pic]
[pic] and [pic]
• Relation between Roots and Coefficients
Let [pic] be the roots of [pic]
[pic]
Tips: [pic]
[pic]
• Transformations of Polynomial Equations
| |Old Roots |New Roots |Put |
|e.g. |[pic] |[pic] |[pic] |
|e.g. |[pic] |[pic] |[pic] |
|e.g. |[pic] |[pic] |[pic] |
| | |[pic] | |
| | |[pic] | |
| | |[pic] | |
| | |[pic] | |
Inequalities
• Triangle Inequality
[pic]
• Inequalities involving absolute values
[pic]
• A.M.[pic]G.M.
[pic]
OR [pic]
• Cauchy – Schwarz’s Inequality
[pic]
OR [pic]
Proof: [pic] [pic]
[pic]
If [pic] the result is trivial.
If [pic]
[pic]
[pic]
Strategies to prove inequalities:
1. By Method of Difference/ Ratio
2. M.I.
3. By Differentiation
4. Use known results
Complex Numbers
• Definitions: [pic] ([pic])
Let [pic] ([pic])
1. [pic], [pic] Imaginary part is not imaginary!
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
• Polar Form
[pic] ([pic])
or
[pic]
e.g. [pic]
Let [pic], [pic]
[pic] [pic], [pic]
[pic] [pic], [pic]
• Geometric Relationship
[pic] is purely imaginary [pic] [pic] [pic] [pic]
[pic] is purely imaginary
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
• Locus Problems
[pic] [pic]
General method: Put [pic]
• DeMoivre’s Theorem
[pic] (n is any integers)
Roots of [pic]:
[pic] [pic]
e.g. Solve [pic]
[pic]
Hence we can factorize [pic] into real linear/quadratic factors
[pic]
Matrices & Determinants
• Arithmetic of Matrices
Addition/Subtraction: [pic]
Scalar Multiplication: [pic]
Multiplication: [pic]
Transpose: [pic], [pic]
• Determinants
[pic]
[pic]
( expanded along 1st row )
* Can be expanded along any row/ column [pic]
Some properties:
1. [pic] Interchanging two rows gives negative sign
2. [pic] Common factor extracted from a row/ column
3.
4. [pic]
[pic]
5. [pic]
• Inverse
Definition: [pic]
Existence: [pic]
[pic]
[pic]
[pic]
• Applications in Cooradinate Transformation
“output” [pic] “input”
transformation matrix
Rotation: [pic]
Reflection: [pic]
Enlargement: [pic]
System of Linear Equations
• Notation
[pic]
[pic]
• Finding Unique Solution by Cramer’s Rule [pic]
[pic]
[pic]
• [pic] & Consistency
| |[pic] |[pic] |
|[pic] |unique solution |unique trivial solution |
| |obtained by |(0, 0, 0) |
| |Cramer’s Rule | |
| |Method of Inverse Matrix | |
| |G. E. | |
|[pic] |no solution (inconsistent) |infinitely many solutions |
| | |(obtained by G. E.) |
| |infinitely many solutions | |
| |(obtained by G. E.) | |
• Gaussian Elimination
Goal: [pic]
Coordinate Geometry
• Angle between two straight lines
[pic]
• Distance from a point to a line
[pic]
• Equations of straight lines
Point-slope form: [pic]
Slope-intercept form: [pic]
• Equations of Circles
Standard form: [pic]
General form [pic]
Centre [pic]
Radius [pic]
• Equations of Parabola
• Equations of Ellipses
[pic]
• Equations of Hyperbola
[pic] [pic]
• Family of Straight Lines / Circles
[pic]
• Locus Problems
Let the movable point / variable point interested be (x, y).
Try to find an equation connecting x and y.
• Equations of Tangents and Chords
[pic][pic] [pic] [pic] [pic]
[pic]
[pic][pic] [pic] [pic] [pic]
-----------------------
Put [pic]
lf [pic] for all n, then [pic].
lf [pic] for all n, then [pic].
[pic]
includes values of x
(1) at which [pic] and [pic]
(2) at which [pic] [pic] or [pic] is undefined
(3) the expression inside absolute value sign is zero
e.g. [pic]
include [pic]
[pic]
completing square
[pic]
[pic]
[pic]
[pic]
Put u = x + 1
du = dx
|x |2 |1 |
|u |3 |2 |
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
Note: In general,
[pic]
[pic]
[pic]
Tips:
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
partial fractions
[pic] use summation notation
[pic]
[pic]
[pic]
[pic]
[pic]
e.g.
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
e.g.
e.g.
[pic] ‘make’ [pic]
[pic] ‘make’ [pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
principal argument
[pic]
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