Kay. Since I've yet to touch on L'Hopital's Rule, I shall do so now.
The most important thing to learn about l'Hôpital's rule is when it should not be used:
Definitely do NOT use it when the limits of the two parts are not both 0, or both infinity. In this case the rule is likely to give a wrong answer!
Example:
limx->0+ (cos x)/x
is positive infinity, because the numerator approaches 1 while the denominator approaches 0. If we incorrectly apply l'Hôpital's rule, we get
limx->0+ (- sin x)/1 = 0.
So you DO use L'Hopital's Rule when you get an indeterminate in the first place...this is inf/inf, 0/0, etc.
Okay, for Trig SUB!!!!! I'm getting pretty good at this, so bear with me....
My trick is: Everytime I see a trig function to an odd power, I take out an even...After this I use an appropriate identity. It's really not all that hard...I have my notecards somewhere...just ask me for them..
OkAY!!!! for things you can comment on....
Does anyone know how to:
1. Divide stuff? like x^2 + x+ 7 all over x-8. the other day BRob tried to do a problem like that, and I failedddd miserably. Easier way??
2. Chasing the Rabbit. One time I ended up with chasing the rabbit, but the answer was something super easy. any hints as to when you should use by parts i.e. chasing the rabbit?
alright. night.
Monday, September 6, 2010
blog 2
alright alright, so this week we had our first real test. it was kindaaaa hard, but i think i did well :) we are still working on integration .. by parts, trig sub, wallis formula, all that good stuff.
so, first i'll tell you what wallice's formula is (by the way idk how to spell it so ignore that)...
if you have an integration problem of sin or cos raised to a power.. this is when you use this
S cos^5(x)
alright, so if your degree is ODD, you do (2/3)(4/5)..(n-1/n) and simply multiply them together. so your answer would be 8/15.
S sin^8(x)
if your degree is EVEN, you do (1/2)(3/4)...(n-1/n) (pi/2). then multiply
so your answer would be (1/2)(3/4)(5/6)(7/8)(pi/2). i don't feel like multiplying it out haha.
HELP:
alright, trig sub. it's pretty much a bunch of formulas telling you what to substitute in and when to do it when you are integrating trig functions.
i know how to do these.. i just tend to mess up cuz i don't memorize when i have to do what. & i also didn't bring my book home to remember to put anything about it on here.. :x so.. could someone maybe go over a few formulas for me?
when to do synthetic division:
when your top function degree is larger than the top.
so say you had S (x^2 + 2x +5)/(x-6)
6 would go in your box, then 1, 2, 5...
i think. if i'm wrong someone please let me know! also, i get kinda lost after that.. i stop and don't remember what to do next, the only thing i remember is that i have to put my remainder over the bottom of the fraction at the end... help please
so, first i'll tell you what wallice's formula is (by the way idk how to spell it so ignore that)...
if you have an integration problem of sin or cos raised to a power.. this is when you use this
S cos^5(x)
alright, so if your degree is ODD, you do (2/3)(4/5)..(n-1/n) and simply multiply them together. so your answer would be 8/15.
S sin^8(x)
if your degree is EVEN, you do (1/2)(3/4)...(n-1/n) (pi/2). then multiply
so your answer would be (1/2)(3/4)(5/6)(7/8)(pi/2). i don't feel like multiplying it out haha.
HELP:
alright, trig sub. it's pretty much a bunch of formulas telling you what to substitute in and when to do it when you are integrating trig functions.
i know how to do these.. i just tend to mess up cuz i don't memorize when i have to do what. & i also didn't bring my book home to remember to put anything about it on here.. :x so.. could someone maybe go over a few formulas for me?
when to do synthetic division:
when your top function degree is larger than the top.
so say you had S (x^2 + 2x +5)/(x-6)
6 would go in your box, then 1, 2, 5...
i think. if i'm wrong someone please let me know! also, i get kinda lost after that.. i stop and don't remember what to do next, the only thing i remember is that i have to put my remainder over the bottom of the fraction at the end... help please
Sunday, September 5, 2010
Post #2
Hello my Calculus BC friends,
TRIG SUBSTITUTION!
Some basic integrals:
S sinu du = -cos u + C
S cosu du = sin u + C
S tan u du = -ln|cos u| + C
S cot u du = ln|sin u| + C
S secu du = ln|sec u + tan u| + C
S cscu du = -ln|csc u + cot u| + C
S sec^2 u du = tan u + C
S csc^2 u du = -cot u + C
Some identities:
sin^2x + cos^2x = 1 .
sin^2x = (1 - cos 2x)/2
cos^2x = (1 + cos 2x)/2
*What I try to do: usually try to take out some kind of squared, then change the to an identity, distribute in, and substitute.
*ALL the Rules:
SIN & COS guidelines:
1. If the power of the sine is odd and positive, save one sine
factor and convert the remaining factors to cosines. Then, expand
and integrate.
2. If the power of the cosine is odd and positive, save one cosine
factor and convert the remaining factors to sines. Then, expand
and integrate.
3. If the powers of both sine and cosine are even and
non negative, make repeated use of the half-angle identities for
sin^2x and cos^2x to convert the integrand to odd powers of the
cosine. Then proceed as in guideline 2.
SEC & TAN guidelines:
1. If the power of the secant is even and positive, save a secantsquared
factor and convert the remaining factors to tangents.
Then expand and integrate.
2. If the power of the tangent is odd and positive, save a secanttangent
factor and convert the remaining factors to secants. Then
expand and integrate.
3. If there are no secant factors and the power of the tangent is
even and positive, convert a tangent-squared factor to a secantsquared
factor, then expand and repeat if necessary.
4. If the integral is of the form S secmx dx, where m is odd and
positive, use integration by parts.
5. If none of the first four guidelines applies, try converting to
Wallis formula:
Only works with sin and cos when going from 0 to pi/2. n is the exponent
when n is ODD: (2/3)(4/5)(6/7)...(n -1)/n
EVEN: (1/2)(3/4)(5/6)...((n-1)/n)(pi/2)
HERE IS WHAT YOU CAN COMMENT ON:
Now I understand everything, but I somehow cannot always work the problems. Does anyone have some kind of trick on how to know when you look at a problem and know you have to either substitute, by part it, or trig sub? Also, do you know something that can help me remember how to do trig sub? (like the steps explained easier or a trick to remember or the steps you follow EVERY time?)
TRIG SUBSTITUTION!
Some basic integrals:
S sinu du = -cos u + C
S cosu du = sin u + C
S tan u du = -ln|cos u| + C
S cot u du = ln|sin u| + C
S secu du = ln|sec u + tan u| + C
S cscu du = -ln|csc u + cot u| + C
S sec^2 u du = tan u + C
S csc^2 u du = -cot u + C
Some identities:
sin^2x + cos^2x = 1 .
sin^2x = (1 - cos 2x)/2
cos^2x = (1 + cos 2x)/2
*What I try to do: usually try to take out some kind of squared, then change the to an identity, distribute in, and substitute.
*ALL the Rules:
SIN & COS guidelines:
1. If the power of the sine is odd and positive, save one sine
factor and convert the remaining factors to cosines. Then, expand
and integrate.
2. If the power of the cosine is odd and positive, save one cosine
factor and convert the remaining factors to sines. Then, expand
and integrate.
3. If the powers of both sine and cosine are even and
non negative, make repeated use of the half-angle identities for
sin^2x and cos^2x to convert the integrand to odd powers of the
cosine. Then proceed as in guideline 2.
SEC & TAN guidelines:
1. If the power of the secant is even and positive, save a secantsquared
factor and convert the remaining factors to tangents.
Then expand and integrate.
2. If the power of the tangent is odd and positive, save a secanttangent
factor and convert the remaining factors to secants. Then
expand and integrate.
3. If there are no secant factors and the power of the tangent is
even and positive, convert a tangent-squared factor to a secantsquared
factor, then expand and repeat if necessary.
4. If the integral is of the form S secmx dx, where m is odd and
positive, use integration by parts.
5. If none of the first four guidelines applies, try converting to
Wallis formula:
Only works with sin and cos when going from 0 to pi/2. n is the exponent
when n is ODD: (2/3)(4/5)(6/7)...(n -1)/n
EVEN: (1/2)(3/4)(5/6)...((n-1)/n)(pi/2)
HERE IS WHAT YOU CAN COMMENT ON:
Now I understand everything, but I somehow cannot always work the problems. Does anyone have some kind of trick on how to know when you look at a problem and know you have to either substitute, by part it, or trig sub? Also, do you know something that can help me remember how to do trig sub? (like the steps explained easier or a trick to remember or the steps you follow EVERY time?)
Post #2
Okay, so after what felt like the longest week ever, it's now time to do the blog. This week in Calculus BC I was kinda discouraged by trig sub because its something i really don't understand. I only do the problems right when i have the formulas in front of me...and after studying them for a week straight, and still getting mixed up on them, i'm finding it almost hopeless.
Hopefully someone can show me their study techniques?
But lets go over a few things...
For trig sub, something i always get wrong is WHEN to actually do the method..so, i believe it is when you can't basically bi-part something? correct?
Also, you should never bi-part or trig sub things when you only have the trig function and its derivative/ integral..just saying. I do it all the time and it is definately the hard way.
I really wish there is something i can explain that i know how to do, but there really isn't..
I guess i'll explain Wallis Formula.
So, you do this when you have sin or cos and the degree is EVEN:
1. Start with 1/2 and multiply the chronological numbers until you get to the exponent.
2. Then multiply by pi/two
3. Add +c
4. Box or circle your answer
When the degree is ODD:
1. Start with 1/2 and multiply the chronological numbers unitl you get to the exponent.
2. Put a + c
3. Box or circle your answer
So, i really feel like a baby and hopefully someone can help me..
i really just need all the helpful hints and basic problems explained to me.
i'm not quite sure why my brain hasn't kicked into school mode yet..
Hopefully someone can show me their study techniques?
But lets go over a few things...
For trig sub, something i always get wrong is WHEN to actually do the method..so, i believe it is when you can't basically bi-part something? correct?
Also, you should never bi-part or trig sub things when you only have the trig function and its derivative/ integral..just saying. I do it all the time and it is definately the hard way.
I really wish there is something i can explain that i know how to do, but there really isn't..
I guess i'll explain Wallis Formula.
So, you do this when you have sin or cos and the degree is EVEN:
1. Start with 1/2 and multiply the chronological numbers until you get to the exponent.
2. Then multiply by pi/two
3. Add +c
4. Box or circle your answer
When the degree is ODD:
1. Start with 1/2 and multiply the chronological numbers unitl you get to the exponent.
2. Put a + c
3. Box or circle your answer
So, i really feel like a baby and hopefully someone can help me..
i really just need all the helpful hints and basic problems explained to me.
i'm not quite sure why my brain hasn't kicked into school mode yet..
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