Hey All,
I posted a final practice exam. I created problems which seemed to match the description of the exam. I'm missing out on the end of the exam at the moment and wouldn't know how closely what I wrote up resembles the real exam. At any given time, you should make sure you have the most up-to-date version number.
Note that the copies in the Dropbox will get updated dynamically. While the copies in my Course Files folder will get updated periodically.
Current Version Numbers and Links to Files:
Final Practice Exam [v1.2.4.2]
Final Practice Exam Solutions [v1.2.4.2]
You can ask me questions via e-mail or posting a comment here. Cheers.
Saturday, December 7, 2013
Friday, December 6, 2013
Pictures
These pictures give a rough idea of how well (or not well) the normal distribution approximates the binomial distribution for increasing values of $n$. Specifically, we have $n$ going from 1, 2, 3, 4, 5, 10, 20, 30, 40, 50, 100, 200, 300, 400, 500. Note that the $y$-axis does not necessarily intersect the origin and that the range of values of both axes are changing as well.
 |
| $n=1$ |
Wednesday, December 4, 2013
Using the central limit theorem
We apply the central limit theorem to integer-valued random variables. In that situation, we assign to every integer $k$ it's own little interval $\left(k-\frac{1}{2},k+\frac{1}{2}\right)$. Thus instead of $P(X=5)$ we'd be looking for $P(4.5 \leq X \leq 5.5)$.
Calculus II for Bio Q&A
Finals are around the corner. Ask your questions! (either for credit or just because)
Here's a list of things to make your comments look cool! I suggest you create your comment in a text file and copy and paste over to the comment field here, or just send me an email.
 |
| The ever useful overlapping circles... |
Here's a list of things to make your comments look cool! I suggest you create your comment in a text file and copy and paste over to the comment field here, or just send me an email.
Wednesday, October 30, 2013
Answering Student Questions
Images in this post were generated using Mathematica 8.0.0.0
"In a compartment model, what doe a determinant of 0 mean?"
On page 612, the book explains that there are three possibilities when the determinant is 0. Matter gets stuck in compartment 1, matter gets stuck in compartment 2, or the matter in the system is a constant.
How do you compute the Hessian matrix of a function $f$?
The formula for the Hessian is on page 551 and the components of the matrix are second derivatives of $f$. If we write $\mathrm{Hess}f(x,y)=\begin{bmatrix} a & b \\ c & d \end{bmatrix} $, then compute $a$ by taking two derivatives in $x$, compute $d$ by taking two derivatives in $y$, and compute $b$ and $c$ by taking one derivative in $x$ and then taking a second derivative in $y$.
"One of the exercises generated a value of 2 for every boundary. Does that mean it is a max and min, or there is none?"
You have to compare the value 2 to the value obtained at the critical point on the interior. If the value of the critical point is greater than 2, then the minimum is attained all along the boundary and the maximum was attained at the critical point. If the value of the critical point is less than 2, then the opposite. In the situation that there are multiple critical points, then you must compare all the values attained.
"In a compartment model, what doe a determinant of 0 mean?"
On page 612, the book explains that there are three possibilities when the determinant is 0. Matter gets stuck in compartment 1, matter gets stuck in compartment 2, or the matter in the system is a constant.
How do you compute the Hessian matrix of a function $f$?
The formula for the Hessian is on page 551 and the components of the matrix are second derivatives of $f$. If we write $\mathrm{Hess}f(x,y)=\begin{bmatrix} a & b \\ c & d \end{bmatrix} $, then compute $a$ by taking two derivatives in $x$, compute $d$ by taking two derivatives in $y$, and compute $b$ and $c$ by taking one derivative in $x$ and then taking a second derivative in $y$.
"One of the exercises generated a value of 2 for every boundary. Does that mean it is a max and min, or there is none?"
You have to compare the value 2 to the value obtained at the critical point on the interior. If the value of the critical point is greater than 2, then the minimum is attained all along the boundary and the maximum was attained at the critical point. If the value of the critical point is less than 2, then the opposite. In the situation that there are multiple critical points, then you must compare all the values attained.
Sunday, September 22, 2013
Calculus II for Bio
Ask a math question of medium difficulty. Try looking through some and see if you can work out the answers. You'll need a writing utensil and some paper. Don't be lazy.
Here are two problems to get things started:
(Without looking in your book) Give an example of two $2\times 2$ matrices which do not commute.
If there was a $2\times 2$ matrix $A$ such that $A \cdot B = B \cdot A $ for all $2\times 2$ matrices $B$, what does $A$ look like?
Here are two problems to get things started:
(Without looking in your book) Give an example of two $2\times 2$ matrices which do not commute.
If there was a $2\times 2$ matrix $A$ such that $A \cdot B = B \cdot A $ for all $2\times 2$ matrices $B$, what does $A$ look like?
Thursday, September 19, 2013
Section 3 Comments 20130919
This section has a relaxed atmosphere which is conducive to learning.
This section is cool.
: Great! I definitely want students to come to a conducive-learning environment.
I don't like the quizzes. The homework is going to give me gray hairs at age 18. Tim/John*, you're a cool TA though.
*Inside joke
The section is helpful, but I don't like the quizzes. We should do more exercises.
Do the quizzes count as a grade?
: I don't like the quizzes either, but you probably need them.
: Yes, the quizzes are factored into your grade.
The differentials involving population are quite difficult.
: We'll have to talk about that some time.
I hate trigonometry.
: There's a saying: "Keep your friends close, and your enemies closer." You'll need trigonometry to answer calculus questions. So while you don't have to like it, you have to use it.
This section is cool.
: Great! I definitely want students to come to a conducive-learning environment.
I don't like the quizzes. The homework is going to give me gray hairs at age 18. Tim/John*, you're a cool TA though.
*Inside joke
The section is helpful, but I don't like the quizzes. We should do more exercises.
Do the quizzes count as a grade?
: I don't like the quizzes either, but you probably need them.
: Yes, the quizzes are factored into your grade.
The differentials involving population are quite difficult.
: We'll have to talk about that some time.
I hate trigonometry.
: There's a saying: "Keep your friends close, and your enemies closer." You'll need trigonometry to answer calculus questions. So while you don't have to like it, you have to use it.
Wednesday, May 16, 2012
Note to Self
My students probably struggled with the last few topics covered in the semester as my coverage on those topics was not as thorough as my coverage on the rest of the topics covered in the semester.
On a related note, I occasionally ponder the possibility of covering less, but in greater detail.
On a different note, I think it's strange how many students ask for review sessions, but when I bring up the idea of extending class time or a separate hour (temporarily ignoring the issue of increasing credit) to my peers, there is the idea that students wouldn't agree to it. The problem with office hours is that students either have other classes schedule during that time or are afraid to come. Office hours by appointment is fine, as long as it isn't abused.
Maybe there should be a game. Played by a professor. Every time the professor says something that might be on the exam, students should say some key phrase like "That's magic!" Perhaps every once in a while, and especially the first week, this should be aided with a pause or a hand signal or a sign (much like an "Applause" sign).
On a related note, I occasionally ponder the possibility of covering less, but in greater detail.
On a different note, I think it's strange how many students ask for review sessions, but when I bring up the idea of extending class time or a separate hour (temporarily ignoring the issue of increasing credit) to my peers, there is the idea that students wouldn't agree to it. The problem with office hours is that students either have other classes schedule during that time or are afraid to come. Office hours by appointment is fine, as long as it isn't abused.
Maybe there should be a game. Played by a professor. Every time the professor says something that might be on the exam, students should say some key phrase like "That's magic!" Perhaps every once in a while, and especially the first week, this should be aided with a pause or a hand signal or a sign (much like an "Applause" sign).
Next TA Meeting
This general idea of removing bias crossed my mind. How should we, if at all, deal with section to section discrepancies? I like the idea of mixing up the exams and having the person's name on the very back page of the last problem. That way, you can keep the scoring page on the front, and when you hand back the exams, you just turn the pile over and they can look for their names without seeing each other's scores. It also reduces section bias. But undoubtedly there is differences for the homework portion and emphasis of material from section to section. Maybe one of the TA emphasizes a certain type of problem and grades with respect to that emphasis. So even unbiased grading on midterm and final exams will be lopsided. I definitely would either scale each section so that all sections are comparable, or assign grades based on individual sections. TAs might expect a certain student to do well or do poorly.
See also next post.
See also next post.
Other Ideas to Focus On
$\lim a_n=0$ does not imply convergence.
Students use the ratio/root test on the endpoints of a power series. It's hard for them to grasp that those are exactly the points where the ratio/root test gives 1, and hence inconclusive. Apply the ratio/root test is a waste of time. Just as bad is when they apply the ratio/root test to the endpoints and get a value other than 1.
$\lim\left(1+\frac{x}{n}\right)^n=e^x$
Students use the ratio/root test on the endpoints of a power series. It's hard for them to grasp that those are exactly the points where the ratio/root test gives 1, and hence inconclusive. Apply the ratio/root test is a waste of time. Just as bad is when they apply the ratio/root test to the endpoints and get a value other than 1.
$\lim\left(1+\frac{x}{n}\right)^n=e^x$
Saturday, May 12, 2012
Mistakes
THE FOLLOWING IS FALSE:
$$\frac{c}{a+b}=\frac{c}{a}+\frac{c}{b}$$
FAKE EXAMPLE OF THIS ERROR BEING MADE:
Student writes:
$$\frac{x^2}{1+x^2}=\frac{x^2}{1\vphantom{x^2}}+\frac{x^2}{x^2}$$
TRY IT WITH NUMBERS:
$$\frac{2}{1+1}=1$$
$$\frac{2}{1}+\frac{2}{1}=4$$
$$\frac{c}{a+b}=\frac{c}{a}+\frac{c}{b}$$
FAKE EXAMPLE OF THIS ERROR BEING MADE:
Student writes:
$$\frac{x^2}{1+x^2}=\frac{x^2}{1\vphantom{x^2}}+\frac{x^2}{x^2}$$
TRY IT WITH NUMBERS:
$$\frac{2}{1+1}=1$$
$$\frac{2}{1}+\frac{2}{1}=4$$
Tuesday, May 8, 2012
Practice Exam 5
I don't feel it is necessary for me to make another practice test. At this point you should have realized they're just homework problems in disguise. Now it's time for you to make your own practice test. Below is a rough guideline.
Instructions: Do the following. Exceptions to the rule are anything mentioned by Dr. Brown to focus on or not focus on. Please show your work and explain whenever possible. It's better to say what's on your mind (provided it's correct), because the graders are not mind readers. Please simplify answers when reasonable, such as $\sin{2\pi/3}$ or $\log(1)$.
1) Midterm 1
2) Midterm 2
3) One problem similar to every different type of "easy" homework problem. By "easy" homework problem, I mean the less conceptual and abstract ones. Do odd-numbered problem so you can check your solution with the back of the book. Alternate about every twenty minutes between easier and harder sections (different for every student).
4) My four practice tests
5) Learn to find exams in other locations. Skip problems when they don't apply. Let me get you started: http://www.math.jhu.edu/~wsw/F11/.
Bonus (for students looking for the extra challenge): One problem similar to every different type of "hard" homework problem.
Instructions: Do the following. Exceptions to the rule are anything mentioned by Dr. Brown to focus on or not focus on. Please show your work and explain whenever possible. It's better to say what's on your mind (provided it's correct), because the graders are not mind readers. Please simplify answers when reasonable, such as $\sin{2\pi/3}$ or $\log(1)$.
1) Midterm 1
2) Midterm 2
3) One problem similar to every different type of "easy" homework problem. By "easy" homework problem, I mean the less conceptual and abstract ones. Do odd-numbered problem so you can check your solution with the back of the book. Alternate about every twenty minutes between easier and harder sections (different for every student).
4) My four practice tests
5) Learn to find exams in other locations. Skip problems when they don't apply. Let me get you started: http://www.math.jhu.edu/~wsw/F11/.
Bonus (for students looking for the extra challenge): One problem similar to every different type of "hard" homework problem.
Monday, May 7, 2012
Compute the Taylor series for x^4-3x^2+1 around a=1
Compute the Taylor series for $f(x) = x^4-3x^2+1$ around $a=1$.
Method 1 Take derivatives.
Step 1: Determine the series coefficients.
Step 1a: Evaluate the function at 1.
$f(1) = -1$
Step 1b: Take the first derivative and evaluate it at 1.
$f^{\prime}(x) = 4x^3-6x$
$f^{\prime}(1) = -2
Step 1c: Take the second derivative and evaluate it at 1.
$f^{\prime\prime}(x) = 12x^2-6$
$f^{\prime\prime}(1) = 6
Step 1d: And so on...
$f^{\prime\prime\prime}(x) = 24x$
$f^{\prime\prime\prime}(1) = 24$
$f^{\prime\prime\prime\prime}(x) = 24$
$f^{\prime\prime\prime\prime}(1) = 24$
Step 2: Lay out the coefficients in front of the appropriate power of $(x-a)$ and divide by the corresponding factorial.
For example, the coefficient of $(x-1)^2$ is the evaluation of the second derivative (2) divided by the factorial of two.
In tabular form we have the following:
At the end of the day we have $$-1-2(x-1)+3(x-1)^2+4(x-1)^3+(x-1)^4$$
[20120507]
Method 2 Potentially the easiest method provided it's allowed by the instructor.
First we recognize that $f$ can be written as $((x-1)+1)^4-3((x-1)+1)^2+1$.
Then we expand the terms to obtain $((x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$ from the first term, $-3((x-1)^2+2(x-1)+1)$ from the second term, and $1$.
Simplifying we get $(x-4)^4+4(x-1)^3+3(x-1)^2-2(x-1)-1$.
[20191223]
Method 3 Theoretical method; arguably harder.
We know the coefficient of $(x-1)^4$ must be 1, because it is the only term that contributes to the highest degree term of $f$.
We expand $(x-1)^4$ to get x^4-4x^3+6x^2-4x+1 (I computed the expansion with the help of Pascal's triangle).
There are no powers of $x^3$ in $f$ so we need to offset it with $4(x-1)^3$. The expansion here is $4(x^3-3x^2+3x-1)$ or $4x^3-12x^2+12x-4$.
Keeping track we have $-6x^2+8x-3$ to worry about. So we add $3(x-1)^2$. The expansion here is $3(x^2-2x+1)$ or $3x^2-6x+3$.
The remaining amount to worry about is $2x$. So we add $-2(x-1)$ or $-2x+2$.
The remaining about to worry about is $2$. So we add $-1$.
Remark: Throughout this method, I use the term "add." It's useful to think in terms of adding negative "x" instead of subtracting "x." My high school teacher taught my fellow students and me that subtraction (Satan) and division (Devil) are evil and so we should instead "add a negative" and "multiply by the inverse," respectively.
[20191223]
Method 1 Take derivatives.
Step 1: Determine the series coefficients.
Step 1a: Evaluate the function at 1.
$f(1) = -1$
Step 1b: Take the first derivative and evaluate it at 1.
$f^{\prime}(x) = 4x^3-6x$
$f^{\prime}(1) = -2
Step 1c: Take the second derivative and evaluate it at 1.
$f^{\prime\prime}(x) = 12x^2-6$
$f^{\prime\prime}(1) = 6
Step 1d: And so on...
$f^{\prime\prime\prime}(x) = 24x$
$f^{\prime\prime\prime}(1) = 24$
$f^{\prime\prime\prime\prime}(x) = 24$
$f^{\prime\prime\prime\prime}(1) = 24$
Step 2: Lay out the coefficients in front of the appropriate power of $(x-a)$ and divide by the corresponding factorial.
For example, the coefficient of $(x-1)^2$ is the evaluation of the second derivative (2) divided by the factorial of two.
In tabular form we have the following:
| Coefficient | Factorial | x term | All Together |
|---|---|---|---|
| $f^{(0)}(1)=f(1)$ | 0! | $(x-1)^0$ | $\frac{-1}{0!}$ |
| $f^{(1)}(1)=f^{\prime}(1)$ | 1! | $(x-1)^1$ | $\frac{-2}{1!}(x-1)$ |
| $f^{(2)}(1)=f^{\prime\prime}(1)$ | 2! | $(x-1)^2$ | $\frac{6}{2!}(x-1)^2$ |
| $f^{(3)}(1)=f^{\prime\prime\prime}(1)$ | 3! | $(x-1)^3$ | $\frac{24}{3!}(x-1)^3$ |
| $f^{(4)}(1)=f^{\prime\prime\prime\prime}(1)$ | 4! | $(x-1)^4$ | $\frac{24}{4!}(x-1)^4$ |
At the end of the day we have $$-1-2(x-1)+3(x-1)^2+4(x-1)^3+(x-1)^4$$
[20120507]
Method 2 Potentially the easiest method provided it's allowed by the instructor.
First we recognize that $f$ can be written as $((x-1)+1)^4-3((x-1)+1)^2+1$.
Then we expand the terms to obtain $((x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$ from the first term, $-3((x-1)^2+2(x-1)+1)$ from the second term, and $1$.
Simplifying we get $(x-4)^4+4(x-1)^3+3(x-1)^2-2(x-1)-1$.
[20191223]
Method 3 Theoretical method; arguably harder.
We know the coefficient of $(x-1)^4$ must be 1, because it is the only term that contributes to the highest degree term of $f$.
We expand $(x-1)^4$ to get x^4-4x^3+6x^2-4x+1 (I computed the expansion with the help of Pascal's triangle).
There are no powers of $x^3$ in $f$ so we need to offset it with $4(x-1)^3$. The expansion here is $4(x^3-3x^2+3x-1)$ or $4x^3-12x^2+12x-4$.
Keeping track we have $-6x^2+8x-3$ to worry about. So we add $3(x-1)^2$. The expansion here is $3(x^2-2x+1)$ or $3x^2-6x+3$.
The remaining amount to worry about is $2x$. So we add $-2(x-1)$ or $-2x+2$.
The remaining about to worry about is $2$. So we add $-1$.
Remark: Throughout this method, I use the term "add." It's useful to think in terms of adding negative "x" instead of subtracting "x." My high school teacher taught my fellow students and me that subtraction (Satan) and division (Devil) are evil and so we should instead "add a negative" and "multiply by the inverse," respectively.
[20191223]
Saturday, April 28, 2012
Practice Exam 4
Practice Test 4
Made available 20120428 11:59 PM.
1) Determine the tangent line of the polar function $r=6\cos\theta$
at the point $(x,y)=(3,3)\in\mathbb{R}^{2}$.
Made available 20120428 11:59 PM.
1) Determine the tangent line of the polar function $r=6\cos\theta$
at the point $(x,y)=(3,3)\in\mathbb{R}^{2}$.
Friday, April 27, 2012
Practice Exam 3
Practice Test 3
Administered on 20120427. Updated and made available online 20120430 6:15 PM.
1) Solve the initial value problem
\[
y^{\prime}+\frac{1}{x}y-\sin x=0
\]
\[
y(\pi)=0
\]
Administered on 20120427. Updated and made available online 20120430 6:15 PM.
1) Solve the initial value problem
\[
y^{\prime}+\frac{1}{x}y-\sin x=0
\]
\[
y(\pi)=0
\]
Monday, April 23, 2012
Practice Exam 2
Practice Exam 2.
Made on 20120420. Partially distributed on 20120420. Made available online 20120423.
No calculators!
#1) Evaluate the integral.
\[
\int\frac{s^{3}}{\sqrt{s^{2}+4}}ds
\]
Made on 20120420. Partially distributed on 20120420. Made available online 20120423.
No calculators!
#1) Evaluate the integral.
\[
\int\frac{s^{3}}{\sqrt{s^{2}+4}}ds
\]
Practice Exam 1
Practice Exam 1
Made on 20120417. Distributed in class on 20120417. Made available online on 20120423.
#1) Evaluate the integral.
\[
\int\frac{6x+1}{(x+1)(2x-1)}dx
\]
Made on 20120417. Distributed in class on 20120417. Made available online on 20120423.
#1) Evaluate the integral.
\[
\int\frac{6x+1}{(x+1)(2x-1)}dx
\]
Sunday, April 15, 2012
Class Discussion due 20120422
What!?! Spring Fair is this weekend. As such, I'll consider reserving a room and leaving a pile of tests on a table for those who want to come in and sit to take the test. The goal is to let you guys come in freely on Friday 3-7 PM. (4/20/2012). Though I recommend when you come, you commit to staying for an hour block. Perhaps I will come in and out to check. But I hope everybody enjoys Spring Fair, minus the one hour they come.
Then Friday 3-7 PM, we will do it again (4/27/2012). I will make the exam available online for those who can't make it.
In addition, I will make a test that you can work on before each Friday. That's 4 practice tests!
Hint 1: It will be based on your homework. Why? 1) That's most likely how Dr. Brown will write his test. 2) If you get a problem wrong, or don't know how to do a problem, then you'll know which problems you'll need to do more of. 3) In order to write more, I have to somehow make it easy to write.
Hint 2 (for first two practice tests): There will be 8 problems. One of the problems will try to ask something from the first half of the course (in minor preparation for the final). One of the problems will involve polar coordinates (Section 10.4). One of the problems will be an improper integral. The remaining five problems will come from Section 11.1 to 11.7.
Advice: Do all the problems from Section 11.7.
Then Friday 3-7 PM, we will do it again (4/27/2012). I will make the exam available online for those who can't make it.
In addition, I will make a test that you can work on before each Friday. That's 4 practice tests!
Hint 1: It will be based on your homework. Why? 1) That's most likely how Dr. Brown will write his test. 2) If you get a problem wrong, or don't know how to do a problem, then you'll know which problems you'll need to do more of. 3) In order to write more, I have to somehow make it easy to write.
Hint 2 (for first two practice tests): There will be 8 problems. One of the problems will try to ask something from the first half of the course (in minor preparation for the final). One of the problems will involve polar coordinates (Section 10.4). One of the problems will be an improper integral. The remaining five problems will come from Section 11.1 to 11.7.
Advice: Do all the problems from Section 11.7.
Wednesday, April 4, 2012
Class Discussion due 20120410
Check your email regarding the Practice Test Session.
For the Homework Buddy system, you're suppose to put A2D (agree to disagree) in the situation where you've discussed the discrepancy and somehow feel your answer is right and the other person won't listen to you. Otherwise, you should typically convince the other person why your answer is right and get them to change it. Future checks of the use of A2D will be more strict. And in general, listing another person as your Homework Buddy should mean ALL your answers match up. Perhaps not the work, but at least the answers. For now, I will also reduce the Homework Buddy bonus from 5 to 4. Until better use of the system occurs
For the Homework Buddy system, you're suppose to put A2D (agree to disagree) in the situation where you've discussed the discrepancy and somehow feel your answer is right and the other person won't listen to you. Otherwise, you should typically convince the other person why your answer is right and get them to change it. Future checks of the use of A2D will be more strict. And in general, listing another person as your Homework Buddy should mean ALL your answers match up. Perhaps not the work, but at least the answers. For now, I will also reduce the Homework Buddy bonus from 5 to 4. Until better use of the system occurs
Tuesday, March 27, 2012
Problem 10.4.40
I made a big mistake in my first section and a small mistake in my second section.
We could solve problems like number 40 as follows.
Problem: When do two curves $r_1(\theta)$ and $r_2(\theta)$ intersect?
We could solve problems like number 40 as follows.
Problem: When do two curves $r_1(\theta)$ and $r_2(\theta)$ intersect?
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