VM266

Chapter 11 Inifinite Series

11.1 Sequences

A sequence is a function f whose domain is the set of positive integrers

The notation for a sequence is as follows.

If f:A→S is a sequence, then a symbol, for example "a", is chosen to represent elements of this sequence.

Then for each k∈A, f(k) is denoted ak, and f itself is denoted ⟨ak⟩k∈A.

Other types of brackets may be encountered, eg. (ak)k∈A and {ak}k∈A.

The latter is discouraged because of the implication that the order of the terms does not matter.

Any expression can be used to denote the domain of f in place of k∈A.

The set A is usually understood to be the set {1,2,3,…,n}.

If this is the case, then it is usual to write ⟨ak⟩k∈A as ⟨ak⟩ or even as ⟨a⟩ if brevity and simplicity improve clarity.

11.2 Convergent or Divergent Series

Theorems and Definitions

Definitions: Convergent and divergent series1. A series is an infinite sum, written . It doesn’t have to start at k = 0.2. The nth partial sum of the series is 3. The series converges to a number L means: . In other words, for n large enough, is the same as L to any number of decimal places you like.4. If converges to L, we sometimes write = L . In other words, the sum is as close to L as you want if you take enough terms.5. Sometimes there is no number L with . In such a case we say that the series diverges. This could happen because the partial sums bounce around, or, in the case of a series like , because the numbers get bigger and bigger without bound.6. Types of series: a) is a positive term series means that for all k.b) An alternating series has terms that switch signs. Every alternating series can be written as either or as where for all k.c) A series is absolutely convergent if the positive term series is convergent. If a series is absolutely convergent, it is convergent. The reverse might not be true. For example, if , then converges but is not absolute convergent, since = diverges. There is a name for this situation, as follows: d) A series is conditionally convergent if converges but diverges. Some basic tips for doing power series problems 1. You MUST start each convergence question with the kth term test: The series is divergent if a) does not exist or b) is not 0. If that limit is 0, you must continue with other tests. So you need to know stuff about limits, as follows. i) exponential functions grow faster than polynomials. and ii) factorials grow faster than exponential functions: and iii) limits pull inside functions: .

11.3 Positive- Term Series

Definitions and Theorems

11.5 Alternating Series and Absolute Convergence

A series in which successive terms have opposite signs is called an alternating series.
The Alternating Series Test (Leibniz's Theorem)
This test is the sufficient convergence test. It's also known as the Leibniz's Theorem for alternating series.
Let {an} be a sequence of positive numbers such that
1. an+1 < an for all n;

2. .
Then the alternating series and both converge.
Absolute and Conditional Convergence
A series is absolutely convergent, if the series is convergent.
If the series is absolutely convergent then it is (just) convergent. The converse of this statement is false.
A series is called conditionally convergent, if the series is convergent but is not absolutely convergent.

11.4 The Ratio and Root Tests

Ratio and Root test

11.6 Power Series

11.7 Power Series Representations of Functions

Chapter 10 Indeterminate Forms and Improper Integrals

10.1 The Indeterminate Forms 0/0 and inf/inf

Cauchy's Formula and L’Hospital’s Rule

10.2 Other Indeterminate Forms

Having a zero x infinity

Guidelines for Investigating the indeterminate forms.

1) Let y = f(x) ^g(x)

2) Take natural logarithms in guideline 1: lny = ln f(x) ^g(x) = g(x) ln f(x)

3) Investigate lin ln y = lim x -> c [g(x) ln f(x)] and conclude the conditions

10.3 Integrals with Infinites limits of Integration

Explanation in having Improper integrals with inifinite limits

10.4 Integrals with Discontinuous Integrands

Subtopic

Chapter 9 Techniques of Integrations

9.1 Integration by Parts

If u = f(x) and v = g(x) and if f' and g' are continuous, then ∫u dv = uv - ∫ v du

9.2 Trigonometric Integrals

Trigonometric Integral Guildlines

9.3 Trigonometric Substitutions

Expressions in Integrand

sqrt(a^2 - x^2)

x = a sin theta

sqrt(a^2 + x^2)

x = a tan theta

sqrt(x^2) - a^2

x = a sec theta

9.4 Integrals of Ratinoal Functions

Guidelines for integrals of rational functions

9.5 Integrals Involving Quadratic Expressions

9.6 Miscellaneous Substitions

If an integrand is a rational expression in sinx and cosx, the following substitutions will produce a rational expression in u: sinx = 2u/(1+u^2), cosx= (1-u^2)/(1+u^2), dx = 2/(1+u^2)du, where u = tan(x/2)

Chapter 13: Plane Curves and Polar Coordinates

13.1 Plane Curves

A plane curve is a set c of ordered pair(f(t),g(t)), where f and g are continuous on an interval I

Let C be the curve consisting of all ordered pairs (f(t),g(t)), where f and g are continuous on an interval I. The equations x =f(t), y =g(t) for t in I, are parametric equations for c with paramet t

13.2 Tangent Lines and Arc Length

Summary of this

https://www.cs.drexel.edu/classes/Calculus/MATH123_Spring03/lecture22.pdf

13.3 Polar Coordinates

The rectangular coordinates (x,y) and polar coordinates (r,theta) of a point P are related as follows: i) x = rcos theta

13.4 Integrals in polar coordinates

http://www.youtube.com/watch?v=I2Z6K_g5kpc

Guidelines for finding the area of an R_theta region

1) sketch the region, leabeling the graph of r = f(theta). Find the smallest value theta = alpha and the largest value theta = beta for points in (r, theta) in the region.2) Sketch a typical circular sector and label its central angle dtheta3)express the area of the setro in guideline as 2 1/2 r^2dtheta.4) apply the limit of sums operator inegral from alpha to beta to the expression in guideline 3 and evaluate the integral

13.5 Polar Equations of Conics

Let F be a fixed point and l a fixed line in a plane. The set of all points P in the plane, such that the ration d(P,F)/d(P,Q) is a positive constant e ith d(P,Q) the distance from P to l, is a conic section. The conic is aparaaola if e =1, an ellipse if 0<e<1, and a hyperbola if e <1.

Chapter 8 Inverse Trigonometric and Hyperbolic Functions

8.1 Inverse Trigonometric Functions

arccosecant y = arccsc x x = csc y x ≤ −1 or 1 ≤ x −π/2 ≤ y < 0 or 0 < y ≤ π/2

arcsecant y = arcsec x x = sec y x ≤ −1 or 1 ≤ x 0 ≤ y < π/2 or π/2 < y ≤ π

arccotangent y = arccot x x = cot y all real numbers 0 < y < π

arctangent y = arctan x x = tan y all real numbers −π/2 < y < π/2

arccosine y = arccos x x = cos y −1 ≤ x ≤ 1 0 ≤ y ≤ π

arcsine y = arcsin x x = sin y −1 ≤ x ≤ 1 −π/2 ≤ y ≤ π/2

8.2 Derivatives and Integrals

dx sin-^1(u) = 1/sqrt(1-u^2)dxu

dxcos^-1(u)=-1/sqt(1-u^2)dxu

dxtan^-1u= 1/(1+u^2)dxu

dx sec^-1u = 1/usqrt(u^2-1)dxu

∫1/sqrt(a^2-u^2)du = arcsin (u/a) + C

∫1/(a^2+u^2)du=(1/a)arctan(u/a)+C

∫1/usqrt(u^2-a^2)du=(1/a)arcsec(u/a)+C

8.3 Hyperbolic Functions

Def.Eq.

8.4 Inverse Hyperbolic Functions

Definitions and equations

Chapter 7 Logarithmic and Exponential

7.1

One-to-one function is a function f domain D and rage R whenever a does not equal to b in D, then f(a) does not equal to f(b) in R.

Inverse function is a one-to-one function with domain D and range R with a condition that y=f(x) if and only x=g(y).

Theorem 7.3 : g(f(x))=x for every x in D and f(g(y)) = y for every y in R.

Domain of f^-1 = range of f and range of f^-1 = domain of f

Guidlines for finding f(inverse) in simple cases

1)Verify that f is a one-to-one function(or that f is increasing or is decreasing) throughout its domain.

2)Solve the equation y=f(x) for x in terms of y, obtaining an equation of the form x=f(inverse)(y)

3)Verify the two conditions: f^-1(f(f(x))=x and f(f^-1(x))=x for every x in domains of f and f^-1, respectively.

Theorem7.6 states that if f is continuous and increasing on [a,b], then f has an inverse function f^-1 that is continuous and increasing on [f(a),f(b)].

Theorem 7.7 states that if a differentiable function f has an inverse function g=f^-1 and if f'(g(c)) does not equal 0, then g is differentiable at c and g'(c) = (1/f'(g(c)))

Theorem 7.8 states that if g is the inverse function of a differentiable function f and if f'(g(x)) does not equal 0 then g'(x)=(1/f'(g(x)))

Section 7.2

$The natural logarithmic function, denoted by ln and the function is for every x > 0$

The natural logarithmic function, denoted by ln and the function is for every x > 0

Theorem 7.11 : If u = g(x) and g is differentiable, then: Dx ln u = 1/u Dx u if g(x) > 0 and Dx ln |u| = 1/u Dx u if g(x) does not equal to 0

Law of natural logarithm (7.12): if p > 0 and >0, then :

i)ln pq = ln p + ln q

ii) ln p/q = ln p - ln q

iii)ln p^r = r ln p for every ration number r

Guidelines for logarithmic differentiation

1) y=f(x)

given

ln y =ln f(x)

take natural logarithms and simplify

Dx[ln y] = Dx [ln f(x)]

differentiate implicity

1/y Dx y = Dx[ln f(x)]

Theorem 7.11

Dx y = f(x) Dx [ln f(x)]

multiply by y = f(x)

Section 7.3

Theorem 7.14 states that for every real number x there corresponds exactly one positive real number y such that ln y =x

Natural exponential function, denoted by exp, is the inverse of the natural logarithmic function.

The letter e denotes the positive real number such that ln e =1

If x is any real number, then e^x = y if and only if ln y =x

Theorem 7.19

i) ln e^x = x for every x

ii) e^ ln x = x for every x >0

Theorem 7.20

i) (e^p)(e^q) = e^p+q

ii) e^p / e^q = e^p-q

iii) (e^p)^r = e^pr

Theorem 7.21 states that Dx e^x = e^x

Theorem 7.22 states that if u = g(x) is differentiable, then Dx e^u = e^u Dx u

Section 7.4

integral 1/u du = log(u)+constant is if u = g(x) =/ 0 and g is differentiable

integral e^u du = e^u+constant is if u = g(x) and g is differentiable

Theorem 7.25

integral tan(u) du = -log(cos(u))+constant

integral cot(u) du = log(sin(u))+constant

intergral sec u du = ln |sec u + tan u| + C

integral csc u du = ln |csc u - cot u| + C

Chapter 12: Toics from Analytic Geometry

12.1 Parabolas

A parabola is the set of all points in a plane equidistant from a fixed point F (the focus) and a fixed line l (the directrix) in the plane.

For an upward or downward graph look at the equation y = ax^2 + bx_c

12.1 Ellipses

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points (the foci ) in the plane is constant

Graph of the equation is x^2/a^2 + y^2/b^2 = 1

12.3 Hyperbolas

A hyperbola is the set of all points in a plne, the difference of whose distances from two fixed points in the plane (the foci) is a positive constant.

x^2/a^2 - y^2/b^2 = 1 is a hyperbola with vertices (+-a,0).

12.4 Rotation of axes

A summary of rotation of axes