511 Basics Handouts

Review of Basic Concepts and Skills

Phyllis R. Nelson

Cal Poly Pomona

EGR 511 - Winter 2013

Phyllis R. Nelson (Cal Poly Pomona) Review of Basic Concepts and Skills EGR 511 - Winter 2013 1 / 52

Outline

1 Introduction

2 Software: how can it help understand and solve math problems?

3 Limit and continuity of a function

4 Differentiability

5 Theorems used for validating algorithms

6 Sets, sequences, and series

7 Applications of series


Introduction

Moore’s law has led to a transformation of engineering practice. Today’sengineers increasingly rely on a variety of software packages for design andanalysis. It is thus essential that engineers have a basic understanding ofboth the capabilities and the possible inaccuracies of the software they use.

The purposes of this set of slides are to

review basic calculus concepts used in numerical modeling

introduce several software packages

demonstrate the use of these software packages

After a brief discussion of the software packages and their use, I will coverthe definitions and theorems in section 1.1 of the text.


Outline

1 Introduction



4 Differentiability





How can software packages help understand math?

Visualization (graph a function)

Symbolic operations (factoring, simplification, integration,differentiation, . . . )

Numerical computations (find a number or set of numbers)

Numerical or symbolic solutions to sets of equations (linear, nonlinear,ordinary differential, partial differential, . . . )

Fitting data to models

These are the topics we will consider this quarter.

The result of one of these processes may not be “goodenough” or even “valid” for your use. You must always checkthat the model is valid for your application!


Useful software packages for engineers

Commercial:

Matlab

Mathematica

Maple

Mathcad

. . .

Free, at least for academic use:

octave (although the symbolic toolbox is limited)

gnuplot

maxima

sage

. . .


Outline

1 Introduction



4 Differentiability





Limit of a function

Definition

A function f defined on a set X of real numbers has the limit L at x0

limx→x0

f(x) = L

if, given any real number ε > 0, there exists a real number δ > 0 such that

|f(x)− L| < ε whenever x ∈ X and 0 < |x− x0| < δ

This definition says nothing about what happens at x = x0.

Example

f(x) =

{1, x 6= 00, x = 0

limx→0

= 1


Limits involving infinity

Definition

A function f defined on a set of real numbers has the limit ∞ at x0

limx→x0

f(x) =∞


f(x) > ε whenever x ∈ X and 0 < |x− x0| < δ

A similar definition can be constructed for a limit of −∞.

Example

limx→0

x−2 =∞ limx→±1

x2 + 2x2 − 1

do not exist


Limits involving infinity - (2)

Example

limx→±1

x2 + 2x2 − 1

Visualization:Use gnuplot to plot this function near x = ±1.

set xrange [-5:5]set yrange [-25:25]plot (x**2+2)/(x**2-1), 0

The left and right limits atx = ±1 are ±∞, so thereis no unique value L.


Limits involving infinity - (3)

Definition

A function f defined on a set of real numbers that includes ∞ has thelimit L as x→∞

limx→∞ f(x) = L


|f(x)− L| < ε whenever x ∈ X and x > δ

Example

limx→∞ e

−x = 0 limx→∞ a−

1x+ a

= a


Continuity of a function

Definition

Let f be a function defined on a set X of real numbers and x0 ∈ X. Thenf is continuous at x0 if

limx→x0

f(x) = f(x0)

The function f is continuous on the set X if it is continuous at eachnumber in X.

Example

At x = 0, f(x) and g(x) have removable singularities.

f(x) =

{1, x 6= 00, x = 0

g(x) =sinxx

h(x) = x−1


“Continuous” is stricter than “limit”!

A function having a limit for all x ∈ X is not necessarily continuous on X!

f(x) =

{1, x 6= 00, x = 0

has a limit at x = 0 but is not continuous there.

g(x) = x− bxc

has no limit if x is an integer, but is bounded by 0 and 1 everywhere.

h(x) = x−2

is infinite at x = 0. Does it have a limit there? Is it continuous there?


Investigate continuity using xmaxima

Example

Is this function continuous?

f(x) =2x2 − 3x+ 1

x− 1

Visualization:Any “interesting” behavior should happen at x = 1, so plot f near there.

This code plots f(−2 < x < 2) in maxima:

f1 : (2*x^2-3*x+1) / (x-1);plot2d(f1, [x,-2,2]);


Continuity example - (2)

The plot is linear.

Could the numeratorcontain the denominatoras a factor?

Symbolic operations:

(%i3) limit(f1,x,1);(%o3) 1(%i4) factor(2*x^2-3*x+1);(%o4) (x - 1) (2 x - 1)(%i5) ratsimp(f1,x);(%o5) 2 x - 1

YES!


Outline

1 Introduction



4 Differentiability





Differentiability

Definition

The function f , which is defined in an open interval containing x0, isdifferentiable and has a derivative f ′ at x0 if the following limit exists:

f ′(x0) = limx→x0

f(x)− f(x0)x− x0

A function having a derivative at each x ∈ X is differentiable on X.

The derivative f ′(x0) gives the slope of the curve f(x) at x = x0. Thus,the line through the point (x0, f(x0)) with slope f ′(x0) is the tangent tothe curve of f(x) vs. x at x = x0.


Investigate differentiability

Example

f(x) = |x| =

x x > 00 x = 0−x x < 0

f ′(x) is 1 for x > 0 and −1 forx < 0, but does not exist at x = 0.


Outline

1 Introduction



4 Differentiability





Rolle’s theorem

Theorem

Suppose f ∈ C[a, b] and f is differentiable on (a, b). If f(a) = f(b), thena number c in (a, b) exists with f ′(c) = 0.

Example

Rolle’s theorem applies for

f(x) = x(x2 − 1) g(x) =√

1− x2 both for x ∈ [−1, 1]

f is continuous and differentiable everywhere, while g is only real forx ∈ [−1, 1] and is not differentiable at x = ±1.

Example

Rolle’s theorem can’t be used for f(x) = |x| if (a, b) includes 0. (Why?)


Investigate Rolle’s theorem using xmaxima

plot2d(x*(x^2-1),[x,-1,1]);

f ′(x) = 3x2 − 1 = 0

at x = ± 1√3

plot2d(sqrt(1-x^2),[x,-1,1]);

g′(x) = − x√1− x2

= 0

at x = 0


Mean value theorem

Theorem

If f ∈ C[a, b] and f is differentiable on (a, b), then a number c in (a, b)exists with

f ′(c) =f(b)− f(a)

b− a

Example

f(x) =√

1− x2 − x

plot2d([sqrt(1-x^2)-x,1-x,-x],[x,-1,1]);


Extreme value theorem

Theorem

If f ∈ C[a, b] then c1, c2 ∈ [a, b] exist with f(c1) ≤ f(x) ≤ f(c2) for allx ∈ [a, b].

If f is differentiable on (a, b), then the numbers c1 and c2 occur either atthe endpoints of [a, b] or where f ′ is zero.

Example

f(x) =√

1− x2 − x

f ′(x) = − x√1− x2

−1

f ′(x) = 0 ⇒ x = − 1√2

plot2d([sqrt(1-x^2)-x],[x,-1,1]);


Generalized Rolle’s theorem

Theorem

Suppose f ∈ C[a, b] is n times differentiable on (a, b). If f(x) = 0 at then+ 1 distinct numbers a ≤ x0 < x1 < . . . < xn ≤ b, then a number c in(x0, xn), and hence in (a, b), exists with f (n)(c) = 0.

This theorem is seldom used excedpt in proofs. I will skip it.


Intermediate value theorem

Theorem

If f ∈ C[a, b] and K is any number between f(a) and f(b), then thereexists a number in (a, b) for which f(c) = K.

Example

f(x) = x3 + 12x

2 − x− 12 x ∈ [−1.5, 1.5]

f(−1.5) = −1.25 f(1.5) = 2.5

For −1.25 ≤ K ≤ 2.5, there are1, 2 or 3 values c.


Outline

1 Introduction



4 Differentiability





Sets and sequences

Definition

A set is a collection of elements.

A sequence is an ordered set.

The order of the elements in a sequence can be described using an index.Thus, a sequence can also be defined as a function whose domain is thepositive integers 1, 2, . . . , n, . . . , N , where N can be infinite.

Example

A sequence of real numbers {xn}∞n=1 where xn is

12n

xn

n!an cos(nx) + bn sin(nx)


Limit of a sequence

Definition

The infinite sequence of real numbers {xn}∞x=0 has the limit x (convergesto x) if for any ε > 0 there exists a positive integer N(x) such that

|xn − x| < ε whenever n > N(x)

Two common notations:

limn→∞xn = x or xn → x as n→∞

Example

limn→∞

n

n+ 1= 1

(1 +

2n

)n→ ? as n→∞


Investigate a sequence using Matlab (octave)

Example (1 +

2n

)n→ ? as n→∞

Visualization:

This time we want to plot individual points, not a continuous curve.Here’s the octave code and resulting plot:

t = 1:1:25;plot((1.+2./t).^t,’x’);

There is a (finite) limit.

e2 ' 7.3891Phyllis R. Nelson (Cal Poly Pomona) Review of Basic Concepts and Skills EGR 511 - Winter 2013 29 / 52

Series

Definition

Given a sequence of real or complex numbers {xn}, generate the sequence{sn} of the partial sums of {xn}

sn =n∑k=1

xn

The sequence {sn} is a series and its limit as n→∞ is called the sum.

Example

{xn} =1

2n−1{sn} =

n∑k=1

12k−1


Investigate a series

{xn} =1

2n−1

In octave

n = 1:1:10;x = 1./2.^(n.-1);plot(n,x,’x’);

The value of xn approaches 0 as n increases, so it is possible that theseries of partial sums converges.


Series example - (2)

{sn} =n∑k=1

12k−1

s = zeros(1,10);s(1) = x(1);for i=2:length(n)s(i) = s(i-1) + x(i);

endfor;plot(n,s,’x’);

The sum of the series appears to be 2 (s(10) = 1.9980). How could weconfirm that?

What happens if the 2 in the sequence definition is replaced by some othernumber? Clearly that number must be > 1 if the series is to converge.(Why?)



Write a function series test in a file series test.m.

function sels = series_test(q,m)# series_test(q,m) Computes the first m partial# sums of the sequence# 1/q^{k-1}

n = 1:1:m;x = 1./q.^(n.-1);s = zeros(1,m);s(1) = x(1);for i=2:ms(i) = s(i-1) + x(i);

endfor;sels = s;

end



Execute this function with q = 2 and m = 10 to test it.

octave:12> ss = series_test(2,10);octave:13> ss(10)ans = 1.9980

Now try other values.

q m s(m)2 15 1.99993 10 1.50004 10 1.33335 10 1.2500

10 10 1.1111



Is there a pattern to these sums?q 2 3 4 5 10

s(∞) 2 32

43

54

109

Perhaps sn → qq−1 as n→∞.

sn =∞∑k=1

1qk−1

= 1 +1q

+1q2

+ . . .

1qs =

∞∑k=2

1qk−1

=1q

+1q2

+ . . . = s− 1

s =1

1− 1q

=q

q − 1


Outline

1 Introduction



4 Differentiability





Integration as the sum of a series

Definition

The Riemann integral of the function f on the interval [a, b] is thefollowing limit, provided it exists:∫ b

af(x) dx = lim

max ∆xi→0

∞∑i=1

f(zi) ∆xi

where the xi satisfy a = x0 ≤ x1 ≤ . . . ≤ xn = b, ∆xi = xi − xi−1, and ziis arbitrarily chosen in the interval [xi−1, xi].

The Riemann sum is a common method of computing the value of anintegral numerically. The most basic approach is to take equally-spacedpoints xi and to choose the zi to be at either the start or the end of eachinterval.


Investigate a Riemann integral with gnuplot

This gnuplot script uses parametric mode and plotting with boxes toshow the rectangles in the Riemann sum.

# riemann.gp - gnuplot script for plotting Riemann sumsset key offset xrange [0:2]set xlabel ’x’set yrange [0:1.5]f(x) = x*(x-1)*(x-2)+1set parametricset trange [0:100]set samples 101plot t/50., f(t/50.), \t/10.+0.05, f(t/10.) with boxes#t/10.-0.05, f(t/10.) with boxes


Riemann integral example - (2)

Run the script with

load ’riemann.gp’

Using the “+” shift gives

while the “-” shift gives


Taylor’s theorem

Theorem

Suppose f ∈ Cn[a, b], that fn+1 exists, and x0 ∈ [a, b]. For everyx ∈ [a, b] there exists a number ξ(x) between x0 and x with

f(x) = Pn(x) +Rn(x)

where

Pn(x) =n∑k=0

f (n)(x0)k!

(x− x0)k Rn(x) =f (n+1)(ξ(x))

(n+ 1)!(x− x0)n+1

Pn(x) is the nth Taylor polynomial for f about x0 and Rn(x) is calledthe remainder term or truncation error. The Taylor series for f aboutx = x0 is limn→∞ Pn(x).

If x0 = 0 the Taylor polynomial is called the Maclaurin polynomial andthe Taylor series, the Maclaurin series.


Approximate a function with a Taylor polynomial

The third Taylor polynomial for f(x) = tanx about x0 = 0 is

P3(x) = f(0) + f ′(0)x+f ′′(0)

2!x2 +

f (3)(0)3!

x3

f(0) = tan(x)|x=0 = 0 f ′(0) = sec2 x∣∣x=0

= 1

f ′′(0) = 2 sec2 x tanx∣∣x=0

= 0

f (3)(0) =[4 sec2 x tanx+ 2 sec4 x

]x=0

= 2

P3(x) = x+x3

3


Taylor polynomial approximation example - (2)

Using maxima, we can see graphically how well P3(x) approximates tan(x).

plot2d([tan(x),x+x^3/3],[x,-1.5,1.5]);



Clearly this series approximation is quite good for “small” x.



The truncation error is

R3(x) =f (4)(x)

4!x4 ξ(x) =

(8 sec2 x tanx+ 16 sec4 x tanx

)ξ(x)

where x0 = 0 ≤ ξ(x) ≤ x.

A plot shows that the maximum possible truncation error (ξ(x) = x)significantly over-estimates the actual truncation error.


Fourier series

There are other sets of basis functions besides the (x− x0)n of theTaylor and Maclaurin series.

The sine and cosine functions are the basis for Fourier series, which canapproximate either a general function on the interval −π < x < π or aperiodic function for all x.

Definition

A function f of real numbers −π < x < π can be represented by theFourier series

f(x) =a0

2+∞∑n=1

[an cosnx+ bn sinnx]

an =1π

∫ π

−πf(x) cosnx dx bn =

1π

∫ π

−πf(x) sinnx dx


Investigate a Fourier series

Example

f(x) = |x| − π < x < π

Since f is an even function, bn = 0 for all n.

a0 = π an>0 =

{− 4πn2 n odd

0 n even

|x| = π

2− 4π

∞∑n=1

cos(2n− 1)x(2n− 1)2


Fourier series example - (2)

Saving the following code in a file FSerDemo.m creates a Matlab oroctave function FSerDemo.

function S = FSerDemo(x)# FSerDemo(x) Calculates the first 5 partial# sums S of the Fourier series# for |x| on [-pi,pi]

n = 5;for k=1:1:nT(:,k) = -4/pi*cos((2*k-1)*x(:))/(2*k-1)^2;

endforS(:,1) = pi/2 + T(:,1);for k=2:1:nS(:,k) = S(:,k-1) + T(:,k);

endfor



fplot(’FSerDemo’,[-pi,pi])



To see the individual elements of the sequence, use the functionFSerTerms.

function T = FSerTerms(x,n)# FSerTerms(x) Calculates the first n terms T# of the Fourier series for# |x| on the interval [-pi,pi]

for k=1:1:nT(:,k) = -4/pi*cos((2*k-1)*x(:))/(2*k-1)^2;

endfor



x = linspace(-pi,pi,151);T = FSerTerms(x,5);plot(x,T)



Finally, compare the fifth partial sum with the actual function.

S = FSerDemo(x);plot(x,S(:,5),’;S(x,5);’,x,abs(x),’;abs(x);’)



To get a better view, look at the difference between the approximationand the function.

plot(x,S(:,5) .- abs(x(:)),’;S(x,5);’,x,abs(x),’;abs(x);’)


511 Basics Handouts

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