Wright State University Lake Campus/20171/MTH2310/log
 Calculus_II/Test_1 (Thur 2 Feb) T1=Test 1 (sections): 1.7, 4.5, 5.6, G,, 5.7, 5.10
 Calculus_II/Test_2 (Thur 9 Mar) T2=Test 2 (sections) 6.1, 6.2,6.3, 6.4,6.6
 Calculus_II/Test_3 (Thur 30 Mar) T3 = Test 3 (sections)
Test 1[edit  edit source]
 See also Calculus_II/Test_1
collapse this with cot and cob 

Monday 9 January 2017[edit  edit source]Announcement: Think about a Wikimedia foundation username. Go to w:Wikipedia:Username policy for hints.SS
Mon 16 January 2017 (UTC)[edit  edit source]Finished problems for Test 1 Start Test 2 on Friday.
Trapezoidal and Simpson's rule[edit  edit source]
the approximation to the integral becomes Note except for the first and last terms, this is essentially: Trapezoidal: Wikipedia:Simpson's rule  should I simplify and import here, or not? January 2526 material moved below[edit  edit source]Friday 19:03, 27 January 2017 (UTC)[edit  edit source]These will be on Thursday's Test 1:
Test 1 is Thur 2 Feb 2017 11:30 am  12:50 pm (Dwyer 150)[edit  edit source]See Calculus_II/Test_1, especially
STUDY GUIDE for TEST 1 (temporary transclusion to Calculus II/Test 1[edit  edit source]{{cotclick to view}} {{#lst:Calculus II/Test 1}} {{cob}} 17:37, 31 January 2017 (UTC) Tuesday (last day before test)[edit  edit source]
Feb 6 2017: Recap Test 1[edit  edit source]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)
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[edit  edit source]
 See also Calculus II/Test 2
collapsed with cot and cob  

Chap 6.1 Applications of integration: More about areas[edit  edit source]Chapter 6.1 Examples 1, 2, 3 (pp.4324) 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)[edit  edit source]Explain (with sketches) you you find the area enclosed by the following curves:
Chapter 6.2 Applications of integration: Volumes[edit  edit source]Chapter 6.2: Examples 1, 2, 3, and 4 (page 438) 26 January 2017 Chapter 6.3 Applications of integration: Volumes by cylindrical shells[edit  edit source]Chapter 6.2: Examples 1  18:24, 13 February 2017 (UTC) (UTC) Chapter 6.4 Applications of integration: Arc length[edit  edit source]? Place this on test as extra credit?[edit  edit source]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
In the last step we replaced t by x. Now integrate WRT t or x. to get the desired result26 January 2017 Infinitesimal is an interesting article, but not useful for this course. floating illusion (not on test)[edit  edit source]
Chap 6.5 Applications of integration: Average value of a function[edit  edit source]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[edit  edit source]On Monday and Wednesday, I will will spend some time on questions that will certainly be on the testGuy vandegrift (discuss • contribs) 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:

Test 3[edit  edit source]
 See Calculus_II/Test_3 for only the Test 3 study guide.
8.1 Infinite sequences and series: Sequences[edit  edit source]
Read Examples 1, 2, 3 page 554 for understanding.
* Examples 4, 5 pp5578 15 February 2017 (UTC)
8.2 Infinite sequences and series: Partial sums[edit  edit source]
* A derivation of S_{N} = Σ^{N−1}_{j=0 }x^{j} = (1x^{N})/(1x) 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[edit  edit source]
Chapters 8.3 Infinite sequences and series: Integral and comparison tests[edit  edit source]
 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: 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 below)
Chapters 8.4 Infinite sequences and series: Other convergence tests[edit  edit source]
 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[edit  edit source]
!!!!! 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[edit  edit source]
* Examples 1,5 pp.601. 2022 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[edit  edit source]
* Womething like this will be on testCarefully study the TaylorMaclurin Series at p604. Sample problem: Find the 538th derivative of 13x^{2122}
* You will certainly be asked to carefully derive the Taylor series about x=0 for either sine, cosine, or exp (e^{x}), or perhaps . We will attempt to do (1+x)^{k}, as this was first done by Newton. This is done on Example 8, page 611).
Two comments helpful for the test are collapsed[edit  edit source]
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} First applicatin is to the Taylor expansion for sin(x): Note that this pattern repeats itself because for any n, 
From Wikipedia's Binomial theorem 

From w:Binomial theorem According to a hidden comment in w:Special:Permalink/766761934#Newton.27s_generalized_binomial_theorem, we cannot write this as 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 610615. (Example 8 was done previously, and example 12 is extremely useful)
 Before Test 3 I want to carefully review for . See this 6min Youtube and/or this excerpt from Wikibooks:
 Show
 If then

 Notice that we had to assume that to avoid dividing by 0 (which leads to the natural logarithm).
Test 4[edit  edit source]
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 (pp63438)
 wikipedia:Dot product wikipedia:Cross product
 Vectors let's look at rotations.Guy vandegrift (discuss • contribs) 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 25 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.[edit  edit source]
See p669 of textbook: Find the distance from a point to given a plane, if the plane is defined as follows:
 is some point on the plane (i.e., three given numbers).
 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 dotproduct. (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, is a unit normal to the plane, then the magnitude of is the distance from the point to the plane. In the figure:
 is any vector connecting the point to somewhere on the plane (the calculated distance should not depend on ).
 The unit normal is given by:
 =
=
If we view the expression in curly brackets as some constant, and drop the subscript "1", then we have the formula for a plane normal to .
Note that the distance, if because the point is already on the plane.
Not on any Test[edit  edit source]
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 builtin 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)^(n1); %converts numbers initially close to 1 to +/ 1
sum=sum+y(n); %adds them up
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)[edit  edit source]
 Let's check out my new invention that will give me enough money to retire:
Testing the levelbeam 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[edit  edit source]
not now 

Problem 3 from Test 2: Find the area enclosed between the line and the cubic over the interval between and . Do this both ways: At left: At right: 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:
... 13 March 2017 (UTC)
