Math 2B Review Sheet « greg.knese

December 5, 2007

Math 2B Review Sheet

Filed under: Math2b — gregknese @ 1:53 pm

Here are a bunch of general questions that you should be able to answer:

General Questions

1. What does the definite integral of a function from $a$ to $b$ represent?

2. Integrals are defined as limits of Riemann sums. Where in this course has this idea been useful? Specifically, there are several places where the most natural approximation to an idea is to write down a sum and then once we notice that the sum is the Riemann sum of something, we can define the idea using an integral.

3. What are the two main purposes of the Fundamental Theorem of Calculus?

4. Substitution is what differentiation rule in reverse?

5. Describe in general terms when substitution might be useful in evaluating an integral.

6. What’s the difference between a definite and indefinite integral?

7. Describe the process for finding the area between two curves.

8. How do you find the area of a washer? (a disk with a concentric interior disk cut out)

9. When does a function have an inverse function?

10. What types of functions are proportional to their rate of growth?

11. How do you find the derivative of an inverse function?

12. If sine, cosine, and tangent do not have inverse functions, then what do we mean by $\sin^{-1}$ , $\cos^{-1}$ , and $\tan^{-1}$ ?

12b.) Why do we have to be careful in problems when using the above inverse trig functions?

13. What is l’Hôpital’s rule good for? When do you use it?

14. Describe when it might be useful to use integration by parts to evaluate and integral.

15. Integration by parts is what differentiation rule in reverse?

16. How do we integrate $\cos^m x \sin^n x$ when $m$ or $n$ is odd?

17. How do we integrate $\tan^m x \sec^n x$ when $m$ is odd or $n$ is even?

18. How do we integrate $\cos^2 x$ and $\sin^2 x$ ?

19. When is it good to use a trig substitution and which trig function should you substitute in which circumstances?

20. Describe all of the different ways that you may need to break up a rational function in order to integrate it. i.e. What do you do if your rational function has higher degree in the numerator than in the denominator? What do you do with double roots in the denominator? What do you do with irreducible quadratic factors in the denominator? What do you do with double irreducible quadratic factors in the denominator?

21. When is an integral improper?

22. How do we define improper integrals?

23. When does a parametric curve have horizontal or vertical tangents?

24. Why does a point in the plane have many different polar coordinates?

25. If you know one set of polar coordinates for a point how do you find all of the others?

26. If you know the polar coordinates of a point, how do you find the $(x,y)$ coordinates?

27. When is it useful to use logarithmic differentiation?

Do you know…?

1. …the formula for the work performed on an object under a changing force?

2. …the formula for the average value of a function on an interval?

3. …the derivative of $e^x$ ? What about $e^{f(x)}$ ?

4. …the derivative of $\ln x$ ? What about $\ln (f(x))$ ?

5. …the derivatives of the inverse trig functions: $\sin^{-1}$ , $\cos^{-1}$ , and $\tan^{-1}$ ?

6. …the formula for integration by parts?

7. …the formula for the arc length of the graph of a function?

8. …the formula for the slope of the tangent line of a parametric curve?

9. …the formula for the arc length of a parametric curve?

10. …the formulas for going back and forth between polar coordinates and cartesian coordinates?

Do you know everything?

If you can answer these questions without help, then you most likely know your stuff.

1. The force applied to an object from $x=0$ to $x=1$ meters is given by $f(x) = x^2 \ln (1/x)$ for $x >0$ and $f(0) = 0$ in Newtons. What is the work done on the object?

2. Find the arc length of the curve $y=\ln x$ for $1\leq x \leq \sqrt{3}$ .

3. Find

$\int_{2}^{\infty} \frac{x^2+2x}{(x+1)^2 (x^2+1)} dx$

Do you know most things?

Here are problems that usually focus on a single idea each.

1. Find polar coordinates for the point $(-2, -5\pi/6)$ with both positive and negative $r$ values.

2. Find the slope of the tangent line to the curve $r=\ln\theta$ at $\theta = e$ .

3. Describe the curve given by the parametric equations

$x = 2\sin t$

$y = 3\cos t$

$0 < t < 2\pi$

Find $dy/dx$ and $d^2y/dx^2$ . For which values of $t$ is the curve increasing/decreasing, concave up/down. Does this agree with the geometric picture of the curve?

4. Find the length of the curve

$x = e^t+e^{-t}$ , $y= 5-2t$ , $0\leq t \leq 3$

5. A bacteria culture grows with constant relative growth rate. After 2 hours there are 600 bacteria and after 8 hours the count is 75,000.

(a) Find the initial population

(b) Find an expression for the population after $t$ hours.

(d) Find the rate of growth after 5 hours

(e) When will the population reach 200,000?

6. Find the length of the curve

$y= \frac{x^3}{6} + \frac{1}{2x}$ , $1/2 \leq x \leq 1$

7. Evaluate or show it diverges:

$\int_{0}^{1} \frac{1}{4y-1} dy$

8. Evaluate or show it diverges:

$\int_{0}^{1} \frac{dx}{\sqrt{1-x^2}}$

9. Write out the form of the partial fractions (do not solve for constants)

(a) $\frac{x}{x^2+3x-4}$

(b) $\frac{2x+1}{(x+2)^3(x^2+4)^2}$

10. Evaluate

$\int_{0}^{1} \frac{x-1}{x^2+3x+2} dx$

11. Evaluate

$\int \frac{du}{u\sqrt{5-u^2}}$

12. Evaluate

$\int \sin 3x \sin 6x dx$

13. Evaluate

$\int \sec^2 x \tan x dx$

14. Evaluate

$\int t\sin (2t) dt$

15. Evaluate

$\int_{1}^{4} \sqrt{t} \ln t dt$

16. Compute:

$\lim_{x \to 0} (1-2x)^{1/x}$

17. Differentiate

$f(x) = \ln (x^2 +10)$

18. Use logarithmic differentiation to find the derivative of

$y = (2x+1)^5(x^4-3)^6$

19. Differentiate

$y = e^{(3x^3)}$

20. Evaluate

$\int x e^{(5x^2)} dx$

21. Find a formula for the inverse of the function

$y = \frac{4x-1}{2x+3}$

22. Find $(f^{-1})' (2)$ for

$f(x) = x^5 -x^3+2x$

23. Find the derivative of

$f(x) = \int_{0}^{x} e^{(t^2)} dt$

Comments (60)

60 Comments »

Jeez Prof, is there anything you DIDN’T include?

Comment by panicking student — December 5, 2007 @ 4:39 pm | Reply
I’ll take that as a rhetorical question.

Comment by gregknese — December 5, 2007 @ 5:27 pm | Reply
HAHAHAAHAHAHA SORRY I COULDN’T HELP BUT LAUGH. =]

Comment by Math2B Student — December 5, 2007 @ 5:42 pm | Reply
Did anyone get the answer to webwork problem 7 (the cable problem)

my answer is 915.3103493

my integral is 2u^(3/2)/.6075 a=1 b =7.6825

HELP!

Comment by Student (status:confused) — December 5, 2007 @ 9:00 pm | Reply
and yes i remembered to multiply by 13.7 at the end

Comment by Student (status:confused) — December 5, 2007 @ 9:00 pm | Reply
I didn’t read the problem carefully enough when I did this in class. The formula gives the height as a function of the distance from the point between the two poles. So, I computed the wrong thing in class. I’ll make this problem extra credit, because I don’t want everyone freaking out over this one problem when it is probably better to be studying for the final.

Comment by gregknese — December 5, 2007 @ 9:12 pm | Reply
looks good prof, thanks =)

Comment by AnotherStudent — December 5, 2007 @ 9:30 pm | Reply
Will you be able to give us some practice problems for the Final from the textbook like you did for midterms one and two? That was real helpful, thx.

Comment by UCI Student — December 5, 2007 @ 9:39 pm | Reply
I’ll try and get to that sometime. These problems should be enough to work on in the meantime I hope.

Comment by gregknese — December 5, 2007 @ 9:50 pm | Reply
Ok thanks, cuz I have the Student Solution Manual for the textbook and it helps a lot to see how each problem is solved. Also, whenever you have time, can u list the exact sections that are going to be included on the Final? Thanks a lot.

Comment by UCI Student — December 5, 2007 @ 9:55 pm | Reply
sooo, will we know the format of the test anytime soon..?
i am sorry to burden you with such a trivial question.. AHHH the test is mondaaaaaay!

Comment by stuuuuuudent — December 6, 2007 @ 1:30 am | Reply
I am absolutely baffled why students are so concerned about the format of an exam. In any case, there will be around 12 multiple choice questions and 4 free response questions.

Comment by gregknese — December 6, 2007 @ 6:56 am | Reply
I guess it is because we were so used to studying for the exams…. instead of the materials! Blame College Board! Don’t you only wish students have a passion for math…?

Comment by Curious — December 6, 2007 @ 9:01 pm | Reply
Wait…Everyone doesn’t have a passion for math? My innocence has been lost.

And, yes, I do blame the college board. I should probably not answer this type of question in the future and constantly change the format of exams.

Comment by gregknese — December 6, 2007 @ 9:10 pm | Reply
oh yeah… btw are u gonna post solutions to this, all that is you need to know worksheet?

Comment by Curious — December 6, 2007 @ 9:16 pm | Reply
Unfortunately I would never have enough time. The problems I recommended from the text are mostly odd problems though. You can of course ask questions about any of the problems above.

Comment by gregknese — December 6, 2007 @ 9:21 pm | Reply
What Section was Question #23 under? “Find the derivative of..”

Comment by Student — December 6, 2007 @ 10:08 pm | Reply
The fundamental theorem of calculus section.

Comment by gregknese — December 7, 2007 @ 1:15 am | Reply
Number 11 has two du’s……. is that a typo or do i do du^2?

Comment by Curious — December 7, 2007 @ 12:13 pm | Reply
It’s a typo, but now it’s gone!

Comment by gregknese — December 7, 2007 @ 12:36 pm | Reply
how would we do #4 after we figure out dx/dt and dy/dt? I can’t simplify it!! Thanks !

Comment by student — December 7, 2007 @ 6:45 pm | Reply
Can you simplify the part under the square root? It should be something like $e^{2t} + 2 + e^{-2t}$ . This can be factored as a perfect square.

Comment by gregknese — December 7, 2007 @ 7:15 pm | Reply
oooh yeah, oops… thanks greg!

Comment by student — December 7, 2007 @ 7:42 pm | Reply
professor, i asssume tgat tgise short answer questions would not be on da final riite?

Comment by studebt — December 8, 2007 @ 7:52 pm | Reply
It’s possible I might put one such question on the exam.

Comment by gregknese — December 8, 2007 @ 7:59 pm | Reply
I cannot figure out this problem for the life of me…

Its number 2 under the “Do you know everything?” section:

“2. Find the arc length of the curve for….”

Comment by Math2B Student — December 9, 2007 @ 11:53 am | Reply
If anyone figured it out…. please help me out!

Comment by Math2B Student — December 9, 2007 @ 11:54 am | Reply
Note: this problem is meant to be hard.
You are integrating:
$\sqrt{1+ 1/x^2} = \frac{\sqrt{1+x^2}}{x}$ right?

This suggests a trig substitution to me. Can you get this far?

Comment by gregknese — December 9, 2007 @ 11:57 am | Reply
Um…I don’t understand how you got that thing up there: sqrt(1+x^2)/x=sqrt(1+1/x^2)…

Comment by Pocahontas — December 9, 2007 @ 1:31 pm | Reply
OMGOODNESS… NOW THIS PROBLEM IS SIMPLE… AHH I FEEL LIKE AN IDIOT. AND POCAHONTAS, TO GET WHAT PROF. KNESE GOT, JUST MAKE COMMON DEMONATORS AND YOU SHOULD PROBABLY FIGURE IT OUT FROM THERE.

THANKS A BUNCH PROF!

Comment by Math2B Student — December 9, 2007 @ 2:58 pm | Reply
YES I GOT IT! It gets kind of tricky for those of you trying to figure it out. All I can say is know your trig. definitions and know how to integrate some special trig. functions (either by memorization or by actual derivation). I’m guessing the derivation is probably preferred. haha

Comment by Math2B Student — December 9, 2007 @ 3:07 pm | Reply
Oh..duh. Thanks!

Comment by Pocahontas — December 9, 2007 @ 3:44 pm | Reply
umm….I still haven’t got it….I’m stuck at ln[csc(thea)-cot(thea)] + tan(thea) from 1 to root 3…

Comment by Alex — December 9, 2007 @ 5:16 pm | Reply
Hmmm…I’m not sure how you got that. Did you substitute tangent as I hinted in comment 28 above? Then you should end up with
$\int_{\pi/4}^{\pi/3} \frac{\sec^3 \theta}{\tan \theta} d\theta$
Have you gotten this far? Or, did you do something altogether different?

Comment by gregknese — December 9, 2007 @ 5:29 pm | Reply
Hmm….nope, you wouldn’t happen to have office hours tomorrow would you?

Comment by Alex — December 9, 2007 @ 6:01 pm | Reply
I’ll try to be in my office all day except for when I go to lunch.

Comment by gregknese — December 9, 2007 @ 6:04 pm | Reply
In number 2 under “Do you know everything?”, when you reach the point detailed in comment #28 (sqrt(1 + 1/x^2)), is it possible to set (1/x) = tan(theta) instead of going through and getting to sec(theta)^3 / tan(theta)? I’m actually having trouble integrating that for some reason… any hints?

Comment by Danny — December 9, 2007 @ 8:25 pm | Reply
Actually your way is better: substituting $1/x = \tan \theta$ produces something quite nice.

As for integrating $\sec^3 \theta/ \tan \theta$ , it helps to multiply the top and bottom by $\tan \theta$ and try to substitute $u=\sec \theta$ .

Comment by gregknese — December 9, 2007 @ 8:40 pm | Reply
For your next quarter’s class I recommend that you find time to put up an answer sheet to all these because I don’t find it helpful if i cant check my work.

Comment by student — December 10, 2007 @ 5:15 am | Reply
i know this is off the subject but it bothers me when people leave comments that are not very polite or many cases just rude. He didn’t even need to post a review sheet but he did. There are nicer ways to say things.

Comment by be nicer — December 10, 2007 @ 10:21 am | Reply
I’m completely stumped on #12…is there something obvious I’m overlooking?

Comment by Pocahontas — December 10, 2007 @ 10:45 am | Reply
Maybe one of these formulas will help:
http://gregknese.wordpress.com/2007/11/24/formulas-i-dodont-expect-you-to-memorize/

Comment by gregknese — December 10, 2007 @ 10:46 am | Reply
NEVERMIND! It’s those dumb formulas ^^

Comment by Pocahontas — December 10, 2007 @ 10:46 am | Reply
For do you know everything #1 you just integrate f(x) right? and if so how?

Comment by Jesus. — December 10, 2007 @ 1:53 pm | Reply
Yes, but you have an improper integral: $\ln x$ is not defined at $x=0$ . So, that is one hitch: you have to integrate from t to 1 and then take a limit as t goes to zero from the right. Integrating $x^2 \ln (1/x) = -x^2 \ln x$ can be done using integration by parts.

Comment by gregknese — December 10, 2007 @ 2:01 pm | Reply
14 and 15 are just integration by parts, or is there some cool quick way to find them.

Comment by Jesus. — December 10, 2007 @ 2:02 pm | Reply
I don’t see any cool quick way. Wait…integration by parts IS a cool quick way.

Comment by gregknese — December 10, 2007 @ 2:04 pm | Reply
Hahaha, you’re a funny guy professor Knese. Gave me a good laugh.

Comment by Jesus. — December 10, 2007 @ 2:13 pm | Reply
I was totally worried about the final, but the problems weren’t that difficult. Its just that were time consuming and somewhat tricky at times.

Comment by Anthony — December 11, 2007 @ 1:15 am | Reply
Hi,

I know this is lame question Greg but because your TA is sick, how long do you think it will take for you to grade all our midterms and send out are grades for the class?

Comment by What the deuce? — December 12, 2007 @ 12:23 pm | Reply
He is sick and grading the multiple choice section. I thought I would raise his spirits by giving him some grading to do. He promised to be finished by Friday. However, he also has to grade the last two quizzes. If things go according to plan you should get your grade on Tuesday.

Comment by gregknese — December 12, 2007 @ 2:52 pm | Reply
Hey its Tuesday…do you know when we are realistically going to get our final grades in the class? Thanks….oh and happy holidays!!

Comment by Grandmother Willow — December 18, 2007 @ 11:18 pm | Reply
I submitted them to webgrades last night. I don’t know where you look to see them though. Can you see your grade anywhere? Maybe try going here:
http://webster.reg.uci.edu:8890/studentaccess/grades.jsp

Comment by gregknese — December 18, 2007 @ 11:38 pm | Reply
yea, its not there. The last grade I have from your class is quiz 6. I wonder if its just my account? I’m going to contact the Registrar’s Office…

Comment by Grandmother Willow — December 19, 2007 @ 11:31 am | Reply
Okkk so I just called and they said that the grades won’t be available for viewing online until Friday at 8 p.m.

Comment by Grandmother Willow — December 19, 2007 @ 11:33 am | Reply
Hi, Greg when do we find out how we did on the final?

Comment by Anthony — December 21, 2007 @ 5:09 pm | Reply
I can email out final exam scores tomorrow.

Comment by gregknese — December 21, 2007 @ 11:00 pm | Reply
Hmmmm I actuallly thought i got a 100 percent on that final I guess not. When can we come and collect our tests..cause i still havent gotten both my midterms, which is my fault for being too lazy to go and get them.

Comment by Anthony — December 22, 2007 @ 11:43 am | Reply
Thanks for a great quarter, professor.

Comment by Level — December 22, 2007 @ 12:48 pm | Reply
No, thank you!

Comment by gregknese — January 2, 2008 @ 12:50 pm | Reply

RSS feed for comments on this post. TrackBack URI

	Leonid on Polynomials with no zeros on t…
	Leonid on AMS citing
	gregknese on Math 2D Spring 2008 Example…
	thuvan on Math 2D Spring 2008 Example…
	gregknese on Math 2D Spring 2008 Final exam…

greg.knese

December 5, 2007