Rules for Limits if x--->infinity
1. degree of top equals degree of bottom-->ANSWER: top coefficient over bottom coefficient
2. degree is bigger than bottom degree-->ANSWER: positive or negative infinity
3. top degree is less than bottom degree-->ANSWER: 0
Also with limits--->if indeterminate form is created
L'Rule! All you do is take derivative of top over derivative of bottome. If indeterminate form is still created then simply repeat.
NOTE: I have been getting a lot better at substitution. With all the practice I'm starting to recognize things that need to be substituted. Like if you have sin and cos, those are derivatives of each other so you would use substitution.
NOTE: Let's not forget that the integrals of some things are on our chart. Most of them are like ln of something.
NOTE: Remember AREA means to INTEGRATE! I saw that one popped up on the packet.
Monday, October 4, 2010
Sunday, October 3, 2010
post 6
l'hospitals rule:
if the limit is in indeterminate form, you use this.
you take the derivative of the top and the bottom of the fraction (separately, do not use quotient rule) and then take the limit of it again.
if you get indeterminate form, repeat as many times as necessary.
this week in calc, we just went over all the ways to integrate pretty much. partial fractions, trig sub, all that good stuff. and we had a test on thursday. and now we have a take home test due this week sometime (i think either thursday/friday) if someone knows when it is due please tell me.
i'm just gonna kinda review randomness today.
sorry if it's short. i'm not really focused too well right now...
partial fractions:
whenever you simplify the bottom of the fraction as much as you can... if you have something like this n/n(n-1)
you'd use the normal rules and just do A/n + B/n-1.
but if you ever have something left in the bottom that is still squared, that's when the rules get tricky.
if your exponent is on the outside of the equation...(example: x/(x-1)^2) then you would do A/(x-1) + B/(x-1)^2
and if your exponent was 7, then you would go all the way up to 7.
no if your exponent is on the inside of the equation....(x/(x^2-1) then you would do like this... A + Bx/(x^2-1)
get it ? hopefully you do.
now i am confused on what convergent and divergent is and how you can tell the difference.
help?
if the limit is in indeterminate form, you use this.
you take the derivative of the top and the bottom of the fraction (separately, do not use quotient rule) and then take the limit of it again.
if you get indeterminate form, repeat as many times as necessary.
this week in calc, we just went over all the ways to integrate pretty much. partial fractions, trig sub, all that good stuff. and we had a test on thursday. and now we have a take home test due this week sometime (i think either thursday/friday) if someone knows when it is due please tell me.
i'm just gonna kinda review randomness today.
sorry if it's short. i'm not really focused too well right now...
partial fractions:
whenever you simplify the bottom of the fraction as much as you can... if you have something like this n/n(n-1)
you'd use the normal rules and just do A/n + B/n-1.
but if you ever have something left in the bottom that is still squared, that's when the rules get tricky.
if your exponent is on the outside of the equation...(example: x/(x-1)^2) then you would do A/(x-1) + B/(x-1)^2
and if your exponent was 7, then you would go all the way up to 7.
no if your exponent is on the inside of the equation....(x/(x^2-1) then you would do like this... A + Bx/(x^2-1)
get it ? hopefully you do.
now i am confused on what convergent and divergent is and how you can tell the difference.
help?
Post # 6
So, lets go through a few throwback/ things that are necessary to be known. This week we prepared for the test and hopefully we all did good on it! Cross your fingers being that exams are coming up shortly!! Ahh, deep breath. So, anyway, I'm going to go over a few things that need to be remembered because they are in all of the parts of what we've been up to..so here it goes…
SUBSTITUTION
Substitution takes the place of the derivative rules for problems such as product rule and quotient rule and should be used PLENTY right now…
The steps to substitution are:
1. Find a derivative inside the interval
2. set u = the non-derivative
3. take the derivative of u
4. substitute back in
e INTEGRATION
- whatever is raised to the e power will be your u, and du will be the derivative of u.
For example:
e^2x-1dxu=2x-1 du=2
rewrite the function as:
1/2{ e^u du, therefore
= 1/2e^2x-1+C is your final answer.
LIMITS:
Rules for Limits:…
1. if the degree of top equals the degree of bottom, the answer is the top coefficient over bottom coefficient
2. if top degree is bigger than bottom degree, the answer is positive or negative infinity
3. if top degree is less than bottom degree, the answer is 0
Monday, September 27, 2010
Post # 5
Okay, so i'm just going to kind of explain the things i don't understand..
1. So, let me start with the improper integrals. I dont understand how to plug in certain things to it? or what you do.
2. Will lo'hopitals rule work for all limits?
3. How do you know if a problem with a squareroot should be solved by substitution, bi-parts..etc.
4. What is the best way to solve a limit.
Okay, so lets get to explaining a little.
When you do the thing with the chart on page A21, you should be aware to make sure all of the important things are being substituted for; as well as integrating if it still has an integral symbol in the equation!
Sorry this blog isn't very informative..but it helps me more when poeple answer my questions then saying what i know. Hopefully someone can help!
1. So, let me start with the improper integrals. I dont understand how to plug in certain things to it? or what you do.
2. Will lo'hopitals rule work for all limits?
3. How do you know if a problem with a squareroot should be solved by substitution, bi-parts..etc.
4. What is the best way to solve a limit.
Okay, so lets get to explaining a little.
When you do the thing with the chart on page A21, you should be aware to make sure all of the important things are being substituted for; as well as integrating if it still has an integral symbol in the equation!
Sorry this blog isn't very informative..but it helps me more when poeple answer my questions then saying what i know. Hopefully someone can help!
Sunday, September 26, 2010
Post #5
Well I didn't bring home my notebook...so lets see what I can remember.
I know we talked about how if you have a definite integral with bounds that have an infinity in it. Also does this apply where you have a discontinuity or something? Well, anyways you replace it with an A or some other letter. Then I think you integrate normally..and then take the limit? I actually was confused on this last week. Can anyone give me an example or explain it better?
Well I will reexplain Lo'Hopital's Rule:
This is when you have a limit. You then plug in the number the limit is approaching into the equation. If you get anything like infinity/infinity, zero/zero, infinity/zero, zero/infinity you need Lo'Hopital's rule. So, you take derivative of the top and derivative of the bottom. Plug in the number from the limit. If you still get anything like infinity/infinity..etc. then you repeat this process. If not then you are done!
One last thing:
How do you know when something is to integrate with substitution?
I know we talked about how if you have a definite integral with bounds that have an infinity in it. Also does this apply where you have a discontinuity or something? Well, anyways you replace it with an A or some other letter. Then I think you integrate normally..and then take the limit? I actually was confused on this last week. Can anyone give me an example or explain it better?
Well I will reexplain Lo'Hopital's Rule:
This is when you have a limit. You then plug in the number the limit is approaching into the equation. If you get anything like infinity/infinity, zero/zero, infinity/zero, zero/infinity you need Lo'Hopital's rule. So, you take derivative of the top and derivative of the bottom. Plug in the number from the limit. If you still get anything like infinity/infinity..etc. then you repeat this process. If not then you are done!
One last thing:
How do you know when something is to integrate with substitution?
Week 5 maybe?
Wellllllll, I forgot my notebook at school on friday so I guess I'll just explain some stuff by memory.
Throwbacks from Calculus 1!
Throwback from Calc. 1 that can help us in physics!
Position, Velocity, Acceleration.
Velocity is the derivative of Position.
Acceleration is the derivative of Velocity.
Physics explains it as:
Velocity is how fast an x (position) is moving.
Acceleration is how fast you are going faster.
Calculus is basically an easier way to do this.
Antiderivative = Integration.
Anytime you see instantaneous rate of change, just take a derivative and plug in for x.
Trig Problems:
tancot = 1
cot/csc = cos
sin(2x) = 2sin(x)cos(x)
sin(-x) = -sin(x)
QUESTIONS, I'm looking at a MAO test right now that I have.
Oblique asymptote?
When you have an integral from 0 to x^2 and give you a value and say take the derivative. How you do that?
How to do f^(-1)'(x).
Any Volume problems.
Throwbacks from Calculus 1!
Throwback from Calc. 1 that can help us in physics!
Position, Velocity, Acceleration.
Velocity is the derivative of Position.
Acceleration is the derivative of Velocity.
Physics explains it as:
Velocity is how fast an x (position) is moving.
Acceleration is how fast you are going faster.
Calculus is basically an easier way to do this.
Antiderivative = Integration.
Anytime you see instantaneous rate of change, just take a derivative and plug in for x.
Trig Problems:
tancot = 1
cot/csc = cos
sin(2x) = 2sin(x)cos(x)
sin(-x) = -sin(x)
QUESTIONS, I'm looking at a MAO test right now that I have.
Oblique asymptote?
When you have an integral from 0 to x^2 and give you a value and say take the derivative. How you do that?
How to do f^(-1)'(x).
Any Volume problems.
Mal's Post
Whatever will I explain on this fine evening????
Hmm...good question.
How about some throwback?
Okay, so derivatives...because I'm a little rusty on that.
Okay. So, my thing is that lately I've been noticing that every time I'm trying to take the derivative of my u in py parts, I'm actually integrating. Which isn't what your supposed to be doing. So here's a little review.
a derivative is a what?
SLOPE. Good! I'm glad we're off to a good start here...
Okay. How do you find a slope/derivative?
Well, you multiply the the variable by your exponent and then subtract one from your exponent. VOILA!!! a derivative.
So if I'm given the position equation x(t)=2t^2+5t and I want to find the velocity at t=4, what do I do? Well, I take the derivative of the equation first then plug in 4 to see just what the slope(velocity) is at that particular point. So:
my derivative equals 4t+5. Now you plug in 4.
16+5=21!!!!! Congrats!!! okay.
Now, if I want to find the acceleration, what do I do? take the derivative twice. The second derivative (hence the word twice)..
so we found the first time that it was 4t+5 was the velocity.
so the second derivative would equal what? well the derivative of 5 is 0...constant derivative=0....and the derivative of 4t is just 4...so you acceleration at any point (even t=4, 5, or 6) is 4.
Another quick review would be the following:
If I want to get back from say acceleration to velocity. what would I do? integrate.
Now I realize that I'm being redundant and reviewing probably what's the easiest things in the big mighty calculus book, but I needed SOMETHING to talk about. Love you guys!
Hmm...good question.
How about some throwback?
Okay, so derivatives...because I'm a little rusty on that.
Okay. So, my thing is that lately I've been noticing that every time I'm trying to take the derivative of my u in py parts, I'm actually integrating. Which isn't what your supposed to be doing. So here's a little review.
a derivative is a what?
SLOPE. Good! I'm glad we're off to a good start here...
Okay. How do you find a slope/derivative?
Well, you multiply the the variable by your exponent and then subtract one from your exponent. VOILA!!! a derivative.
So if I'm given the position equation x(t)=2t^2+5t and I want to find the velocity at t=4, what do I do? Well, I take the derivative of the equation first then plug in 4 to see just what the slope(velocity) is at that particular point. So:
my derivative equals 4t+5. Now you plug in 4.
16+5=21!!!!! Congrats!!! okay.
Now, if I want to find the acceleration, what do I do? take the derivative twice. The second derivative (hence the word twice)..
so we found the first time that it was 4t+5 was the velocity.
so the second derivative would equal what? well the derivative of 5 is 0...constant derivative=0....and the derivative of 4t is just 4...so you acceleration at any point (even t=4, 5, or 6) is 4.
Another quick review would be the following:
If I want to get back from say acceleration to velocity. what would I do? integrate.
Now I realize that I'm being redundant and reviewing probably what's the easiest things in the big mighty calculus book, but I needed SOMETHING to talk about. Love you guys!
Sunday, September 19, 2010
Mal's Post
So, partial fractions were the main focus this week, along with sharpening our integration skills. Some things you need to remember when it comes to partial fractions, is that there are basically 4 cases, including synthetic division, that you can use when regular substitution fails, and by parts I think too. So, basically, when you see a fraction...
1. See if it can be solved using synthetic division (i.e. degree of top greater than degree of bottom)
2. If synthetic won't work, then Factor the Bottom!
Once you're dong factoring, you're gonna want to split it up with different letters over all the factors...so A B C, and so on.
However. If you have say a (x+1)^2, you have to put that particular factor over Cx+D...don't ask why, it's just the way of the gods..
So once you decide what to do after factoring, you go through and plug in numbers for x (after finding common denominators) that cancel out the other letters, till you have values for all letters (variables) used.
Once finding those values you go BACK, again, and plug those in in the appropriate places. Once you get that, then you FINALLY integrate. Yayy!! finally! (Most of the times you should get a lot of natural logs...just saying. Got it? good.
Okay...a question...what the heck is this integration table thing?? Unfortunately I was absent that day, so I was unable to get a good explanation....so if someone could do a problem or two and explain each step, it'd be much appreciated....:DD Thanks!!!
1. See if it can be solved using synthetic division (i.e. degree of top greater than degree of bottom)
2. If synthetic won't work, then Factor the Bottom!
Once you're dong factoring, you're gonna want to split it up with different letters over all the factors...so A B C, and so on.
However. If you have say a (x+1)^2, you have to put that particular factor over Cx+D...don't ask why, it's just the way of the gods..
So once you decide what to do after factoring, you go through and plug in numbers for x (after finding common denominators) that cancel out the other letters, till you have values for all letters (variables) used.
Once finding those values you go BACK, again, and plug those in in the appropriate places. Once you get that, then you FINALLY integrate. Yayy!! finally! (Most of the times you should get a lot of natural logs...just saying. Got it? good.
Okay...a question...what the heck is this integration table thing?? Unfortunately I was absent that day, so I was unable to get a good explanation....so if someone could do a problem or two and explain each step, it'd be much appreciated....:DD Thanks!!!
post 4
sorrryyyy i missed post 3. i fell asleep before i did it, haha. anyways...
this week in calc we learned partial fractions and went over more trig sub! yayyyy. and we learned about integration tables on friday. which is pretty cool that all you have to do is plug it into a formula.
partial fractions is used when you pretty much have no other choice. and it has to be fraction. and your top degree is smaller then your bottom degree.
becuase when your top degree is larger than your bottom degree, you would use synthetic division, which may i say makes life so easy. haha
anyways, here's how you do partial fractions...
1. you factor the bottom.
2. you do a/first term, b/second term, c/third term..etc & set that equal to your equation****
3. you then do common denominator and get rid of all the fractions.
4. you plug in CONVENIENT x values.
5. solve for a, b, c, d and what not.
6. then you go back to step 2 and plug in all the numbers you got.
7. integrate & solve
****now there is a tricky part for step two. if you have something squared when you factor. like x^5/(x+1)^2
you would do this
a/(x+1) + b/(x+1)^2
... and if it was cubed, or to the fourth or whatever, you would go all the way up to that degree.
ALSO, if you have something inside of it squared after factoring.. like this
x^5/x(x^2+1)
then you would put the term that has a squared in it like this...
a/x + (bx+c)/(x^2+1)
and you would do that for whatever letter you are at... for example if you already had an a & b, you would do cx + d/ whatever.
get it? k good :)
can someone go over synthetic division again. for some reason i keep forgetting it.
this week in calc we learned partial fractions and went over more trig sub! yayyyy. and we learned about integration tables on friday. which is pretty cool that all you have to do is plug it into a formula.
partial fractions is used when you pretty much have no other choice. and it has to be fraction. and your top degree is smaller then your bottom degree.
becuase when your top degree is larger than your bottom degree, you would use synthetic division, which may i say makes life so easy. haha
anyways, here's how you do partial fractions...
1. you factor the bottom.
2. you do a/first term, b/second term, c/third term..etc & set that equal to your equation****
3. you then do common denominator and get rid of all the fractions.
4. you plug in CONVENIENT x values.
5. solve for a, b, c, d and what not.
6. then you go back to step 2 and plug in all the numbers you got.
7. integrate & solve
****now there is a tricky part for step two. if you have something squared when you factor. like x^5/(x+1)^2
you would do this
a/(x+1) + b/(x+1)^2
... and if it was cubed, or to the fourth or whatever, you would go all the way up to that degree.
ALSO, if you have something inside of it squared after factoring.. like this
x^5/x(x^2+1)
then you would put the term that has a squared in it like this...
a/x + (bx+c)/(x^2+1)
and you would do that for whatever letter you are at... for example if you already had an a & b, you would do cx + d/ whatever.
get it? k good :)
can someone go over synthetic division again. for some reason i keep forgetting it.
Week Number Four
This week we reviewed trig. substitution integration and we also learned how to solve integration using partial fractions.
You can actually use partial fractions to break up fractions even if you aren't using it for integration.
Some simple steps for partial fraction integration:
1) First of all, make sure it isn't any other type of integration.
2) If you can factor the top or bottom then do so (if you can cancel anything then do so).
3) Once you've ruled out everything else, you then split the bottom factors into A/(factor1) + B/(factor2) +... = original.
4) Get common denominators and add.
5) Pick a convenient value for x (one that would give you zero once plugged into a factor) and plug in.
6) Solve for A, B, ... .
7) You then take the values of your variables and plug back in to when you first broke up the fraction.
8)Integrate! It will almost always be natural log integration. Remember that any number in front of a natural log is also it's exponent.
Question:
For integration using charts like we did in class on friday, how exactly do I know which formula to plug in to if the problem I'm facing doesn't have all of the necessary components. I.e. if the formula has an x and my problem doesn't.
You can actually use partial fractions to break up fractions even if you aren't using it for integration.
Some simple steps for partial fraction integration:
1) First of all, make sure it isn't any other type of integration.
2) If you can factor the top or bottom then do so (if you can cancel anything then do so).
3) Once you've ruled out everything else, you then split the bottom factors into A/(factor1) + B/(factor2) +... = original.
4) Get common denominators and add.
5) Pick a convenient value for x (one that would give you zero once plugged into a factor) and plug in.
6) Solve for A, B, ... .
7) You then take the values of your variables and plug back in to when you first broke up the fraction.
8)Integrate! It will almost always be natural log integration. Remember that any number in front of a natural log is also it's exponent.
Question:
For integration using charts like we did in class on friday, how exactly do I know which formula to plug in to if the problem I'm facing doesn't have all of the necessary components. I.e. if the formula has an x and my problem doesn't.
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