Notes

Winter Semester
Calculus 2 ( 1014 )

5.4 Indefinite Integrals and Net Change Theorem
- When we work in definite integrals we use F(b) - F(a) but with indefinite integrals it works differently
Instead of going through F(b) - F(a) we can just begin changing it into its antiderivative.
- Note though after we find the antiderivative with must add a constant +C at the end
- There are multiple indefinite integral identities in the chart.

Net Change Theorem
- The theorem states that when a quantity changes the final value is equal to the initial value + integral of the rate of change which is commonly a definite integral (Meaning it will equal a number since definite integrals will always give a value)

5.5 U-Substitution
- Substitution rule for integration means we can put another variable in the place of a value
u = g(x) meaning du = g'(x)
Its easiest to think of dx and du as differentials
You can decide which value to be u by using LIPET
Logarithmic
Inverse
Polynomials
Exponential
Trigonometric

Substitution with indefinite integrals

Steps to solve!
- First start by deciding on what value will be equal to u
- Next we find du (the derivative of u) and after solve for dx


- We rewrite and input the variable u and value of du or dx
- Move the coefficient outside of the integral and begin solving by finding the antiderivatives
- Input the u value back and add +C to your answer

Substitution with definite integrals
Note! Whenever you use u substitution you need to change the upper and lower bounds in respect to u
since without it would be in respect to x still
- We solve it normally calculating F(b) - F(a)
We can follow the steps above used to indefinite integrals nearly the exact same.

Symmetry
- If there is symmetry you can expect that one will be equal to the other


6.1 Areas between curves
- We begin using Riemann's sum again
but in this case they're split up into integrals
- We integrate them normally such as solving definite integrals in respect to x

Integrating in respect to y
- There isn't anything that differs too greatly other than instead of xr - xl the values change to y and you solve it as a normal definite integral

6.2 Volumes
- Again we are using Riemann's sums and integrals but we are adding a twist with including radius

Volumes of Solids in revolution
- Now we will be solving both inner and outside for area
In an example of a disk
Here we use A = pi(outer radius)^2 - pi(inner radius)^2
these then will be rewritten as a definite integral to then solve

Finding volume with cross-sectional area
- Cross section is a shape or what will be left if you cut a solid shape in half.
- This isn't any different than solving like with the others but broken into more parts



A cross section of a solid is a plane figure obtained by the intersection of that solid with a plane. The cross section of an object therefore represents an infinitesimal "slice" of a solid, and may be different depending on the orientation of the slicing plane.



6.4 Work
- We know work from physics its a force
W = F * d
F = ma or F = mg most commonly
it can also be written as m ( d^2s/dt^2) for F = ma

Now lets express this in integrals!
Work = upper b lower a f(x)dx
- We treat it as a definite integral and solve
Now we know that f = kx when applying hooke's law
where k is always the spring constant

6.5 Average value of a function
- Commonly when we solve for the average of something we add and then divide by the amount of numbers we added
- but for functions we write it as x = (b - a)/ n (n being any number)


- for integrals we will rewrite it as f avg = 1/b-a (upper b lower a) f(x)dx
a being the value of x and b the value of y
- Then we can simply solve for the average!

7.1 Integration by parts
- IBP short form is a way for us to solve integrals when we struggle and don't entirely know the antiderivative off the top of our heads
- the formula for integration by parts looks as such
integral sign u dv is equal to and can be written as
= u(v) integral sign v du

Indefinite integrals
- We need to identify u, du, dv, v
u can be picked using lipet and dv is whatever is left over
- We can then apply all the rules we know of integration and at the end remember to add +C


Definite integrals
- This works nearly similar to normal definite integrals but just like indefinite integrals
we need to identify the values and input them
- After we can solve as usual with F(b) - F(a)

Reduction formulas!
- Whenever we may have a trigonometric function to the power we can reduce because one may equal another such as
cos^2 x = 1 - sin^2 x
- This way we can cancel variables out to make it easier to solve
- Then we can solve normally


7.2 Trigonometric Integrals
- Trigonometric integrals differ from normal integrals due to using trig identities
- There are many trig identities that correlate with another meaning we can sometimes simplify to cancel out variables

Integrals of Powers of Sine and Cosine
- These work similarly to where we use U- Substitution to solve the integral and cancel
- We also can convert trig identities or factor or separate sin and cos to powers to change it into either




making it easier to substitute or identify a u value

Integrals of Powers of Secant and Tangent
- Secant and Tangent work similarly to sine and cosine power integrals not differing too greatly
- We again convert, factor/separate to make it easier for us to solve the integral

Using Product Identities
- These are examples of product identities that can help when evaluating trig integrals
- Sin(A)Cos(B) = 1/2 [sin(A-B) + sin(A+B)]
- Sin(A)Sin(B) = 1/2 [cos(A-B ) cos(A+B )]
- Cos(A)Cos(B) = 1/2 [cos(A-B ) cos(A+B )]

7.3 Trigonometric substitution
- This works somewhat similar to u substitution
- The only difference is you use trig identities in this case
- This type of substitution is called Inverse substitution




7.4 Integration of Rational functions by partial fractions

Method of Partial fractions
- first off lets imagine a rational function f(x) = P(x)/Q(x)
where P and Q are polynomials

In case 1 where Q(x) is the product of distinct linear factors
- We need to factor the denominator Q(x) to then be able to express it as A/(ax - b)^i or Ax + B / (ax^2+bx+c)


- We can factor the denominator to make it simpler to solve for A, B, C
- To determine the values of them we multiply both side of the equation
- Then by expanding on the right side we can rewrite into standard form showing us that the values of the left are equal to the right side
- In this case x^2 = (everything in the brackets corresponding to its variable) and the same for x and the constant



In case 2 where Q(x) is a product of linear factors some of which are repeated
- It works similarly to case 1 but instead the denominators are different where it goes
A/a + B/a^2 and C/the entire function aka C

The main thing that we're shown in 7.4 is that we are correlating what we're given to an expression/ identity to then substitute it in and factoring the denominator to be rid of it. From there we can use u substitution or other methods we've learned to solve.

7.7
The midpoint rule is used when you can't get a precise point
It is when you do the riemann sums and just divide the number by 2
We can use X = b - a / n and Xi = 1/2 ( Xi-1 + Xi)

But with the trapezoidal rule works a tad different
X = b-a/n and Xi = a+i(X)
though it still continues with using riemann sums

ERROR BOUNDS
ET = K(b-a)^3 / 12n^2 is used for the trapezoidal rule
EM = K(b-a)^3/24n^2 is used for midpoint rule
and we recognize that when the n increases the accurate approximations
and the errors on the left and right endpoints are in opposite signs and decrease by a factor of 2 when we double n value


We also know that midpoint and trapezoidal rule are the most accurate compared to endpoint approx.
The errors are opposite signs for trapezoidal and midpoint and decrease by a factor of 4 when we double n value
Lastly the size of the error for a midpoint rule is half the size of a error in trapezoidal rule

7.8
() means excluded where it is really close but not yet
[] means included where it is inside the graph or range
for examples
[1,2) shows us that everything from 1 - 1.9
Convergent : if the limit exists
Divergent : if the limit does not exist


IMPROPER INTEGRALS
In the case that we get a finite number (any number other than infinite) we say
the integral converges but when we get infinity it diverges

Type 1 : interval is infinite
- if f is continuous on [a,b) but not [a,b]
ex. f is discontinuous at b
it can be expressed in
if the limit exists in finite terms

- if f is continuous on (a,b] but not on [a,b]
ex. f is discontinuous at a then
it can be expressed in
if the limit exists in finite terms

- if f is discontinuous at x = c with a < c < b then



- if upper bound N and lower bound a f(x)dx exists for all numbers where N >= a (greater than and/or equal) then



- if upper bound a and lower bound N f(x)dx exists for all numbers where N<= a (less than and/or equal) then



- if f is defined for every real number x then for any real number a



We can use these rules and apply them to solve



Type 2 : f has an infinite discontinuity in [a,b]

- if f is continuous on [a,b) and is discontinuous at b then


if the limit exists as a finite number

- if f is continuous on (a,b] and is discontinuous at a then


if this limit exists as a finite number


- if f has a discontinuity at c where a < c < b and
both upper bound c lower bound a f(x)dx and upper bound b lower bound c f(x)dx are convergent


9.1 Differential Equations
- a differential equation contains unknown functions and some of its derivatives
- its used to predict future behavior on the basis of how current values change
- the order of a differential equation = the highest order of the derivative appearing in the equation
It is commonly used in population growth
- dP/dt = kP
t = time
P(t) = # of individual in the population at time t
k = proportionality constant
P(t) = Ce^kt and then P'(t) = C(ke^kt) = k(Ce^kt) = kP(t)

- when it comes to spring force ( -kx) we can also show it as m (d^2 x / dt^2)
and can then rewrite it as d^2x/ dt^2 = -(k/m) x

A general differential equations
- for example y' = xy
y is an unknown function of x
the function f is called a solution of a differential equation, if the equation is satisfied when y = f(x) and its derivatives are substituted into the equation that means we can use f (the solution) and rewrite the equation
when we are considering the differential equation
f'(x) = xf(x)
as we are solving
y' = f(x)
y' = x^3
y = x^4/4 +C
where C is a constant
- initial condition is a particular solution that satisfies a condition of the format y(t0) = y0
which could also be explained as f(x) = y (it is the same meaning but different variables)
- initial value problem is the problem of finding a solution that satisfies the initial condition


9.2 direction fields and Euler's method

- Sometimes we can't solve most differential equations of obtaining an explicit formula for the solution
but despite us not having an explicit solution we can still learn a lot about the solution through a
graphical approach (direction fields) and Euler's method
- for ex. y' = x+y and y(0) = 1
we're told to sketch the graph of the solution for the Initial value problem (IVP) and we don't know the formula for the solution
so how can we sketch the graph?
the equation y' = x + y gives us the assumption that the slope is at any point (x,y) on the graph
(this is known as the solution curve) which is equal to the sum of x- and y- coordinates
since the curve passes through the point (0,1) its slope must be equal to 1 ( 0 + 1 = 1 )
then to guide us into sketching the curve we draw a short line segments at a number of points (x,y ) with slope x+y
- when we draw the curve its parallel to nearby line segments and suppose we have a
first order differential equation y' F(x,y)
where F(x,y) is some expression in x and y. The differential equation states that the slope of a solution curve
at a point (x,y) on the curve is F(x,y). By drawing the short line segments with slope F(x,y) at multiple points (x,y)
the result is called a direction field and it indicates the direction the solution curve is heading
- a differential equation in the form of y' = f(y) that an independent variable is missing from the right side of the equation is called autonomous
for these equations we note. the slopes corresponding to two different points with the same y coordinate must be equal
meaning if we know one solution to an autonomous differential equation then we have the knowledge of many others just by shifting the graph of the known solution left or right across the y axis


9.3 Separable Equations
- separable equation is a first order differential equation that is expressed with dy/dx
dy/dx = g(x) f(y) the name separable comes from the fact the expression on the right side can be "separated"
into a product of a function of x and a function of y
example:
dy/dx = g(x) / h(y) where h(y) = 1/f(y)
h(y) dy = g(x) dx where we are separating the y's on one side and x's on the other
which we can then integrate both sides and use chain rule to justify the procedure: if h and g satisfy


10.1 Curves defined by Parametric equations
- Lets imagine a particle is moving along the curve C it is impossible to describe C
with an equation of the form y = f(x) because C fails the Vertical line test
Recall vertical line test :
It should only touch the imaginary line one unless its a straight line at 0 or not a function
But the x and y coordinates of the particle are functions of time t and
so we can write x = f(t) and y = g(t) and this pair of equations is a convenient way of describing the curve!

- A parametric equation is a curve with a 3rd variable ( a parameter)
in the example beforehand the third variable was t
each value of t determines at point (x,y) which can be plotted
- As t varies the point (x,y) = (f(t), g(t)) varies and traces put a curve which we can then call a parametric curve
We're usually given a table to help us plot for each value of t
- In general the curve of a parametric equation will look as
x = f(t) , y = g(t) , a<= t <= b
it has an initial point f(a), g(a) and a terminal point of f(b), g(b)

10.2 Calculus with Parametric Curves

Tangent
- assume f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve
x = f(t) and y = g(t) where y is also a differentiable function of x
dy/dx = dy/dt/dx/dt if dx/dt is not equal to 0
by using chain rule dy/dt = dy/dx * dx/dt which resulted in the above

- parametric curve has a vertical tangent when dx/dt = 0 and dy/dt is not equal to 0 of
d^2y/dx^2 = d/dx (dy/dx) = d/dt * (dy/dx) / dx/dt

(if both dx/dt = 0 and dy/dt = 0 we have to use another way to determine the slopes)
ex. A curve C is defined by the parametric equations x = t^2 , y = t^3 - 3t
a) show that C has two tangents at the point (3,0) and find their equations
- We notice the x = 3 for t + - sqrt 3 and in both cases y = t(t^2 - 3) = 0
therefore the point (3,0) on C arises from two values of the parameter
t = sqrt 3 and - sqrt 3 indicating to us C crosses itself at (3,0)
dy/dx = dy/dt / dx/dt = 3t^2 -3 / 2t
- we then input the values of t and solve giving us the two tangent lines
y = sqrt 3 (x - 3) and y = -sqrt 3 (x - 3)

b) find the points on C where the tangent is horizontal and vertical
- C has a horizontal tangent when dy/dx = 0 and dx/dt is not equal to 0
since dy/dt = 3t^2 - 3 this happens when t^2 = 1 that is t = +- 1
the corresponding points on C are ( 1, -2 ) and (1 , 2)
C has a vertical tangent when dx/dt = 2t = 0 that is t=0 ( note that dy/dt is not equal to 0 there)
The corresponding point on C is (0,0)

c) determine where the curve is concave upward or downward
- To determine concavity we calculate the second derivative
the curve is concave upward when t > 0 and concave downward when t < 0

d) Sketch the curve
- Using the info we collect from part b and c to help us sketch the figure

Arc length
- We already know how to find the length (L) of a curve C with the given form y = F(x)
if F' is continuous then L = definite integral sqrt 1 + (dy/dx)^2 dx
suppose that C can also be described by the parametric equations x = f(t) and y = g(t)
where dx/dt = f'(t) > 0 this tells us that C has traversed once ( from left to right as t increases from alpha to beta) and
f(alpha) = a and f(beta) = b We then input everything and solve
- using Leibniz notation we have the following result
if a curve C is described by the parametric equations x = f(t), y = g(t)
where f' and g' are continuous on [alpha,beta] and C is traversed exactly once as t increases from alpha to beta
then the length of C is L = definite integral sqrt (dx/dt)^2 + (dy/dt)^2 dt
- Note that the formula in theorem 5 is consistent with the general formula L = integral ds
ds = sqrt (dx/dt)^2 + (dy/dt)^2 dt

10.3 Polar Coordinates
- A coordinate system shows us a point in the plane by an ordered pair of numbers known as coordinates
Usually we use Cartesian coordinates which are directed distances from two perpendicular axes
but we can describe the coordinate system by using the polar coordinate system which is more convenient
- We choose a point on the plane known as the pole (or origin) and is labeled O (not zero its letter o )
then we draw a half line starting at O calling this the polar axis
the axis is drawn horizontally to the right and corresponds to the positive x-axis in cartesian coordinates
- If P is any other point in the plane let r be the distance from O to P and let theta be the angle (in radians)
between the polar axis and the line OP
- point P is represented by the ordered pair (r,theta) and r, theta are called polar coordinates of P
we use the convention that an angle is positive if measured in
the counterclockwise direction form the polar axis and negative in the clockwise direction
- if P = O then r = 0 and we agree that (0,theta) represents the pole for any value of theta
- in a case where r is negative we have (-r,theta) and (r,theta) where they both lie on the same line through O and at
the same distance of r from O but on the opposite sides of O (explaining why one is positive and another negative)

- In the Cartesian system everyone point has only one representation but in the polar system each point has many representations where it can be the same point but written in many other ways
- If the point P has cartesian coordinates (x,y) and polar coordinates (r,theta) then
we have cos(theta) = x/r and sin(theta) = y/r
so to find the cartesian coordinates (x,y) when the polar coordinates (r,theta) are known we use the equations
x = r cos() and y = r sin()
to find the polar coordinates of (r,theta) when we know the cartesian coordinates (x,y) are known we use the equations
r^2 = x^2 + y^2 ( aka a^2 +b^2 = c^2 ) and tan() = y/x

Polar curves
- The graph of a polar equation r = f() or more general F(r, theta) = 0
has all points P that have at least one polar representation whose coordinates satisfy the equation
- A couple complete curves are
cardioid because its shaped like a heart
four leaved rose is a curve of four loops

11.1 Infinite sequences
- A infinite sequence or just a sequence is a list of numbers written in a definite (specific) order
n is a variable used as a placeholder for a number
- A sequence can be defined as a function f(n) whose domain is the set of positive integers
f(n) = a of n
- Some sequences can be defined by giving a formula for the nth term (basically examples)
a) the following are three equivalent descriptions of this sequence

b) the definition {n/n+1} upper bound infinity and lower bound n = 2
indicates that the formula for the nth term is an = n/n+1 and we start the sequence with n = 2

c) the sequence {sqrt 3 , sqrt 4 , sqrt 5 , sqrt 6 , ...} can be described by { sqrt n+2 } upper bound infinity lower bound n = 1
if we start with n = 1. Equivalently we could start with n = 3 and write { sqrt n } upper bound infinity lower bound n = 3

d) the definition {(-1)^n (n+1)/3^n } upper bound infinity and lower bound n = 0

The limit of a sequence
- Its best to plot the terms on a number line or graph to visually see
In fact the difference can be made as small as we like by taking n sufficiently large
- We indicate this by writing lim n/n+1 = 1
which can be rewritten in the notation of lim an = L ( a finite number)
this means that the terms of the sequence {an} approach L as n becomes large

- A sequence has a limit (L) and we write lim an = L
if the limit exists we say the sequence converges otherwise it diverges ( meaning if it goes to infinity it diverges)

- A convergent sequence it can be shown in all these different ways


Monotonic and Bounded sequences
- A sequence {an} is called increasing if an < an+1 for all n>= 1
and it is called decreasing if an > an+1 and n>= 1
- A sequence is known as monotonic if it is either increasing or decreasing

- A sequence {an} is bounded above if there is a number M such that
an <= M for all n >= 1
- a sequence bounded below if there is a number m such that
m <= an for all n>= 1
- if a sequence is bounded above and below then it is called a bounded sequence
- Not every bounded sequence is convergent for ex. an = (-1)^n satisfies -1<= an <= 1 but is divergent
- Not every monotonic sequence is convergent ex. (an = n -> infinity)
BUT IF A SEQUENCE IS BOTH BOUNDED AND MONOTONIC THEN IT MUST BE CONVERGENT

11.2 Series
- given an infinite sequence we consider the partial sums
which forms a new sequence {Sn} which may or may not have a limit

- if lim Sn exists then we call it the sum of the infinite series
if the sequence is convergent and lim sn = s exists as a real number then the series an is convergent and we rewrite it
the number s is called the sum of the series. if the sequence {Sn} is divergent then the series is called divergent

Sum of a geometric series
- an example of a infinite series is the geometric series where each term is obtained from
preceding one by multiplying it by the common ratio (r)
for ex if r = 1 it continues to infinity meaning the limit DNE and the geometric series diverges
if r is not equal to 1 we get 2 equations and by subtracting them
sn - rsn = a - ar^n -> sn = a(1-r^n)/1-r
if -1 < r < 1 we know the r^n = 0 as n -> infinity
so by solving for the limit the geometric series is convergent with the sum being a/1-r
- this can all be summarized as
A geometric series is convergent is r < 1
and its sum is a/ 1-r
but it is divergent when r > or = 1
- To find the sum of the geometric series you find the common ratio and compare to see if converges or diverges
and from there substitute it into a/1-r
r (being the ratio)
a(being the first term)

Test for divergence
- recall if an series is divergent then the sn sequence is divergent
- with a harmonic series we can use partial sums and compare the results to the rules to determine it diverges
- if the series an is convergent then lim an = 0 but this is not always true we cannot conclude that an is convergent
though the thereom states if lim an DNE or if lim an not equal to 0 then the series is divergent
but in this ex/case the series is not divergent so it must be converging and the lim must equal 0

Properties of Convergent series
- if an and bn are convergent series then so are the series can (where c is a constant), (an + bn) and (an - bn)
it can then be written different ways





11.4 The comparison tests
These tests are used to compare a given series with a series that is known to converge or diverge
- The direct comparison test supposes that series an and bn both have positive terms
i) if bn is convergent then an <= bn for all n (values) then an is also convergent
ii) if bn is divergent and an >= bn for all n then an is also divergent
- when using comparison test we must have one known series (bn) for the purpose of comparison
most of the time we use one of these series
A p-series 1/n^p converges if p>1 and diverges if p <= 1
A geometric series ar^n-1 converges if r < 1 and diverges if r > = 1
A harmonic series 1/n = 1 + 1/2 + 1/3 + 1/4 diverges


11.5 Alternating series and Absolute convergence

Alternating series
- an alternating series is a series whose terms are alternately positive and negative
( going from positive and negative )
the nth term of an alternating series is of the form
an = (-1)^n-1 bn or an = (-1)^n bn
where bn is a positive number (in fact bn = |an|)

- The following test says that if the absolute value of the terms of an alternating series
decreases toward 0 then the series converges
Alternating series test if the alternating series
(negative and positive or from positive to negative)
satisfies the conditions
i) bn+1 <= bn for all n
ii) lim bn = 0
then the series is convergent

- the alternating harmonic series satisfies
i) bn+1 < bn because 1/n+1 < 1/n
ii) lim bn = lim 1/n = 0
therefore the series is convergent

Absolute Convergence and Conditional Convergence
- a series an is called absolutely convergent if the series of absolute values an is convergent
note that if an is a series with positive terms than |an| = an and so absolute convergence is the same as convergence in this case
- to restate this an is absolutely convergent then it is convergent
absolutely convergent -> convergent
- the alternating series is absolutely convergent because the p series is convergent
- Conditional convergence is if it is convergent but not absolutely convergent
this is when the series an converges but |an| diverges ( the lines mean the absolute value)

The Ratio Test
- the following test is very useful in determining whether a given series is absolutely convergent
i) if lim |an+1 / an| = L < 1 then the series an is absolutely convergent ( therefore convergent )
ii) if lim | an+1 / an | = L > 1 or lim | an+1 / an | = infinity
then the series is divergent
iii) if lim |an+1/an| = 1 the ratio test is inconclusive that is no conclusion can be drawn
about the convergence or divergence of an
- although the ratio test works in some examples an easier method is to use the test for divergence
since an = n^n / n! = n*n*n...n/1*2*3...n >= n
if follows that an does not approach 0 as n-> infinity
therefore the given series is divergent by the test of divergence

11.8 Power series
- a series of the form
cn (x-a)^n = c0 + c1 (x-a) + c2 (x-a)^2
is called a power series in (x-a) or a power series centered at a or a power series about a
x : a variable
the cn's : constants called the coefficients of the series
a : the center of the power series

- for a power series cn (x-a)^n there are only three possibilities
i) the series converges only x = a R=0
ii) there is a positive number R such that the series converges if |x-a| < R and diverges |x-a| > R
iii) the series converges for all x R = infinity
R in case (ii) is called the radius of convergence of the power series

- the interval of convergence of a power series is the interval that consists of all values of x for which the series converges
in case (i) the interval consists of just a single point a
in case (ii) note that the inequality |x-a|<R can be rewritten as a-R <x<a+R
diverges when x = a - R
converges when x = a + R
in case (iii) the interval is (-infinity, infinity)
Note: in case (ii) there are four possibilities for the interval of convergence
(a-R,a+R) (a-R,a+R] [a-R,a+R) [a-R,a+R]

11.9 Representations of Functions as Power series
- An intro to the representation of functions as power series using the geometric series
Examples of how to rewrite and manipulate functions to express them as power series, as well as finding their interval of convergence
• A geometric series ar^n-1 that as a/1-r converges if |r| < 1; diverges if |r|>= 1
let a = 1 and r = x then the geometric series becomes 1/1-x = 1+x+x^2+x^3+... x^n |x|<1
therefore we represent the function f(x) = 1/(1-x) as a sum of a power series
when the graph is sketched we notice as n increases sn(x)
becomes a better approximation for f(x) for -1 < x < 1

Differentiation and Integration of Power series
- The sum of a power series is a function f(x) = cn(x-a)^n whose domain is the interval of convergence of the series
- We would like to be able to differentiate and integrate such functions and the next theorem says that we can do so by
so called term by term differentiation and integration
- If the power series cn(x-a)^n has radius convergence R>0 then the function f defined by f(x) = c0 + c1(x-a) + c2(x-a)^2
is differentiable and therefore continuous on the interval (a-R, a+R) and
i) f'(x) = c1+2c2(x-a) + 3c3(x-a)^2 + ncn (x-a)^n-1
ii) integral f(x)dx = C+c0 (x-a) + c1 (x-a)^2/2 + c2 (x-a)^3/3 = C+cn(x-a)^n+1/n+1
the radius of convergence of the power series in equations i and ii are both R


11.10 Taylor and Maclaurin Series
(the ! means factorial and a factorial means to multiply the number by decreasing positive integers)
- if f has a power series representation at a
i.e f(x) = cn(x-a)^n (a Taylor series)
with radius of convergence R, | x - a | < R then cn = f^(n)(a)/n!
- Different form of Taylor series
f(x) = f^(n)(a)/n! * (x-a)^n
= f(a) + f'(a)/1! * (x-a) + f"(a)/2! * (x-a)^2
Note: when a = 0 the series is called Maclaurin series

We collect in the following table for future reference some of the important Maclaurin series
that we have derived in this section and the preceding one!









New Taylor series from old
- If a function has a power series representation at a then the series is uniquely determined.
that is no matter how a power series representation for a function f is obtained it must be the Taylor series of f
-We can then use new taylor series representations by manipulating series from the table rather than using coefficient formula

The n th degree Taylor Polynomial
- if f has derivatives of all orders it is true that
f(x) = f^(n)(a)/n! (x-a)^n
when f(x) is the limit of the sequence of partial sums

In the case of the Taylor series the partial sums are




notice that Tn is a polynomial of degree n called the nth degree Taylor polynomial of f at a
the nth degree taylor polynomial can be used as an approximation to f(x)
- for instance for an exponential function f(x) = e^x the result shows that the taylor polynomials at 0
(or Maclaurin polynomials) with n = 1,2,3
- in general f(x) is the sum of its Taylor series if
f(x) = lim Tn (x)
if we let Rn (x) = f(x) - Tn(x) so that f(x) = Tn(x)+Rn(x)
then Rn(x) is called the remainder of the Taylor series. If we can somehow show that the lim Rn(x) = 0
it follows that lim Tn(x) = lim [f(x) - Rn(x)] = f(x) - lim Rn(x) = f(x)


11.11 and 11.05

- Taylor polynomials can be used to approximate functions , how physicists and engineers use these in such fields as relativity, optics, blackbody radiation, electric dipoles, the velocity of water waves and building highways across a desert
Tn(x) is called the nth degree polynomial of f(x) at a
Tn(x) -> f(x) as n -> infinity -> f(x) ~ Tn(x)
( the wavy line indicates approximate)

- The partial sum of a taylor series is





Notice that Tn is a polynomial of degree n called the nth degree taylor polynomial of f at a
- In general f(x) is the sum of its Taylor series if f(x) = lim Tn(x)
if we let Rn(x) = f(x) - Tn(x) so that f(x) = Tn(x) + Rn(x)
then Rn(x) is called the remainder of the taylor series. if we can somehow show that the
error lim |Rn(x)| = 0 then it follows that
lim Tn(x) = lim [f(x)-Rn(x)] = f(x) - lim Rn(x) = f(x)

- Absolute error |Rn(x)| = |f(x) - Tn(x)|
Taylor inequality if |f^(n+1)(x)| <= M where M is a positive constant then
|Rn(x)| <= M/(n+1)! |x-a|^n+1

- Alternating series estimate
if s = (-1)^n-1 bn where bn>0 is the sum of an alternating series that satisfies
i) bn+1 <= bn
ii) lim bn = 0
then |Rn| = |s-sn|<= bn+1