ECM1702 - Calculus and Geometry

Revision Notes

Joshua Byrne

Autumn 2011

Contents

1 The Real Numbers 11.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Set Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Open/Closed Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Coordinate Geometry 12.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Relationship between m and the Angle of Inclination . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Distance Between Two Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Distance Between a Point and a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Conic Sections 23.1 Expressing Curves in Cartesian and Parametric Form . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Quadratic Equations of the Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3 Finding Tangents and Normals to the Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4 Directrices, Foci and Eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.5 Identifying a Conic Section by its Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 Functions 44.1 Domains, Codomains and Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.2 Algebra with Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.3 Composite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.4 Odd and Even Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.5 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.6 Monotonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.7 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.8 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.9 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.10 Piecewise Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5 Sequences, Series and Limits 75.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.2 Defining Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.3 Convergence of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.4 Series and The Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.5 Series and Partial Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.6 Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.7 Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.8 Properties of Convergent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.9 Standard Series Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.10 Tests for Convergence and Divergence of a Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.11 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6 Limits and Continuity of Functions 126.1 Definition of a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 Left and Right Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

7 Differentiation 137.1 Differentiation by First Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.2 Continuity and Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.3 Rules of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.4 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.5 Computing Higher Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.6 Parametric Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.7 Logarithmic Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.8 Differentiation of Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.9 Leibniz Rule for Repeated Differentiation of Products . . . . . . . . . . . . . . . . . . . . . . . . 15

8 Applications of Differentiation 158.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158.2 Extrema of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158.3 Convexity and Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158.4 Limits using Calculus - L’Hopital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.5 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.6 Rolle’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.7 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.8 Maclaurin Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.9 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

9 Curve Sketching 18

These notes were compiled using Thomas’ Calculus, lecture notes uploaded to ELE, and internet resources.

1 The Real Numbers

1.1 Notation

Natural Numbers 1, 2, 3... set NIntegers ...-2, -1, 0, 1, 2... set ZRational Numbers Any numbers that can be written

a

bset Q

Real Numbers Includes√

2, π set R

x is an element of the set R: x ∈ R

Absolute Value: abs(x) = | x |

1.2 Set Notation

• The set of x in A is such that x satisfies some property B: {x ∈ A : x satisfied B}

• A is a subset of B: A ⊂ B

• Intersection - in A and B: A ∩ B

• Union - in A or B: A ∪ B

• Complement - not in A: A′: A′ = {x ∈ R : x /∈ A}

1.3 Open/Closed Intervals

• Open interval: (a, b) = {x : a < x < b}

• Closed interval: [a, b] = {x : a ≤ x ≤ b}

• Half opened/closed interval: (a, b] = {x : a < x ≤ b}

2 Coordinate Geometry

We can represent points in the plane as pairs (x, y) ∈ R2 - this is very similar to the notation used in set theory.

For example, {(x, y) ∈ R2 : f(x, y) = 0} means the set of points that satisfy f(x, y) = 0.

2.1 Linear Equations

The following are all versions of writing linear equations.

• {(x, y) : ax+ by + c = 0}

• y = mx+ c

• y − y1 = m(x− x1)

ax+ by + c = 0 is preferred.

2.2 Relationship between m and the Angle of Inclination

m =increase in y

increase in x= tan θ, where θ is the angle between the positive real axis and the line.

Parallel lines have the same gradient. Perpendicular lines have gradients that, when multiplied together equal-1. (i.e. If m1 is the gradient of a tangent, and m2 is the gradient of a normal to that tangent, m1 ×m2 = −1)

2.3 Distance Between Two Points

The distance between two points,P1 : (x1, y1) and P2 : (x2, y2), on a co-ordinate axis is calculated using theformula | P1P2 |=

√(x2 − x1)2 + (y2 − y1)2

1

2.4 Distance Between a Point and a Line

The shortest distance between a point P : (x0, y0) and a line L : ax+by+c = 0 will always be the perpendiculardistance. The distance PN (where N is the point on L where the perpendicular line meets) can be calculated

using the equation | PN |= ax0 + by0 + c√a2 + b2

3 Conic Sections

The conic sections are a family of curves including parabolas, ellipses, circles and hyperbolas.

3.1 Expressing Curves in Cartesian and Parametric Form

3.1.1 The Parabola

• The parabola with the y-axis as its line of symmetry has the Cartesian equation 4ay = x2. This parabolacan also be expressed by the parametric equations x = 2at, y = at2.

• The parabola with the x-axis as its line of symmetry has the Cartesian equation y2 = 4ax. This parabolacan also be expressed by the parametric equations y = 2at, x = at2.

3.1.2 The Ellipse

The ellipse, which has centre (0, 0), has the Cartesian equationx2

a2+y2

b2= 1. This ellipse can also be expressed

by the parametric equations x = a cos θ, y = b sin θ.

3.1.3 The Circle

The circle, which has centre (0, 0), has the Cartesian equation x2 + y2 = a2. This circle can also be expressedby the parametric equations x = a cos θ, y = a sin θ.

3.1.4 The Hyperbola

The hyperbola, which has centre (0, 0), has the Cartesian equationx2

a2− y

2

b2= 1. This hyperbola has asymptotes

at y = ± bax and can also be expressed by the parametric equations x = a sec θ, y = b tan θ.

3.2 Quadratic Equations of the Conic Sections

All conic sections can be written as an equation of the form Ax2 +Bxy + Cy2 +Dx+ Ey + F = 0.For example, a circle with centre (a, b), radius r has the Cartesian equation (x− a)2 + (y− b)2 = r2, which canbe multiplied out to give the equation x2 − 2ax+ y2 − 2by + a2 + b2 − r2 = 0.

3.3 Finding Tangents and Normals to the Curves

There are two ways that the tangent and normal to a curve can be found - by using the parametric equationsof the curve, and by using y = mx+ c. This will focus on the method using parametric equations.

3.3.1 Finding the Tangent and Normal to a Curve using Parametric Equations

Bold type gives the general stages for finding the tangent and normal to any curve with knownparametric equations.Italicised type shows an example of finding the tangent and normal to a parabola.

1. Ascertain the Cartesian and parametric equations of the curve.The Cartesian equation of the parabola is y2 = 4ax and the parametric equations are y = 2at, x = at2.

2

2. Use differentiation to find m - find dydx using the chain rule.

Firstly, differentiate y = 2at.

y = 2atdy

dx= 2a

Now differentiate x = at2.

x = at2dy

dx= 2at

Now find dydx .

dy

dx=dy

dt× dt

dx=

2a

2at=

1

t

Therefore m =1

t

3. Use y − y1 = m(x− x1) to find the equation of the tangent.Substitute values for x, y and m into the equation y − y1 = m(x− x1) giving:

y − 2at =1

t(x− at2)

ty − 2at2 = x− at2

ty = x+ at2, which is the equation of the tangent.

4. To find the equation of the normal, first the value of m must be determined.

For two perpendicular lines (y = m1x+ c1 and y = m2x+ c2), m1 ×m2 = −1. Therefore m2 =−1

m1. Let

the gradient of the tangent,1

t= m1. This means m2 =

−11t

= −t.

5. Use y − y1 = m(x− x1) to find the equation of the normal.Substitute values for x, y and m into the equation y − y1 = m(x− x1) giving:

y − 2at = −t(x− at2)

y − 2at = −tx+ at3

y = −tx+ 2at+ at3, which is the equation of the normal.

3.4 Directrices, Foci and Eccentricity

Conics can be described in terms of the locus of points whose distance to a fixed line D (the directrix) and thepoint F (Focus) are in constant ratio e (Eccentricity).

F

P(x, y)

N

D

Figure 1: The relationship between the curve, the focus F and the directrix D

3

Consider P (x, y) such that| PF || PN |

= e, or | PF |2= e2 | PN |2.

N is the point on the directrix to give the shortest distance PN. So we use | PN |2=(ax+ by + c)2

a2 + b2, where D,

the directrix, has equation ax+ by + c = 0.

Theorem: Given any D, F and e, the following statements hold for the locus of points P (x, y) such that| PF |= e | PN |.

If e < 1, the conic is an ellipse.If e = 1, the conic is a parabola.If e > 1, the conic is a hyperbola.

3.5 Identifying a Conic Section by its Equation

Remember all conic sections have an equation in the form Ax2 +Bxy +Cy2 +Dx+Ey + F = 0, it is possibleto identify which conic we have an equation for by calculating B2 − 4AC.

If B2 − 4AC = 0, it is a parabola.If B2 − 4AC < 0, it is an ellipse.If B2 − 4AC > 0, it is a hyperbola.

4 Functions

4.1 Domains, Codomains and Ranges

What can go into a function is the DOMAIN.What may possibly come out is the CODOMAIN.What actually comes out is the RANGE.

The set D is the domain.The set R is the codomain/range.

For a real function defined by a formula, the maximal domain is the largest subset of D.

It is required that for every x, there is only ONE value of f(x). i.e. The function must be a ONE-TO-ONEfunction or a MANY-TO-ONE function.

4.2 Algebra with Functions

It is possible to combine functions as if they were numbers, as long as they have a common domain. i.e. Thefollowing rules are all defined on D(f) ∩D(g).

• (f + g)(x) = f(x) + g(x)

• (f − g)(x) = f(x)− g(x)

• fg(x) = f(x)g(x)

• f

g(x) =

f(x)

g(x)

4.3 Composite Functions

These are usually known as ‘function of a function’. We write f(g(x)) = (f ◦ g)(x).The domain of f ◦ g is the set of x ∈ D(g) such that f(x) ∈ D(f).

Composite functions are sometimes written as f2(x).

4

4.4 Odd and Even Functions

If f(x) is an odd function, then f(−x) = −f(x).For example, f(x) = x3 and f(x) = sin(x) are odd functions - when plotted on a graph, they are symmetricunder rotation by π.

If f(x) is an even function, then f(−x) = f(x).For example, f(x) = x2 and f(x) = cos(x) are even functions - when plotted on a graph, they are symmetricabout the y-axis.

Most functions are neither odd nor even, but any function f(x) can be decomposed into E(f(x)) + O(f(x)),where:

E(f(x)) =f(x) + f(−x)

2and O(f(x)) =

f(x)− f(−x))

2.

Example of Using Odd and Even FunctionsSuppose f(x) is an odd function and g(x) is an even function, then is (f ◦ g)(x) an even function, odd functionor neither?

Let f(x) = y, then f(−x) = −y.Let g(x) = z, then g(−z) = z.

Remember: (f ◦ g)(x) = f(g(x))

(f ◦ g)(x) = f(z) = y(f ◦ g)(−x) = f(z) = y

Therefore (f ◦ g)(x) is an EVEN function!

4.5 Periodic Functions

A function f(x) is periodic if there is a T > 0 such that f(x+ T ) = f(x) for all x.

T is the period of f(x) if it is the smallest such T .

Examples of periodic functions are sin(x), cos(x) and tan(x).

4.6 Monotonic Functions

• f(x) is monotonic increasing if f(x1) ≤ f(x2) for all x1 < x2. Examples of monotonic increasing functionsare ex and x3.

• f(x) is strictly monotonic increasing if f(x1) < f(x2) for all x1 < x2.

• f(x) is monotonic decreasing if f(x1) ≥ f(x2) for all x1 > x2. Examples of monotonic decreasing functionsare e−x and −x3

• f(x) is strictly monotonic decreasing if f(x1) > f(x2) for all x1 > x2.

4.7 Trigonometric Functions

f(x) (f(x))−1

sin(x) csc(x)

cos(x) sec(x)

tan(x) cot(x)

If a function can be written as y(x) = A sin(ωx + α) then y is a sinusoidal function of x with amplitude A,angular frequency ω and phase α.

The period P can be calculated as P =2π

ω.

5

4.8 Hyperbolic Functions

The Hyperbolic Sine The Hyperbolic Cosine

sinh(x) =ex − e−x

2cosh(x) =

ex + e−x

2

Note that sinh(x) and cosh(x) are the odd and even parts of ex respectively.

f(x) (f(x))−1

sinh(x) csch(x)

cosh(x) sech(x)

tanh(x) coth(x)

4.8.1 Osbourne’s Rule

All trigonometric identities become hyperbolic trigonometric identities, EXCEPT that the product of sineschanges sign. For example:

Circular Hyperbolic

cos2(x) + sin2(x) = 1 cosh2(x)− sinh2(x) = 1

1 + tan2(x) = sec2(x) 1− tanh2(x) = sech2(x)

sin(2x) = 2 sin(x) cos(x) sinh(2x) = 2 sinh(x) cosh(x)

4.9 Inverse Functions

Given that f(x) is a function with domain D and range R, the inverse function f−1(x) would have domain Rand range D - i.e. f−1(x) has the domain of f(x) as the range and vice versa. The inverse function will onlyexist if f(x) is a ONE TO ONE function.

Inverse functions have the following property. (f ◦ f−1)(x) = (f−1 ◦ f)(x) = x

4.9.1 Finding the Inverse Function

There are two methods for finding the inverse of a function: the graphical method and the algebraic method.

Graphical MethodTo find the inverse of a function graphically, reflect the function in the line y = x.

Algebraic Method To find the inverse of a function using algebra, use the following steps.

1. If necessary, find a subset of the domain on which the map is ONE TO ONE.

2. Solve y = f(x) for x in terms of y.

3. Interchange x and y to obtain y = f−1(x).

When the function is not one to one on the domain, we need to restrict the function to a domain (i.e. asub-domain) in which it IS one to one.

4.9.2 Inverse Trigonometric Functions

As only one to one functions have an inverse, trigonometric functions as they stand do not have inverses.Therefore we must restrict the domain of a trigonometric function in order to find its inverse.The following table gives the domains on which it is possible to compute inverse functions for the associatedtrigonometric function.

6

Function Domain Rangecos(x) R [-1, 1]arccos(x) [-1, 1] [0, π]sin(x) R [-1, 1]arcsin(x) [-1, 1] [−π2 , π

2 ]tan(x) R Rarctan(x) R (−π2 , π

2 )cot(x) R Rarccot(x) R (0, π)

These standard domains can be used to find inverses for other domains.

4.9.3 Inverse Logarithmic Functions

If y = ax, then we can say that x = loga y (the logarithm of y to the base a).If y = ex, then we can say that x = loge y = ln y (the natural logarithm of y).

Note that ax = ex ln a and therefore loga y =ln y

ln a, which is the ’change of base’ rule.

4.10 Piecewise Functions

Piecewise functions are functions that have different formulae in different regions of the domain.

For example, f(x) =

0 x ≤ −1√

1− x2 −1 < x < 1x x ≥ 1

is a piecewise function.

5 Sequences, Series and Limits

5.1 Sequences

• A sequence is simply a list of real numbers - a1, a2, a3, ... = {an}∞n=1.

• an is a general term in the sequence - the nth term.

• a1 is the first term in the sequence.

• An infinite sequence is one that continues indefinitely.

• A finite sequence is one with a finite number of terms.

5.2 Defining Sequences

It is possible to specify a series in a number of different ways:

1. By specifying a general term: Usually known as the nth term - for example, a general term of asequence may look like n 7→ 2n+ 1. The general term should also specify the range of n.

2. By giving a recursion formula: For a sequence a1, a2, a3, ..., a recursion formula is one that requiresthe computation of all previous terms in order to find the value of an. For example, a recursion formulamay look like an+1 = 5an + 3.

3. By specifying the sequence using other means: For example, the sequence {3, 1, 4, 1, 5, 9, 2, 6, 5,3, 5} are the digits of π.

It is sometimes possible to express a recursion formula as a formula for a general term. For example:

an+1 =−5an2n+ 1

with a1 = 1 can be expressed as an+1 =2n(−5)n(n!)

(2n+ 1)!

However this is NOT always possible.

7

5.3 Convergence of Sequences

We say that an infinite sequence, an, converges to a limit L if for every ε (ε - epsilon, a very small amount)there is an N such that L− ε < an < L+ ε for all n > N .

In such a case, we write an → L as n → ∞ or equivalently limn→∞

an = L. If there is no such L, then we

can say the sequence diverges or alternatively the limit does not exist.

5.3.1 Monotonic and Bounded Sequences

We can say that a sequence (finite or infinite) is:

• Strict increasing if an < an+1 for all n

• Strict decreasing if an > an+1 for all n

• Increasing if an ≤ an+1 for all n

• Decreasing if an ≥ an+1 for all n

A sequence is bounded above if there is a number M (known as an upper bound) such that an ≤ M for alln ∈ N .Similarly, a sequence is bounded below if there is a number M (known as an lower bound) such that an ≥Mfor all n ∈ N .

Some Important Remarks

• If an is bounded, this does not imply that an has a limit.

• If an has a limit, this does imply that an is bounded.

• If an is unbounded, this implies that an has NO limit.

Theorem: If an infinite sequence is increasing and bounded above, then it must converge to a finite limit.

Proof: There must be an infinite number of values of the sequence in a finite interval. But owing to itbeing increasing, it cannot ‘return’ to a value after passing it; hence it must have a limit.

5.3.2 The Sandwich Theorem

Theorem: Suppose bn and cn converge to L as n→∞ and suppose that there is an N such that bn ≤ an ≤ cnfor all n ≥ N . Then an → L as n→∞.

The Sandwich theorem allows the computation of the limit of an expression by trapping the expression be-tween two other expressions which have limits that are easier to compute.

5.3.3 Limit Laws for Sequences

If limn→∞

an = A and limn→∞

bn = B, with A,B ∈ R, then:

• limn→∞

(an ± bn) = limn→∞

an ± limn→∞

bn = A±B

• limn→∞

(anbn) = limn→∞

an limn→∞

bn = AB

• limn→∞

anbn

=limn→∞

an

limn→∞

bn=A

Bproviding that B 6= 0.

If an+1 = f(an) and an → L, then L = f(L).

5.3.4 Infinite Limits

If a sequence an is such that an > 0 for sufficiently large n and 1an→ 0 we say that an → ∞. Similarly, if

an < 0 for sufficiently large n and 1an→ 0, we say that an → −∞. In these cases, an diverges to infinity.

8

5.3.5 Comparison Test for Sequences

Suppose that an and bn are sequences and there is an N such that an ≤ bn for all n ≥ N . Then, if the limitsexist, lim

n→∞an ≤ lim

n→∞bn. Also, if an →∞, then bn →∞.

If an does not converge, then it either diverges to ±∞ or it oscillates (e.g. an = (−1)n oscillates).

5.3.6 Standard Sequence Limits

The following are standard sequence limits and may be quoted without proof:

1. If p > 0 then limn→∞

1

np= 0 and lim

n→∞np =∞.

2. If r ∈ R and | r |< 1, then limn→∞

rn = 0.

If r ∈ R and | r |> 1, then limn→∞

rn =∞.

If r ∈ R and r = 1, then limn→∞

rn = 1 (constant sequence).

If r ∈ R and r = −1, then limn→∞

rn diverges (oscillates).

3. If a ∈ R and b > 0, then limn→∞

(lnn)a

nb= 0.

4. If a ∈ R and | p |> 1 then, limn→∞

na

pn= 0.

5. If c ∈ R, then limn→∞

cn

n!= 0.

6. If q ∈ R, then limn→∞

nq

n!= 0.

7. If a > 0, then limn→∞

a1n = 1 and moreover lim

n→∞n

1n = 1.

8. If x ∈ R, then limn→∞

(1 + xn )n = ex.

WARNING: Do not write limn→∞

an = bn as this does not make sense!

5.4 Series and The Sigma Notation

For a general sequence a1, a2, a3, a4 ... ak−1, ak, ak+1 ..., the sum of the first n terms, a1 + a2 + a3 + ...+ an

can be writtenn∑k=1

ak.

That is to say,n∑k=1

ak = a1 + a2 + a3 + ...+ an.

We let the index k in the term ak take, in turn, the values 1, 2, 3 ... n. These sums are called series.

We can also sum infinite sequences, giving infinite series. For example,∞∑k=1

ak = a1 + a2 + a3 + .... An-

other way of writing this would be∞∑k=1

ak = limn→∞

n∑k=1

ak, wheren∑k=1

ak is called a partial sum.

In order to use the Sigma notation, we must find a suitable expression to represent the general term ak.

NOTE: Just as a limit may not exist, it is entirely possible that an infinite series may not be computable, in

particular if limn→∞

n∑k=1

ak does not exist. In this case, the series is said to diverge - otherwise it converges. More

often than not, we can just work out whether a series converges or diverges, rather than actually computingthe limit that the partial sums tend to.

9

5.5 Series and Partial Sums

Consider the infinite series∞∑k=1

ak = a1 + a2 + a3 + .... Obviously, it is not possible to calculate the sum of the

entire series all at once. Therefore we can form a sequence of the sums of some terms where:

S1 = a1S2 = a1 + a2S3 = a1 + a2 + a3...Sn = a1 + a2 + a3 + ...+ an

So, Sn is the sum of the first n terms of the infinite series - Sn is called the nth partial sum of the series.

An infinite series∞∑k=1

ak is convergent if the sequence of partial sums S1, S2, S3, S4 ... Sk−1, Sk, Sk+1 ... in

which Sk =n∑k=1

ak is convergent. If not, the series is divergent.

We define the sum of an infinite number of terms to be the limit of the sequence of partial sums as n→∞.

5.6 Geometric Series

Geometric series are of the form a+ ar+ ar2 + ...+ arn−1 + ... =∞∑n=1

arn−1 where r can be positive or negative.

In a geometric series, a is the first term, and r is the common ratio.

For |r| 6= 1 we can see whether a series converges using the formula Sn =a(1− rn)

1− r.

• If |r| < 1, then rn → 0 as n→∞, leaving Sn →a

1− r- the series converges.

• If |r| > 1, then |rn| → ∞ as n→∞ - the series diverges.

• If r = 1, then Sn = a+ a(1) + a(1)2 + ...+ a(1)n−1 = na and limn→∞

Sn = ±∞ - the series diverges.

• If r = −1, then the series diverges as the nth partial sum is either a or 0.

5.7 Arithmetic Series

Arithmetic series are of the form a+ (a+ d) + (a+ 2d) + ...+ (a+ (n− 1)d) =∞∑n=1

a+ (n− 1)d where each term

differs from the previous term by the same amount, d. In an arithmetic series, a is the first term, and d is thecommon difference.

5.7.1 Summing Arithmetic Series

•n∑k=1

k =1

2n(n+ 1)

•n∑k=1

a+ (k − 1)d =n

2(2a+ (n− 1)d)

For higher powers:

•n∑k=1

k2 =1

6n(n+ 1)(2n+ 1)

•n∑k=1

k3 =1

4n2(n+ 1)2

5.8 Properties of Convergent Series

•n∑k=1

(ak + bk) =n∑k=1

ak +n∑k=1

bk

If both series converge (absolutely),

10

•n∑k=1

(ak − bk) =n∑k=1

ak −n∑k=1

bk

•n∑k=1

c(ak) = cn∑k=1

ak

BUTn∑k=1

(akbk) 6=n∑k=1

akn∑k=1

bk.

5.9 Standard Series Limits

An infinite series∞∑n=1

an converges if the sequence of partial sums Sk =k∑

n=1an has a limit as k →∞.

The following are standard series limits and may be quoted without proof:

1. If an =1

np, then

∞∑n=1

an converges for p > 1 and diverges otherwise.

2. If an = rn, then∞∑n=1

an converges for |r| < 1 but diverges otherwise.

We say that∞∑n=1

an converges absolutely if∞∑n=1|an| converges, and if

∞∑n=1

an converges absolutely, then it con-

verges. However, there are series that converge but do not converge absolutely.

5.10 Tests for Convergence and Divergence of a Series

Often, we cannot find the limit of the nth partial sum as n→∞. However we can determine whether the serieswill converge or diverge using one or some of the following tests.

5.10.1 Divergence Test

If an 9 0 as n→∞, then∞∑n=1

an diverges.

NOTE: If an → 0 as n→∞, then the sequence may converge or diverge.

5.10.2 Leibniz Theorem (The Alternating Series Test)

This theorem states that∞∑n=1

(−1)nan will converge if the following three conditions are satisfied.

1. an ≥ 0 for all n.

2. an is a monotonically decreasing sequence - that is an+1 ≤ an for all n.

3. an → 0 as n→∞.

5.10.3 Ratio Test

Consider the limit L = limn→∞

∣∣∣∣an+1

an

∣∣∣∣.• If L converges and L < 1, then

∞∑n=1

an converges.

• If L converges and L > 1, then∞∑n=1

an diverges.

• If L does not converge, there is no conclusion.

• If L = 1, then there is no conclusion.

5.10.4 Comparison Test

For sequences an and bn:

• If the series∞∑n=1

bn diverges, and an ≥ bn ≥ 0 for all n, then∞∑n=1

an diverges.

• if the series∞∑n=1

bn converges, and bn ≥ an ≥ 0 for all n, then∞∑n=1

an converges.

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5.10.5 Order In Which To Use Tests

Unless it is obvious which test to use, apply the tests in the following order:

1. Divergence Test

2. Leibniz Theorem

3. Ratio Test

4. Comparison Test

5.11 Power Series

A power series is a sum of terms which contains an unknown variable x raised to a power - a example of this isthe binomial series expansion.

A power series has the general form a0 + a1x+ a2x2 + · · · =

∞∑n=0

anxn.

The convergence of a power series depends on the values of x chosen.

5.11.1 The Radius of Convergence

For a power series, there exists a number R, known as the radius of convergence, such that:

• If |x| < R, the series is absolutely convergent.

• If |x| > R, the series is divergent.

• At x = R and x = −R, the series may be convergent or divergent.

5.11.2 Calculating the Radius of Convergence

For a power series∞∑n=0

anxn, the radius of convergence can be calculated using the ratio test. This gives the

result that R = limn→∞

∣∣∣∣ anan+1

∣∣∣∣5.11.3 The General Power Series

The general power series at x = x0 is a0 + a1(x− x0) + a2(x− x0)2 + · · · =∞∑n=0

an(x− x0)n.

The same procedure for finding the radius of convergence applies - in this case:

• If |x− x0| < R, the series is absolutely convergent.

• If |x− x0| > R, the series is divergent.

• At |x− x0| = R, the series may be convergent or divergent.

6 Limits and Continuity of Functions

6.1 Definition of a Limit

For a function f(x), if the values of f(x) can be made as close as possible to L, by taking values of x sufficientlyclose to a (but not equal to a), then write lim

n→af(x) = L.

NOTE: It is not necessary for the function to be defined at a to have a limit there.

6.2 Left and Right Limits

If limn→a

f(x) = L converges, then it can be said that limn→a−

f(x) = limn→a+

f(x) = L.

Also, if limn→a−

f(x) 6= limn→a+

f(x), then limn→a

f(x) does not exist.

These rules can be used to determine whether a function is continuous at x = a, especially with piecewisefunctions.

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6.2.1 Limit Laws for Functions

Let a ∈ R, limn→a

f(x) = F and limn→a

g(x) = G, then:

• limn→abf(x)± g(x)c = lim

n→af(x)± lim

n→ag(x) = F ±G

• limn→abf(x)g(x)c = [ lim

n→af(x)][ lim

n→ag(x)] = FG

• limn→a

[f(x)

g(x)] =

limn→a

f(x)

limn→a

g(x)=F

Gproviding G 6= 0.

7 Differentiation

7.1 Differentiation by First Principles

If f(x) is defined in some interval containing x, thenf(x+ h)− f(x)

his the gradient of the line passing through

points (x, f(x)) and (x+h, f(x+h)). If we take the limit as h→ 0, then we obtain the differential of f(x) withrespect to x at x = a, if the limit exists.

Thus,df

dx= limh→0

f(x+ h)− f(x)

h.

7.2 Continuity and Differentiation

If f(x) is differentiable at x = a, then f(x) is continuous at x = a.

The converse is NOT true. There are two main ways that a function can be continuous, yet not differen-tiable at x = a.

• At corner points.

• At points of vertical tangency.

At these points the gradient of the tangents have different limits from the left and right, and thus the limitdefines that the derivative does not exist.

7.3 Rules of Differentiation

7.3.1 Standard Differentiation Results

For any a ∈ R:

• d

dxxa = axa−1

• d

dxln(ax) =

1

x

• d

dxeax = aeax

• d

dxsin ax = a cos ax

• d

dxcos ax = −a sin ax

• d

dxtan ax = a sec2 ax

• d

dxarcsin

x

a=

1√a2 − x2

• d

dxarccos

x

a=

−1√a2 − x2

13

• d

dxarctan

x

a=

a

a2 + x2

• d

dxsinh ax = a cosh ax

• d

dxcosh ax = a sinh ax

• d

dxax = ln a ax

7.3.2 Other Rules for Differentiation

For the two functions u(x) and v(x):

Linearity Rule:d(u+ v)

dx=du

dx+dv

dx

Product Rule:d(uv)

dx= v

du

dx+ u

dv

dx

Quotient Rule:du

vdx

=vdu

dx− udv

dxv2

If y = y(u) and u = u(x):

Chain Rule:dy

dx=dy

du

du

dx

7.4 Implicit Differentiation

There are two ways to define a function, explicitly and implicitly.Explicit definition of a function: These can be written as y = f(x). For example, y = 2x+ 3 is an explicitequation.Implicit definition of a function: These can be written as f(x, y) = a where a is any number, possibly 0.For example y − 2x = 3 is an implicit definition of y as a function of x.

Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.Consider the implicit function f(x, y) = 0. When differentiating this function, let y be a function of x. Then

differentiate with respect to x giving an equation in x, y, anddy

dx. Finally rearrange to make

dy

dxthe subject.

7.5 Computing Higher Derivatives

This involves repeated differentiation of a function.

• d

dxy =

dy

dx

• d

dx

(dy

dx

)=d2y

dx2

• d

dx

(d2y

dx2

)=d3y

dx3

etc.

7.6 Parametric Differentiation

We can differentiate x = g(t) to givedg

dtand y = f(t) to give

df

dt.

Thendy

dx=

dfdtdgdt

=df

dt

dt

dg.

Also, by chain rule, we can compute higher derivatives -d2y

dx2=

d

dt

(dfdtdgdt

)dt

dx

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7.7 Logarithmic Differentiation

If y = ax, write this as ln y = x ln a and then differentiate this implicitly.

7.8 Differentiation of Inverse Functions

If y = f(x), then let g = f−1. Then g(f(x)) = x, and so g′(f(x))f ′(x) = 1. This implies that g′(y)f ′(x) = 1.

This gives the formuladx

dy=

(dy

dx

)−1=

1

dy

dx

.

7.9 Leibniz Rule for Repeated Differentiation of Products

Suppose y(x) = f(x)g(x) (i.e. two functions multiplied together), thendny

dxn=

n∑r=0

(nr

)f (r)g(n−r).

8 Applications of Differentiation

8.1 Linearization

Differentiation can be used to find the ‘best fit’ straight line to any curve. For example, if y = f(x), then the

equation of a tangential line at x = a is given byy − f(a)

x− a= f ′(a).

Therefore the function L(x) = f ′(a)(x− a) + f(a) is the linearization of f(x) at x = a.

The linearization of any straight line is simply the straight line itself - this is the only function with thisproperty.

8.2 Extrema of Functions

The extrema of a function is the maxima (maximum points) and minima (minimum points).

8.2.1 Extreme Value Theorem for Continuous Functions

Theorem: If f(x) is continuous at every value in the interval [a, b], then f takes on both its maximum andminimum values in this interval.In other words, there are points xmax ∈ [a, b] and xmin ∈ [a, b] such that f(xmin) ≤ f(x) ≤ f(xmax) for allx ∈ [a, b].If the interval is not closed, or is infinite, then this theorem does not hold.

8.2.2 Local Extrema

Local extrema comprise of local maxima and local minima. A local maximum is a maximum point within agiven interval - it need not be the global maximum. Similarly, a local minimum is a minimum point withina given interval.

8.2.3 Derivatives and Extrema

Suppose a function f(x) is continuous and differentiable for x ∈ [a, b]. If there is a local extremum at a point csuch that a < c < b, then f ′(c) = 0.If there is no such point, then the function is locally monotonic for x ∈ [a, b].

8.3 Convexity and Concavity

A function f is said to be concave if any chord joining two points on the curve lies above the curve. Similarly,if any chord joining two points lies below the curve, the function is said to be convex.If f ′′(x) < 0 for x ∈ [a, b] then the function is concave for the interval [a, b].If f ′′(x) > 0 for x ∈ [a, b] then the function is convex for the interval [a, b].

15

8.3.1 Critical Points

The critical points of a function f(x) are defined to be the set of points where:

• f ′(x) = 0

• f ′(x) is undefined.

Note that not all critical points are extrema.

Classification of Critical Points where f ′(x) = 0

• If f ′′(x) > 0 then x is a local minimum.

• If f ′′(x) < 0 then x is a local maximum.

• If f ′(x) = 0 and f ′′(x) = 0, then the critical point may be a local maximum, local minimum or neither.

• If f ′′(x) = 0 and f ′′′(x) 6= 0, then x is a point of inflexion, where the function changes from concave toconvex, or vice versa. Note, in this case f ′(x) need not equal zero.

8.4 Limits using Calculus - L’Hopital’s Rule

It is possible to determine the limit of a ratio of two functions whose limits are both zero using calculus.If the two functions are differentiable, then we can apply L’Hopital’s Rule, which states that

limx→a

f(x)

g(x)= limx→a

f ′(x)

g′(x).

L’Hopital’s Rule can be used successively to evaluate the limit of a ratio of two functions as it follows that

limx→a

f(x)

g(x)= limx→a

f ′(x)

g′(x)= limx→a

f ′′(x)

g′′(x)= . . . etc.

8.5 Intermediate Value Theorem

Theorem: A function that is continuous for x ∈ [a, b] will take on every value between f(a) and f(b).

8.5.1 Finding the Root of an Equation

The intermediate value theorem tells us that if a function is continuous for x ∈ [a, b], then every interval inwhich f(x) changes sign must contain a root of the equation f(x) = 0.

8.6 Rolle’s Theorem

Theorem: Suppose that f(x) is continuous for x ∈ [a, b] and is differentiable for x ∈ (a, b) and f(a) = f(b),then there is at least one value c ∈ [a, b] where f ′(c) = 0.If f(x) is constant (i.e. such as the line y = 4) then clearly f ′(c) = 0 for all a < c < b.If f(x) is not constant, then f(x) must attain its extrema at some a < c < b - this can be shown using theextreme value theorem.For this theorem to be applicable, it is crucial that the function is differentiable for the interval specified.

8.7 Mean Value Theorem

Theorem: Suppose that f(x) is continuous for x ∈ [a, b] and is differentiable for x ∈ (a, b), then there is at

least one value of c ∈ (a, b) such thatf(b)− f(a)

b− a= f ′(c).

8.7.1 Corollaries of the Mean Value Theorem

If f ′(x) = 0 for x ∈ (a, b) then f(x) is constant.

The Constant Difference Theorem

Theorem: If f ′(x) = g′(x) at each x ∈ (a, b) then f(x) = g(x) + c.The constant difference theorem has a simple geometric interpretation. It tells us that if f and g have the samederivative for an interval then there is a constant c such that f(x) = g(x) + c for each x in the interval. Thatis, the graphs of f and g can be obtained from one another by a vertical translation of c.

16

8.8 Maclaurin Series

If a function f(x) can be differentiated n times at x = 0, then the nth term of the polynomial for the MaclaurinSeries expansion for f(x) at x = 0 is

pn(x) = f(0) + f ′(0)x+f ′′(0)x2

2!+f ′′′(0)x3

3!+ · · ·+ f (n)(0)xn

n!=

n∑k=0

f (k)(0)xk

k!.

8.8.1 The Radius of Convergence

The convergence set for a power series of x is called the interval of convergence. If the convergence set is:

• The single value x = 0, we say that the radius of convergence is 0.

• The interval (−∞,∞), we say that the radius of convergence is +∞.

• The interval (−R,R), we say that the radius of convergence is R.

8.9 Taylor Series

Whilst the Maclaurin Series finds a series expansion for x = 0, the Taylor Series can be used to find a seriesexpansion at x = a. If a function can be differentiated n times at x = a, then the then the nth term of thepolynomial for the Taylor Series expansion for f(x) at x = a is

pn(x) = f(a) + f ′(a)(x− a) +f ′′(a)(x− a)2

2!+f ′′′(a)(x− a)3

3!+ · · ·+ f (n)(a)(x− a)n

n!=

n∑k=0

f (k)(a)(x− a)k

k!.

If the expansion has no index n, at which the polynomial is stopped, the result is the Taylor series expansionfor f(x) - i.e. the series is infinite.

8.9.1 The Radius of Convergence

For any power series with terms in (x - a), exactly one of the following must be true.

1. The series converges only at x = a. In this case the radius of convergence is 0.

2. The series converges absolutely for all real values of x. In this case, the interval of convergence is (−∞,∞)and the radius of convergence is +∞.

3. The series converges absolutely for all real values of x in some open interval (a−R, a+R), and divergesfor all values outside this interval. For values x = a−R and x = a+R, the series may converge or diverge.In this case, the interval of convergence is (a−R, a+R), and the radius of convergence is R.

8.9.2 The Remainder Theorem of a Taylor Polynomial

We can find a measure of the accuracy of approximating a function f(x) to its nth Taylor polynomial pn(x).We use the idea of a remainder, giving f(x) = pn(x) + Rn(x), where the value |Rn(x)| is the error associatedwith the approximation. Finding a bound for Rn(x) gives an indication of the accuracy of the approximationf(x) ≈ pn(x).

8.9.3 Taylor’s Theorem

Theorem: If the function f is differentiable n+ 1 times in an open interval I containing a, then for each x inI, there exists a number c between x and a such that

f(x) = f(a) + f ′(a)(x− a) +f ′′(a)(x− a)2

2!+f ′′′(a)(x− a)3

3!+ · · ·+ f (n)(a)(x− a)n

n!+Rn(x)

where Rn(x) =fn+1(c)

(n+ 1)!(x− a)n+1.

Taylor’s theorem is a generalisation of the mean value theorem. If Rn(x) → 0 as x → ∞, for all x ∈ I,we say that the Taylor series generated by the function f at x = a converges to the original function f for the

interval I, and therefore we can state that f(x) =∞∑k=0

f (k)(a)(x− a)k

k!- i.e. the remainder tends to zero so it

need not be included.

17

8.9.4 The Remainder Estimation Theorem

Theorem: If there is a positive constant M such that |f (n+1)(t)| ≤M for all t between x and a inclusive, then

the remainder term Rn(x) in Taylor’s Theorem satisfies the inequality |Rn(x)| ≤M |x− a|n+1

(n+ 1)!.

If this condition holds for every n and the other conditions of Taylor’s Theorem are satisfied by f , then theseries converges to f(x).

9 Curve Sketching

This is a list of criteria to take into consideration when sketching a curve.

• Points where y = 0

• Points where x = 0

• Local maxima and minima (turning points wheredy

dx= 0)

• Ifdy

dx> 0, the function is increasing

• Ifdy

dx< 0, the function is decreasing

• Points of inflexion, whered2y

dx2= 0 but

d3y

dx36= 0

• Odd/Even function

• Periodic function

• Limits as x→ ±∞

• Limits as y → ±∞

• Asymptotes (are there any restrictions on values for x and y?)

• Symmetry

These may be considered in any order.

18