# Wright State University Lake Campus/2017-1/MTH2310/log

## Test 1

collapse this with cot and cob

### Monday 9 January 2017

• Assignment:
3. Open your private wiki account wsul### and click edit. You will see instructions enclosed in the "hide" tags: <!-- hidden text-->
2. Fill in PHY1120 as instructed when y
4. Create a word document called Enrollment and put it in your dropbox. For now, it should contain only your wsul### (046-056). Also place your noun### password so you won't forget it.
5. Paste this into your Wikiversity user account

[[Wright State University Lake Campus/2017-1/Phy1120]] - links to roster, with all important links at the top.

### Mon 16 January 2017 (UTC)

Finished problems for Test 1

Start Test 2 on Friday.

• Introduce MATLAB code.
• Sec6.1 p433: Examples 2,3,4...?

#### Trapezoidal and Simpson's rule

${\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a)\left[{\frac {f(a)+f(b)}{2}}\right].}$

For a domain discretized into N equally spaced panels, or N+1 grid points a = x1 < x2 < ... < xN+1 = b, where the grid spacing is

${\displaystyle h={\frac {(b-a)}{N}}\equiv \Delta x}$

the approximation to the integral becomes

${\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum _{k=1}^{N}{\frac {f(x_{k+1})+f(x_{k})}{2}}\cdot \Delta x}$ ${\displaystyle {}=\left[{\frac {1}{2}}f(x_{1})+f(x_{2})+f(x_{3})+f(x_{4})+\dotsb +f(x_{N})+{\frac {1}{2}}f(x_{N+1})\right]\cdot \Delta x}$

Note except for the first and last terms, this is essentially:

${\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum {\bar {f}}\Delta x}$

Trapezoidal:

Wikipedia:Simpson's rule - should I simplify and import here, or not?

### Friday 19:03, 27 January 2017 (UTC)

These will be on Thursday's Test 1:

• Section 5.7 (pp. 389-392): Examples 1, 2, and 4. You need not memorize the half-angle formulas.
• Sec 5.10 (p413): Example 1: p415

### Test 1 is Thur 2 Feb 2017 11:30 am - 12:50 pm (Dwyer 150)

See Calculus_II/Test_1, especially

### STUDY GUIDE for TEST 1 (temporary transclusion to Calculus II/Test 1

{{cot|click to view}} {{#lst:Calculus II/Test 1}} {{cob}}

### 17:37, 31 January 2017 (UTC) Tuesday (last day before test)

1. Long division of polynomials: http://www.purplemath.com/modules/polydiv2.htm
2. The rest looks "cook-booky", meaning that they are tricks with no explanation as to how they were discovered. NOT INTERESTING! NOT ON ZHANG'S TEST!
3. I want to come back to an essential concept: Look at the two equations in two unknowns for A and B. These are linear because A and B appear only to the first power (or zeroth power), and because AB is not present either. Linear equations are easy to solve, the answer is always either 0, 1, or more solutions. Matlab can solve 100 equations in 100 unknowns very quickly using matrices. See ##https://www.khanacademy.org/math/precalculus/precalc-matrices/solving-equations-with-inverse-matrices/e/writing-systems-of-equations-as-matrix-equations

### Feb 6 2017: Recap Test 1

These links are best played at double speed:

So I googled matlab symbolic integration and found this:

syms x
int(-2*x/(1 + x^2)^2)


Changed it to:

clc;clear all; close all; %clears memories.
syms x fun %Symbolic integration nees symbols not variables
fun = sym(exp(x)) %A trick I vaguely remember.  We need everythign to be a symbol.
int(   fun/(fun^2+3)     )


Output:

ans = (3^(1/2)*atan((3^(1/2)*exp(x))/3))/3  )


## Test 2

collapsed with cot and cob

### Chap 6.1 Applications of integration: More about areas

Chapter 6.1 Examples 1, 2, 3 (pp.432-4)

Also look at Examples 4 and 5 (only examples with the pencil icon are likely to appear on the test).

#### Matlab exercise (not on test)

Explain (with sketches) you you find the area enclosed by the following curves:

${\displaystyle y=\ln x^{2},\quad xy=3,\quad x=3,\quad x=4}$

MATLAB code (discussed in class)
clear all; close all; clc;
x = linspace(.5,10)
y = log(x.^2);
plot(x,y,'-');hold on;
y = 3./x;
plot(x,y,'-');
x(1:100)=3
y=linspace(-5,5)
plot(x,y,'-');hold on;
x(1:100)=4
y=linspace(-5,5);hold on;
plot(x,y,'-');


### Chapter 6.2 Applications of integration: Volumes

Chapter 6.2: Examples 1, 2, 3, and 4 (page 438) ---26 January 2017

### Chapter 6.3 Applications of integration: Volumes by cylindrical shells

Chapter 6.2: Examples 1 -- 18:24, 13 February 2017 (UTC) (UTC)

### Chapter 6.4 Applications of integration: Arc length

#### ? Place this on test as extra credit?

Here, Δs2≈Δx2+Δy2 is only an approximation because the segments are not infinitesimally small

Why Wikipedia (though great) is not enough: w:Special:Permalink/744766508#Derivation documents an important derivation of w:Arc length that was added in 2006. First,it is not sufficiently complete for introductory students, and second, it was removed as can be seen in this 2017 version of the article: w:Special:Permalink/754122488

${\displaystyle ds={\sqrt {dx^{2}+dy^{2}}}}$ leads both:
${\displaystyle {\frac {ds}{dt}}={\sqrt {{\frac {dx^{2}}{dt^{2}}}+{\frac {dy^{2}}{dt^{2}}}}}\Rightarrow {\frac {ds}{dx}}={\sqrt {1+\left(y\,'\right)^{2}}}}$

In the last step we replaced t by x. Now integrate WRT t or x. to get the desired result---26 January 2017

Infinitesimal is an interesting article, but not useful for this course.

### floating illusion (not on test)

click to expand (not on test)
fullFileName = 'C:\Users\Zach\Pictures\GIFs\AnimatedGifs\electricity.gif';
[gifImage cmap] = imread(fullFileName, 'Frames', 'all');
size(gifImage); implay(gifImage);


### Chap 6.5 Applications of integration: Average value of a function

Examples 1 and 3 (For possible extra credit study the proof above p461) 10,13 February 2017 (UTC)

### Problems CERTAIN to be on Test II

On Monday and Wednesday, I will will spend some time on questions that will certainly be on the test--Guy vandegrift (discusscontribs) Monday, 6 March 2017 (UTC)

Area between curves EXAMPLE 2 Section 6.1 p.433. I like this question because it emphasizes that an integral is a Reimann Sum, here of rectangles.

Using the washer method EXAMPLE 4 Section 6.2 p.442. Here, the integral is a Riemann sum of "little volumes", not "little rectangles".

NOT IN BOOK BUT ON TEST: Use the known circumference of the unit circle to generate an expression for a definite integral from x=0 to x=1. Do not solve the integral, but someone needs to verify that it is correct:

${\displaystyle \int _{0}^{1}{\sqrt {(1+(f')^{2}}}dx}$ where ${\displaystyle y=f(x)}$ and ${\displaystyle x^{2}+y^{2}=1\Rightarrow \int _{0}^{1}{\frac {dx}{\sqrt {1+x^{2}}}}={\frac {\pi }{2}}}$
(extra credit: explain why this is the formula for the arclength ${\displaystyle \left(\int d\ell =\int {\sqrt {dx^{2}+dy^{2}}}\right)}$).

## Test 3

### 8.1 Infinite sequences and series: Sequences

Read Examples 1, 2, 3 page 554 for understanding.

*  Examples 4, 5 pp557-8 15 February 2017 (UTC)

### 8.2 Infinite sequences and series: Partial sums

*  Examples 1, 3, 5 p565-568

*  A derivation of SN = ΣN−1j=0  xj   = (1-xN)/(1-x) will certainly be on the test (note error in previous version). If you don't do well on the rest of the test, I will grade it carefully, so don't make any mistakes? The step at the bottom of page 566 of your textbook is breathtakingly beautiful. Also, when can you get a finite result in the limit as N goes to infinity? You will need to know that series and the fact that it is called a w:power series.

We will carefully read Examples 4 and 7, but I don't see a good exam questions for them. Let us replace example 4 by a simpler one. See Talk:Sequences_and_series

### Skim three sections

#### Chapters 8.3 Infinite sequences and series: Integral and comparison tests

Riemann Sum
• Example 1 of 8.3 (p. 577) is instructive, though the integral is too tricky for an exam 23 February 2017

*  Know how to do the integral test on page 577 (see examples 1, 2, 3). I will give you a different integral, and there will be plenty of partial credit for setting up the integral without solving the integral. This is an essential skill because of the link to the Riemann Sum: ${\displaystyle \sum f_{n}\approx \int f(x)dx}$ if the function is smooth enough that Δx=1 is a reasonable approximation (i.e., the rectangles in the figure can have unit thickness). (See review of ${\displaystyle \int _{1}^{\infty }{\frac {dx}{x^{p}}}}$ below)

#### Chapters 8.4 Infinite sequences and series: Other convergence tests

• The alternating series test is intuitively simple: If the points hop back and forth by smaller and smaller amounts, the it converges. p598
• Therem 1 about absolute convergence on page 588 is important, but is not used much by engineers.
• The Ratio test of 589 is useful for knowing when a Power series converges absolutely.

#### Chapters 8.5 Infinite sequences and series: Power series

!!!!! Examples 4 and 5 page 596' 18:20, 20 March 2017 (UTC)

• You need to know what a power series is, see see page 592

### 8.6 Infinite sequences and series: Representations of functions as power series

*  Examples 1,5 pp.601.        20-22 February 2017

• Example 6 was interesting, but I messed up the graph ln(1+x) on the board and misued the log function on Excel. Will not be on test
• We will not do example 7 because it seems so hard to understand: (play double time). Also search for "worry" halfway down this page. The best way to remember this is to use these triangles..

### 8.7 Infinite sequences and series:Taylor and Maclaurin Series

Womething like this will be on testCarefully study the Taylor-Maclurin Series at p604. Sample problem: Find the 538-th derivative of 13x2122

You will certainly be asked to carefully derive the Taylor series about x=0 for either sine, cosine, or exp (ex), or perhaps ${\displaystyle {\sqrt {1+x}}}$. We will attempt to do (1+x)k, as this was first done by Newton. This is done on Example 8, page 611).

steps to derive Taylor Series for sine

First we do it for a = 0, and show why this makes sense:If the series expansion is true then the following is true

${\displaystyle c_{n}={\frac {f^{(n)}(0)}{n!}}}$

First applicatin is to the Taylor expansion for sin(x):

${\displaystyle \sin(0)=0}$
${\displaystyle {\frac {d}{dx}}\sin(0)=\cos(0)=1}$
${\displaystyle {\frac {d^{2}}{dx^{2}}}\sin(0)=-\sin(0)=0}$
${\displaystyle {\frac {d^{3}}{dx^{3}}}\sin(0)=-\cos(0)=-1}$
${\displaystyle {\frac {d^{4}}{dx^{4}}}\sin(0)=\sin(0)=0}$

Note that this pattern repeats itself because for any n,

${\displaystyle {\frac {d^{n+4}}{dx^{n+4}}}\sin(\theta )={\frac {d^{n}}{dx^{n}}}\sin(\theta )}$

From Wikipedia's Binomial theorem
${\displaystyle {r \choose k}={\frac {r\,(r-1)\cdots (r-k+1)}{k!}}={\frac {(r)_{k}}{k!}},}$

According to a hidden comment in w:Special:Permalink/766761934#Newton.27s_generalized_binomial_theorem, we cannot write this as ${\displaystyle {\frac {r!}{k!(r-k)!}}}$ because the definition of ! does not hold for negative numbers. See also w:Gamma function

Time permitting: Section 8.7 examples 6,7,8, 12 pages 610-615. (Example 8 was done previously, and example 12 is extremely useful)

• Before Test 3 I want to carefully review ${\displaystyle \int _{1}^{\infty }{\frac {dx}{x^{p}}}}$ for ${\displaystyle p\neq 1}$. See this 6-min Youtube and/or this excerpt from Wikibooks:
Show ${\displaystyle \int \limits _{1}^{\infty }{\frac {dx}{x^{p}}}={\begin{cases}{\frac {1}{p-1}},&{\text{if }}p>1\\{\text{diverges}},&{\text{if }}p\leq 1\end{cases}}}$
If ${\displaystyle p\neq 1}$ then
${\displaystyle \int \limits _{1}^{\infty }{\frac {dx}{x^{p}}}}$     ${\displaystyle =\lim _{b\to \infty }\int \limits _{1}^{b}x^{-p}dx}$
${\displaystyle =\lim _{b\to \infty }\left({\frac {x^{-p+1}}{-p+1}}\right){\Bigg |}_{1}^{b}=-{\frac {1}{1-p}}\lim _{b\to \infty }\left(b^{-p+1}-1\right)}$
${\displaystyle ={\begin{cases}{\frac {1}{p-1}},&{\text{if }}p>1\\{\text{diverges}},&{\text{if }}p<1\end{cases}}}$
Notice that we had to assume that ${\displaystyle p\neq 1}$ to avoid dividing by 0 (which leads to the natural logarithm).

## Test 4

If we follow the previous course, Test 4 might include material on the power series.

• All the examples in Chapter 9.1 might be on the test (pp634-38)
• wikipedia:Dot product wikipedia:Cross product
• Vectors let's look at rotations.--Guy vandegrift (discusscontribs) 16:56, 5 April 2017 (UTC)
• These examples from Chapter 9 look like good candidates for Test 4:
• Example 1 p664: Find the equation of a line (in x,y,z coordinates) in a given direction through a given point.
• Example 2 p665: Find the equation of a line (in x,y,z coordinates) through two points.
• Example 4 p667: Find the equation of a plane perpendicular to a given direction, passing through a given point.
• Polar coordinates Appendix H page A55
• Examples 2-5 are easy and at least one of these will be on the test.
• Example 6 is moderately difficult and will be on the test in one form or another.
• The whole class stopped at A58 and nothing on or after Example 7 will be covered.

### Example 8 Challenge question "extra credit' and not a lot of it.

Here the dotted line is the plane, and the normal is B, and r1r0 is A. The point r1 is tail vector A.

See p669 of textbook: Find the distance from a point ${\displaystyle {\vec {r}}_{1}=\langle x_{1},y_{1},z_{1}\rangle }$ to given a plane, if the plane is defined as follows:

${\displaystyle {\vec {r}}_{0}=\langle x_{0},y_{0},z_{0}\rangle }$ is some point on the plane (i.e., three given numbers).
${\displaystyle {\vec {n}}=\langle a,b,c\rangle }$ is normal to the plane (again, three given numbers).

Now draw the point and the plane in from a specific angle in which the given point and the normal lie in the plane of the paper (board), and use facts about dot-product. (I never bothered to define comp and proj as described on 652; I was familiar with comp but never heard of proj, but it is an easy concept to grasp if you understand this problem).

Rule: If, ${\displaystyle {\hat {n}}={\vec {n}}/n}$ is a unit normal to the plane, then the magnitude of ${\displaystyle {\vec {R}}\cdot {\hat {n}}}$ is the distance from the point ${\displaystyle {\vec {r}}=\langle x_{1},y_{1},z_{1}\rangle }$ to the plane. In the figure:
1. ${\displaystyle {\mathcal {A}}={\vec {R}}={\vec {r}}_{1}-{\vec {r}}_{0}}$ is any vector connecting the point to somewhere on the plane (the calculated distance should not depend on ${\displaystyle {\vec {r}}_{0}}$ ).
2. The unit normal is given by: ${\displaystyle {\hat {n}}={\frac {\langle a,b,c\rangle }{\sqrt {a^{2}+b^{2}+c^{2}}}}}$
3. ${\displaystyle {\vec {R}}\cdot {\hat {n}}=\langle x_{1}-x_{0},y_{1}-y_{0},z_{1}-x_{0}\rangle \cdot {\frac {\langle a,b,c\rangle }{\sqrt {a^{2}+b^{2}+c^{2}}}}}$${\displaystyle {\frac {a(x_{1}-x_{0})+b(y_{1}-y_{0})+c(z_{1}-x_{0})}{\sqrt {a^{2}+b^{2}+c^{2}}}}}$

${\displaystyle {\frac {ax_{1}+by_{1}+cz_{1}-\{ax_{0}+by_{0}+cz_{0}\}}{\sqrt {a^{2}+b^{2}+c^{2}}}}}$

If we view the expression in curly brackets ${\displaystyle \{ax_{0}+by_{0}+cz_{0}\}=-d}$ as some constant, and drop the subscript "1", then we have the formula for a plane normal to ${\displaystyle {\hat {n}}}$.

${\displaystyle ax+by+cz+d=0}$

Note that the distance, ${\displaystyle |{\vec {R}}\cdot {\hat {n}}|=0}$ if ${\displaystyle ax_{1}+by_{1}+cz_{1}+d=0}$ because the point is already on the plane.

## Not on any Test

Matlab:"regulated" alternating series (not on test)
clear all; close all; clc; %This clears various memories in scripts
N=10%Number of terms to keep (easily switched)
alpha = 5/N %small parameter

%Next we the  built-in function linspace to make a vector of numbers evenly
%spaced from 1 to N (N of them) N numbers ranging from 1 to N
x = linspace(1,N,N) ;

sum = 0 %initialize the sum to be zero
for n=1:N
temp=exp(-alpha*n); %I like to use temporary variables, here called temp
y(n)=temp*(-1)^(n-1); %converts numbers initially close to 1 to +/- 1
end

bar(y)%makes bar graph

Y=y' %Capital Y is the transpose, and easier to print into command window
fprintf('\n The sum is\n%f',sum);


### Taylor expansion: How careful do we have to be in measuring height? (Not very)

• Let's check out my new invention that will give me enough money to retire:
Testing the level-beam ruler
clear all;close all; clc; %empties  various memories
x=[
2
3
4
5]

y= [
4
6
7
8];
p = polyfit(x,y,2)
%Google searche: matlab polynomial curve fitting
plot(x,y,'o',x,polyval(p,x),'blue')
grid on


## Final Exam

not now
${\displaystyle (\pi {\mathcal {R}}^{2})d{\mathfrak {h}}}$ This needs to be rotated 90 degrees and relabeled.
${\displaystyle dV={\mathfrak {h}}(2\pi {\mathcal {R}}\cdot d{\mathcal {R}})}$

Problem 3 from Test 2: Find the area enclosed between the line ${\displaystyle y=4x}$ and the cubic ${\displaystyle y=x^{3}}$ over the interval between ${\displaystyle x=0}$ and ${\displaystyle x=2}$. Do this both ways:

At left: ${\displaystyle dV={\mathcal {A}}d{\mathfrak {h}}=(\pi {\mathcal {R}}^{2})d{\mathfrak {h}}}$

At right: ${\displaystyle dV={\mathfrak {h}}d{\mathcal {A}}={\mathfrak {h}}(2\pi {\mathcal {R}}d{\mathcal {R}})}$

I like this problem because it shows two ways to do this rotation. Both are insightful for understanding 2 and three dimensional integrals (i.e. integrals over volumes and areas).

Also: It is good to use these ideas to show how the volumes for area and circumfrence of a circle are related, as well as for the volume and surface area of a sphere:

${\displaystyle {\frac {dA}{dr}}=C{\text{ where }}A=\pi r^{2}{\text{ and }}C=2\pi r}$

${\displaystyle {\frac {dV}{dr}}=A{\text{ where }}V={\frac {4}{3}}\pi r^{3}{\text{ and }}A=4\pi r^{2}}$ ... 13 March 2017 (UTC)

<div style="text-align: right; direction: ltr; margin-left: 1em;"><small>text</small></div>
<div style="text-align: right; direction: ltr; margin-left: 1em;">text</div>