This week in calculus, we learned about different types of integration. The first type of substitution we learned was change of variables. This means that in the problem, there will be at least 2 different variables.
The second type we learned was area between curves. The formula for this is bSa top eq.-bottom eq.
To find a and b you have to set the equations equal to each other and solve. If the area is on the y axis, then a and b need to be y values so the equations can be solved for x. If the area is on the x axis, a and b need to be x values so the equation can be solved for y.
One of the easiest types of integration we learned this week is e integration. When dealing with 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-1dx
u=2x-1 du=2
now you rewrite the function as:
1/2{ e^u du, therefore
1/2e^2x-1+C will be the final answer.
Another type of integration that we learned is ln integration. An example of ln integration would be:
{ 3/xdx= 3{1/xdx
3 ln |x| +C
Another type of ln integration would be:
{1/5x-1dx
u= 5x-1 du=5dx
1/5 { 5/5x-1 dx
1/5 ln |5x-1| dx +C
I finally have gotten the hang of infinite and definite integrals after taking those quizes this week. The only things i am not very good at is change in variables substitution and area between curves substitution. i understand e and ln substitution pretty well. I just do not know how to break things up in the other types of substitution and i have trouble knowing when to plug numbers in and finding where some of the numbers are coming from. If anyone can help me understand this it would be a huge help. Thanks :).
Saturday, November 21, 2009
Post #14
Week #14 Calculus
So this week in Calculus we learned a few odd and in things about integration and how to use it etc.
So far my two favorite things about Calculus are ln integration and e integration.
ln is so easy because most of the time, whenever there is a fraction, the bottom will be your u and the top will be the derivative of that. Even if the top isn't the derivative, most of the time you only have to add a number in front of the integration symbol to balance it out.
So let's say... what is the equation that models the area under the curve 1/(x-1). Well, you have to take the integral of this but you run into a problem whenever you can't split it up into two fractions. So, you go to ln. The derivative of ln(x) is simply 1/x. So in this case, set u to the bottom and du is 1dx. Now its the integral of 1/u du which is simply ln(u). Substituting back in for u will give you ln(x-1) + c as your answer.
These can get much more fun but they are always so simple. That's why I like them :-).
e^x integration is just as simple. Whenever working with e^x integration, your u is always set to the exponent of the e. The derivative of e^x is simply e^x times the derivative of x. So when working with these, really all you have to worry about it balancing it out with the derivative of x.
For example, e^(2x). This would become (1/2)e^(2x) + c, The reason behind the (1/2) is because if you take the derivative of e^(2x) it is 2e^(2x). However, we don't want that 2. So we take it out by putting a (1/2) in front, just like we did for other integrals.
Natural log and e integration are really easy and easy to identify... I hope everyone else likes it as much as me.
The only issue I've had with these types of integration is doing really well with the simplifying...but Brandi said its okay. I suppose we will get better with it.
-John
So this week in Calculus we learned a few odd and in things about integration and how to use it etc.
So far my two favorite things about Calculus are ln integration and e integration.
ln is so easy because most of the time, whenever there is a fraction, the bottom will be your u and the top will be the derivative of that. Even if the top isn't the derivative, most of the time you only have to add a number in front of the integration symbol to balance it out.
So let's say... what is the equation that models the area under the curve 1/(x-1). Well, you have to take the integral of this but you run into a problem whenever you can't split it up into two fractions. So, you go to ln. The derivative of ln(x) is simply 1/x. So in this case, set u to the bottom and du is 1dx. Now its the integral of 1/u du which is simply ln(u). Substituting back in for u will give you ln(x-1) + c as your answer.
These can get much more fun but they are always so simple. That's why I like them :-).
e^x integration is just as simple. Whenever working with e^x integration, your u is always set to the exponent of the e. The derivative of e^x is simply e^x times the derivative of x. So when working with these, really all you have to worry about it balancing it out with the derivative of x.
For example, e^(2x). This would become (1/2)e^(2x) + c, The reason behind the (1/2) is because if you take the derivative of e^(2x) it is 2e^(2x). However, we don't want that 2. So we take it out by putting a (1/2) in front, just like we did for other integrals.
Natural log and e integration are really easy and easy to identify... I hope everyone else likes it as much as me.
The only issue I've had with these types of integration is doing really well with the simplifying...but Brandi said its okay. I suppose we will get better with it.
-John
Tuesday, November 17, 2009
Houston, we have a problem...
Soo, I realized something that may effect how I look at "Area Between Curves"
I don't remember the rules of graphing!
I know it's in my Advanced Math notebook, but I only found one of the...3ish?
Can someone explain a few for me?
Please and thank you!
I don't remember the rules of graphing!
I know it's in my Advanced Math notebook, but I only found one of the...3ish?
Can someone explain a few for me?
Please and thank you!
Post 13
Hi
Forgot Sunday, was going to submit Monday, but sick…so…yay BLOG!
Integration!!!! (exclamation marks) (did I spell that right?...oh wll)
INDEFINITE: So I actually got this: you add one to the exponent and divide by your new exponent and voila!!!!
And for the trig ones…you’re going to be seriously out of luck if you don’t know your DERIVATIVE FORMULAS!!! Muy importante.
So, examples, shall we? (I’m just going to use my “S” for integral in lack of an actual cool integral button on my stupid computer…wait..I just found it!!!!)
∫ 3x2 dx
Add 1 to the exponent giving you 3, and divide by, you guessed it, 3!
3x3
3
Which then simplifies to :
x3 + C
*only put the +C when doing indefinite integration, NOT DEFINITE (thought I’d stress that).
You know an integral is DEFINITE when it has these little numbers on the side of ∫. You basically do the same thing as in indefinite, but you have to subtract whatever your top number plugged in is minus your bottom number plugged in. Make sense? Not really. Not coherent at the moment.
So example!!!!! (again with the exclamation marks!..gah!...no!...it’s like an instant reflex!!!.....jeez us!)
0∫3 x3
Integrate:
¼ x4 l30
Plug in 3 and you get 81/4, and 0 gives you 0
So your answer is 81/4!! (Exclamation points..)
This past week we also did LRAM and MRAM and RRAM and all that other good stuff and I got it, but I just need to actually MEMORIZE the formulas, you know?
So, I do believe that is about it for everything I know. Now for the questions:
1. Can someone explain to me what Substitution is since I missed that?
2. Also, I need a good example of a definite integration with absolute value…kind of confusing.
THANKS MUCHO!!!! (exclamation points…from now on I’m using *)
Forgot Sunday, was going to submit Monday, but sick…so…yay BLOG!
Integration!!!! (exclamation marks) (did I spell that right?...oh wll)
INDEFINITE: So I actually got this: you add one to the exponent and divide by your new exponent and voila!!!!
And for the trig ones…you’re going to be seriously out of luck if you don’t know your DERIVATIVE FORMULAS!!! Muy importante.
So, examples, shall we? (I’m just going to use my “S” for integral in lack of an actual cool integral button on my stupid computer…wait..I just found it!!!!)
∫ 3x2 dx
Add 1 to the exponent giving you 3, and divide by, you guessed it, 3!
3x3
3
Which then simplifies to :
x3 + C
*only put the +C when doing indefinite integration, NOT DEFINITE (thought I’d stress that).
You know an integral is DEFINITE when it has these little numbers on the side of ∫. You basically do the same thing as in indefinite, but you have to subtract whatever your top number plugged in is minus your bottom number plugged in. Make sense? Not really. Not coherent at the moment.
So example!!!!! (again with the exclamation marks!..gah!...no!...it’s like an instant reflex!!!.....jeez us!)
0∫3 x3
Integrate:
¼ x4 l30
Plug in 3 and you get 81/4, and 0 gives you 0
So your answer is 81/4!! (Exclamation points..)
This past week we also did LRAM and MRAM and RRAM and all that other good stuff and I got it, but I just need to actually MEMORIZE the formulas, you know?
So, I do believe that is about it for everything I know. Now for the questions:
1. Can someone explain to me what Substitution is since I missed that?
2. Also, I need a good example of a definite integration with absolute value…kind of confusing.
THANKS MUCHO!!!! (exclamation points…from now on I’m using *)
Monday, November 16, 2009
Post #13
This week, we really started to get into integration. We learned how to approximate the area under a curve (which is integration) by using four different methods: LRAM, RRAM, MRAM, and Trapezoidal. I feel as though everyone has a hang of it so I am really not going to go into much detail. The biggest thing is probably going to be substitution. We just started learning it and the matter is a little "iffy".
Substitution:
1. Label the parts of your integral. (x^2)(2x) u= (x^2) du=(2x)
2. Integrate (u du). (1/2)(u^2) +c
3. Plug u back in. (1/2)((x^2)^2) +c
Your final answer should be (1/2)(x^4) +c or (x^4)/2 +c
For substitution to work, one must recognize derivative properties that applies the product rule, quotient rule, or chain rule. Say if a problem asks you to find the integral of (x^3)(x). You would have to bring out a 1/3 then multiply your x by 3, making it look like (1/3) integrate (x^3)(3x). Then you just follow yours steps of substitution, but make sure to leave the 1/3 out until the last step then you distribute it in.
Apparently, I'm missing my Second Derivative Test notes. Can anyone explain to me the Second Derivative Test?
Substitution:
1. Label the parts of your integral. (x^2)(2x) u= (x^2) du=(2x)
2. Integrate (u du). (1/2)(u^2) +c
3. Plug u back in. (1/2)((x^2)^2) +c
Your final answer should be (1/2)(x^4) +c or (x^4)/2 +c
For substitution to work, one must recognize derivative properties that applies the product rule, quotient rule, or chain rule. Say if a problem asks you to find the integral of (x^3)(x). You would have to bring out a 1/3 then multiply your x by 3, making it look like (1/3) integrate (x^3)(3x). Then you just follow yours steps of substitution, but make sure to leave the 1/3 out until the last step then you distribute it in.
Apparently, I'm missing my Second Derivative Test notes. Can anyone explain to me the Second Derivative Test?
Homework for 11/16??
Okay, I'm majorly confused!
For tonight's homework I got to number 11.
I attempted all of the others, but I'm still confused on how to get u for number three (I got it from calcchat, but I want to know HOW) and how tan^2xsec^2x(dx) is u=tanx and du=sec^2x(dx) on number 5. What happened to the ^2 on the tan? I thought dx/dx(tanx)=sec^2x???
Also....once I got to number 11..I died. I can't figure out how to do anything after figuring out what u and du is.
Can someone explain what do to next?
For tonight's homework I got to number 11.
I attempted all of the others, but I'm still confused on how to get u for number three (I got it from calcchat, but I want to know HOW) and how tan^2xsec^2x(dx) is u=tanx and du=sec^2x(dx) on number 5. What happened to the ^2 on the tan? I thought dx/dx(tanx)=sec^2x???
Also....once I got to number 11..I died. I can't figure out how to do anything after figuring out what u and du is.
Can someone explain what do to next?
Post #13
Alright, so the main things we did this week were the limits test and integration. And I am quite happy because I did pretty good on my limits test, but integration on the other hand, I am not too sure about. The pictures that Mrs. Robinson drew for us about integration help explain what it is a lot better, but I cannot really do that on here, so I am just going to explain it in words and examples. As usual.
Integration uses Riemann sums, which is the approximation of area by using rectangles or trapezoids. Integration is just finding the area of something with a curve that you would not normally be able to get.
So there are four different methods of integration, LRAM, RRAM, MRAM, and trapezoidal.
The first formula you need to know is x=(b-a)/n [a,b] with n subintervals. You will need to know this because each of the next formulas require that you know what x is.
LRAM- left hand approximation. (this puts the rectangles used to find the area on the left side of the curve) x[f(a)+f(a+x)+...f(b)]
RRAM- right hand approximation. (this puts the rectangles used to find the area on the right side of the curve) x[f(a+x)+...f(b)]
MRAM- approximation from the middle. (this puts the rectangles right on top of the curve, so that the curve goes through the middle of each one) x[f(mid)+f(mid)+...]
Trapezoidal- this does not use squares, instead it uses trapezoids to eliminate most of the empty space inside the curve, and I think this is the most accurate. x/2[f(a)+2f(a+x)+2f(a+2x)+...f(b)]
The thing i'm having problems with is the stuff we learned friday, the indefinite and definite integration. The main thing is that I guess I haven't done enough problems to be able to do it easily, but I can do them when I look at my notes and at the example problems.
Integration uses Riemann sums, which is the approximation of area by using rectangles or trapezoids. Integration is just finding the area of something with a curve that you would not normally be able to get.
So there are four different methods of integration, LRAM, RRAM, MRAM, and trapezoidal.
The first formula you need to know is x=(b-a)/n [a,b] with n subintervals. You will need to know this because each of the next formulas require that you know what x is.
LRAM- left hand approximation. (this puts the rectangles used to find the area on the left side of the curve) x[f(a)+f(a+x)+...f(b)]
RRAM- right hand approximation. (this puts the rectangles used to find the area on the right side of the curve) x[f(a+x)+...f(b)]
MRAM- approximation from the middle. (this puts the rectangles right on top of the curve, so that the curve goes through the middle of each one) x[f(mid)+f(mid)+...]
Trapezoidal- this does not use squares, instead it uses trapezoids to eliminate most of the empty space inside the curve, and I think this is the most accurate. x/2[f(a)+2f(a+x)+2f(a+2x)+...f(b)]
The thing i'm having problems with is the stuff we learned friday, the indefinite and definite integration. The main thing is that I guess I haven't done enough problems to be able to do it easily, but I can do them when I look at my notes and at the example problems.
Sunday, November 15, 2009
Ash's 13th Post
Wow, lucky number 13, eh?
I'm so glad I was up writing, because I forgot all about this. 0=]
Alright, this week, we worked on Integration a bit. I sorta understand it, but I have to be glued to my notes in order to remember what to do. I really hope that John was right when he said you understand it with more practice.
I would attempt to explain it, but I don't want to sound like a fool, so let's move on to something I understand better! :)
Hmm....how about...Implicit Derivatives!!
1. Don’t move anything! Just take the Derivative like normal, except that after EACH term, add a d(insert term here)/dx. Don’t forget to do it on BOTH SIDES!
2. Then solve for d(insert term here)/dx!
Very very simple
Example!
Find the Implicit Derivative of 6y^2+3x=7x^3-8y^2
1. 12y(dy/dx) + 3(dx/dx) = 21x^2(dx/dx) - 16y (dy/dx)
2. 12y(dy/dx) + 16y(dy/dx) = 21x^2(dx/dx)-3(dx/dx)
3. 28y(dy/dx) = 21x^2(dx/dx)-3(dx/dx)
4. (Dy/dx) = (21x^2(dx/dx)-3(dx/dx)) / 28y
For what I don't understand:
I THINK I understand integration a little bit, I just would like someone to go over the homework for this weekend and see if I made any major mistakes. I have trouble with Definite Integration more than Indefinite for some reason...I stopped half-way through the assignment because I really don't think I was doing them right. Can someone help me with that? :)
Oh! I almost forgot! I, for some reason, also have trouble with LRAM, RRAM, MRAM, and Trapezoidal Integration. Not sure why, I just do. I think the graphs blew my mind and I think I can't do it anymore because of the graph? I'm not sure, but I also need help with those. v.v
Anndd…linearization! I’m iffy on that…can someone explain (short and sweet) the simple steps?
Thanks!!!!
I'm so glad I was up writing, because I forgot all about this. 0=]
Alright, this week, we worked on Integration a bit. I sorta understand it, but I have to be glued to my notes in order to remember what to do. I really hope that John was right when he said you understand it with more practice.
I would attempt to explain it, but I don't want to sound like a fool, so let's move on to something I understand better! :)
Hmm....how about...Implicit Derivatives!!
1. Don’t move anything! Just take the Derivative like normal, except that after EACH term, add a d(insert term here)/dx. Don’t forget to do it on BOTH SIDES!
2. Then solve for d(insert term here)/dx!
Very very simple
Example!
Find the Implicit Derivative of 6y^2+3x=7x^3-8y^2
1. 12y(dy/dx) + 3(dx/dx) = 21x^2(dx/dx) - 16y (dy/dx)
2. 12y(dy/dx) + 16y(dy/dx) = 21x^2(dx/dx)-3(dx/dx)
3. 28y(dy/dx) = 21x^2(dx/dx)-3(dx/dx)
4. (Dy/dx) = (21x^2(dx/dx)-3(dx/dx)) / 28y
For what I don't understand:
I THINK I understand integration a little bit, I just would like someone to go over the homework for this weekend and see if I made any major mistakes. I have trouble with Definite Integration more than Indefinite for some reason...I stopped half-way through the assignment because I really don't think I was doing them right. Can someone help me with that? :)
Oh! I almost forgot! I, for some reason, also have trouble with LRAM, RRAM, MRAM, and Trapezoidal Integration. Not sure why, I just do. I think the graphs blew my mind and I think I can't do it anymore because of the graph? I'm not sure, but I also need help with those. v.v
Anndd…linearization! I’m iffy on that…can someone explain (short and sweet) the simple steps?
Thanks!!!!
Posting...#13
This week in calc we started off by having our limits test which i would have liked to do better on. Then we learned integration which involves Lram Rram Mram and trapezoid. Integration is the are under a curve and you can approximate it with the four i just listed Mram and trapezoid are the most precise out of the four.
LRAM-left hand approximation
x[f(a)+f(a+x)+...f(b)]
RRAM-right hand approximation
x[f(a+x)+...f(b)]
MRAM-midpoint approximation
x[f(mid)+f(mid)+...]
Trapezoid-approximates using a trapezoid
x/2[f(a)+2f(a+x)+2f(a+2x)+...f(b)]
An example of integration is:
x^2-3 [1,4] n=3
First you find your delta x which is 1
LRAM 1[f(1)+f(2)+f(3)]
1[-2+1+6]
1[5]=5
RRAM 1[f(2)+f(3)+f(4)]
1[1+6+13]=20
MRAM 1,2,3,4
2+1/2=3/2
3+2/2=5/2
4+3/2=7/2
1[-3/4+13/4=37/4]=47/4
Trapezoid
1/2[f(1)+2f(2)+2f(3)+f(4)]
1/2[-2+2+12+13]=25/2
What I have the most trouble with is MRAM and trapezoid. Mram just because i am very bad at fractions and trapezoid becuase i don't really understand how it works and how to find it so if someone can help me with that thanks.
LRAM-left hand approximation
x[f(a)+f(a+x)+...f(b)]
RRAM-right hand approximation
x[f(a+x)+...f(b)]
MRAM-midpoint approximation
x[f(mid)+f(mid)+...]
Trapezoid-approximates using a trapezoid
x/2[f(a)+2f(a+x)+2f(a+2x)+...f(b)]
An example of integration is:
x^2-3 [1,4] n=3
First you find your delta x which is 1
LRAM 1[f(1)+f(2)+f(3)]
1[-2+1+6]
1[5]=5
RRAM 1[f(2)+f(3)+f(4)]
1[1+6+13]=20
MRAM 1,2,3,4
2+1/2=3/2
3+2/2=5/2
4+3/2=7/2
1[-3/4+13/4=37/4]=47/4
Trapezoid
1/2[f(1)+2f(2)+2f(3)+f(4)]
1/2[-2+2+12+13]=25/2
What I have the most trouble with is MRAM and trapezoid. Mram just because i am very bad at fractions and trapezoid becuase i don't really understand how it works and how to find it so if someone can help me with that thanks.
post number 13
this week was busy. we learned about intergration.
there is two different types of intergration - definite and infenite
infinite - you get and equation
X^n = x^(n+1)/(n+1) plus C
This is pretty much doing the oppiset of taking a derivative and is not that hard. Will just take time getting use to. Also when taking the Intergral of something that is infinite you must remember to put the (+c) otherwise the whole answer is wrong.
Definite - is just F(a) +F(b)= #
you take the intergral of both your a and b (x,y) and then plug it in and get an answer. The difference from this one and infinte is that infinite is an actual equation and definite is an answer.
Also one thing that i now understand is limits.
when degree of top and bottom are the same it is the top coefficient over the bottom coefficient . When top is bigger then bottom the answer is infinity and when bottom is bigger then top the answer is zero.
One thing that i having trouble with though is the LRAM, RRAM stuff. I am lost on this and need help
there is two different types of intergration - definite and infenite
infinite - you get and equation
X^n = x^(n+1)/(n+1) plus C
This is pretty much doing the oppiset of taking a derivative and is not that hard. Will just take time getting use to. Also when taking the Intergral of something that is infinite you must remember to put the (+c) otherwise the whole answer is wrong.
Definite - is just F(a) +F(b)= #
you take the intergral of both your a and b (x,y) and then plug it in and get an answer. The difference from this one and infinte is that infinite is an actual equation and definite is an answer.
Also one thing that i now understand is limits.
when degree of top and bottom are the same it is the top coefficient over the bottom coefficient . When top is bigger then bottom the answer is infinity and when bottom is bigger then top the answer is zero.
One thing that i having trouble with though is the LRAM, RRAM stuff. I am lost on this and need help
WEEK #13
This week was a busy learning one. We had off on monday, and a limits test on Tuesday. On Wednesday. We learned about intergration and about LRAM, RRAM, MRAM, and Trapezoidal to find the Riemann sums (approximation of area under a curve using rectangles or trapezoids. Friday we learned about the "real" integration. An integration, denoted by the S looking symbol, is basically looking at f '(x) and finding f(x).
There are two types of Integration: Indefinite and Definite.
Indefinite - answer is an equation.
Definite - answer is a number.
For indefinite: all the same properties of derivativtes apply, just backwards.
For indefinite polynomials: S(x^n)dx = [x^(n+1)]/[n+1] +C.
Example: S(2x^2)dx = 2(x^3/3) +c = (2/3)x^3 +C.
For other indefinites: basically just do the reverse of derivatives.
Example: S(sinx)dx = -cosx +C
You can also rewrite functions to make your life easier.
Example: S(1/x^3)dx can also be written as S(x^-3).
Definite integration:
it is the area formula on an interval [a,b]
aSb (f(x))dx = F(b) - F(a) = A NUMBER
Example:
3S0 (x^2)dx = (1/2)x^3 (30) = (1/3)(3)^3 - 1/2(0)^3 = 9
There are two types of Integration: Indefinite and Definite.
Indefinite - answer is an equation.
Definite - answer is a number.
For indefinite: all the same properties of derivativtes apply, just backwards.
For indefinite polynomials: S(x^n)dx = [x^(n+1)]/[n+1] +C.
Example: S(2x^2)dx = 2(x^3/3) +c = (2/3)x^3 +C.
For other indefinites: basically just do the reverse of derivatives.
Example: S(sinx)dx = -cosx +C
You can also rewrite functions to make your life easier.
Example: S(1/x^3)dx can also be written as S(x^-3).
Definite integration:
it is the area formula on an interval [a,b]
aSb (f(x))dx = F(b) - F(a) = A NUMBER
Example:
3S0 (x^2)dx = (1/2)x^3 (30) = (1/3)(3)^3 - 1/2(0)^3 = 9
post 13
Another week down and one more to go before Thanksgiving holidays. This week we learned about different types of integration. The first one is to find the area under a curve.
LRAM(left hand approximation)- delta x[f(a)+ f(a+x)...f(b-x0)]
RRAM(right hand approximation)- delta x[f(a+x)+...+f(b)]
MRAM(middle approximation)- delta x[f(mid) + f(mid)+...]
Trapezoidal- (delta x)/2[f(a)+2f(a+x)+2f(a+2x)+...f(b)]
The next two types of integration are indefinite and definite. The answer for indefinite integration is an equation. But on the other hand the answer for a definite intergration is a number.
Indefinite Integration-Sx^n(dx)={(x^n+1)/(n+1)}+C
Definine Integration-bSa f(x)dx=F(b)-F(a)=number
For old stuff that I know limit rule is not hard at all.
1. If the biggest exponent on the top and bottom are equal then the limit is the coefficient of the highest exponent on the top over the the coefficient of the highest exponent on the bottom.
2. If the biggest exponent is on the top then the limit is infinity.
3. If the bigger exponent is on the bottom then the limit is zero.
The steps for related rates are
1. Pick out all variables
2. Pick out all equations
3. Pick out what you are looking for
4. Sketch a graph and label
5. Create an equation with your variables
6. Take the derivative respecting time
7. Substitute back into the derivative
8. Solve
And the steps for tangent lines are
1. Take the derivative of the equation
2. Plug in the x value which gives you your slope
3. Use the slope you get and the point given and plug into slope intercept form (y-y1)=slope(x-x1)
*If a point is not given and only an x value is given plug the x value into the original which will give you a y value creating a point.
For what I am having problems with are minor problems that i am making in the work of he problems. I know for the most part how to work the problems I am just making small minor errors when working the problems which can only be fixed by working problems.
LRAM(left hand approximation)- delta x[f(a)+ f(a+x)...f(b-x0)]
RRAM(right hand approximation)- delta x[f(a+x)+...+f(b)]
MRAM(middle approximation)- delta x[f(mid) + f(mid)+...]
Trapezoidal- (delta x)/2[f(a)+2f(a+x)+2f(a+2x)+...f(b)]
The next two types of integration are indefinite and definite. The answer for indefinite integration is an equation. But on the other hand the answer for a definite intergration is a number.
Indefinite Integration-Sx^n(dx)={(x^n+1)/(n+1)}+C
Definine Integration-bSa f(x)dx=F(b)-F(a)=number
For old stuff that I know limit rule is not hard at all.
1. If the biggest exponent on the top and bottom are equal then the limit is the coefficient of the highest exponent on the top over the the coefficient of the highest exponent on the bottom.
2. If the biggest exponent is on the top then the limit is infinity.
3. If the bigger exponent is on the bottom then the limit is zero.
The steps for related rates are
1. Pick out all variables
2. Pick out all equations
3. Pick out what you are looking for
4. Sketch a graph and label
5. Create an equation with your variables
6. Take the derivative respecting time
7. Substitute back into the derivative
8. Solve
And the steps for tangent lines are
1. Take the derivative of the equation
2. Plug in the x value which gives you your slope
3. Use the slope you get and the point given and plug into slope intercept form (y-y1)=slope(x-x1)
*If a point is not given and only an x value is given plug the x value into the original which will give you a y value creating a point.
For what I am having problems with are minor problems that i am making in the work of he problems. I know for the most part how to work the problems I am just making small minor errors when working the problems which can only be fixed by working problems.
post 13
week 13 been like the first 12 weeks. were learning stuff and i know it the first day and after that, its gone. this week we went over limits because we had a quiz on wednesday, we learned something integrals
the symbol for an integral is a long skinny capital S. there are 2 different kinds of integrals, definite and indefinite.
indefinite intergrals
an indefinite integral is just the symbol with an equation behind it. taking an integral is just like taking the derivative backwards. if you are given x^3, instead of subtracting one from the exponent.. you would add and instead of multiplying the coefficient by the exponent.. you would divide. and no matter what problem it is, you must mark the end of your problem with + C. it will be counted completely wrong if you dont. your answer for indefinite integrals will always be an eqn.
the symbol for an integral is a long skinny capital S. there are 2 different kinds of integrals, definite and indefinite.
indefinite intergrals
an indefinite integral is just the symbol with an equation behind it. taking an integral is just like taking the derivative backwards. if you are given x^3, instead of subtracting one from the exponent.. you would add and instead of multiplying the coefficient by the exponent.. you would divide. and no matter what problem it is, you must mark the end of your problem with + C. it will be counted completely wrong if you dont. your answer for indefinite integrals will always be an eqn.
definite intergrals
a definite integral is pretty much the same except there will be two numbers found at the top and bottom of your integral symbol (the long skinny "s") and after your equation.. it will be marked with dx, but you pretty much ignore that. you treat it the same, you take the derivative backwards.. the only thing is that you pllug in your top number "b" into the derivative and your bottom number "a" into the derivative. once you do that.. you subtract f'(b) - f'(a), then you have your answer. you do not have to mark this answer with + C. your answer for definite integrals will always be number
limit rule is easy. 1. if the top and bottom exponents are the same, the answer is the top coefficient over the bottom coefficient.
2. if the top exponent is bigger than the bottom exponent, the answer is infinity.
3. if the top is less than the bottom, it goes to 0.
im not too good at angles of elevation and optimization still troubles me
a definite integral is pretty much the same except there will be two numbers found at the top and bottom of your integral symbol (the long skinny "s") and after your equation.. it will be marked with dx, but you pretty much ignore that. you treat it the same, you take the derivative backwards.. the only thing is that you pllug in your top number "b" into the derivative and your bottom number "a" into the derivative. once you do that.. you subtract f'(b) - f'(a), then you have your answer. you do not have to mark this answer with + C. your answer for definite integrals will always be number
limit rule is easy. 1. if the top and bottom exponents are the same, the answer is the top coefficient over the bottom coefficient.
2. if the top exponent is bigger than the bottom exponent, the answer is infinity.
3. if the top is less than the bottom, it goes to 0.
im not too good at angles of elevation and optimization still troubles me
Post #13
This week in calculus, we learned integration. Integration is the area under a curve.
LRAM is a left hand approximation
delta x [f(a)+f(a+delta x) + .... f(b- delta x)]
RRAM is a right hand approximation
delta x [ f (a + delta x) + .... + f(b)]
MRAM
delta x [f(mid) + f (mid) + .... ]
Trapezoidal
delta x/ 2 [f (a) + 2f(a+delta x) + 2f (a+ 2 delta x) + ... f(b)]
To find delta x: b-a/n [a, b] with n subintervals
EXAMPLE: x^2-3 [1, 4] n=3
delta x = 4-1/3 = 1
LRAM: 1 [ f(1) + f(2) +f(3) ]
1[ -2 +1+6]
1[5] = 5
RRAM: 1 [ f(2) +f(3) + f(4)]
1[ 1 + 6 + 13] = 20
MRAM: 1 2 3 4
2+1/ 2 = 3/2
3+2/ 2 = 5/2
4+3/2 = 7/2
1[ f( 3/2) + f(5/2) + f(7/2)]
1[-3/4 + 13/4 + 37/4] = 47/4
Trapezoidal: 1/2[ f(1) + 2f(2) + 2f(3) + f(4)]
1/2 [ -2 + 2(1) + 2(6) + 13] = 25/2
I also learned this week that the answer to a derivative problem is never DNE..
I have not quite caught on to the other integration we learned (opposite of a derivative) especially the definite ones. I understand what we have to do and not to forget +c on the indefinite ones but it takes me a while to actually arrive at an answer. I guess this will just come with practice but if someone thinks they can help me catch on faster I would appreciate it.
Post Number Thirteen
So yet another week of calculus down and I keep seeming to do worse.
This week we had a test on limits, easy right? Yep I thought so too, but I guess I thought wrong. I psych myself out way too much. So let’s go over limit rules.
Infinite limits:
If the top degree equals the bottom degree the answer is the top coefficient over the bottom coefficient.
Ex: lim x>infinity x^2/3x^2 = 1/3
If the top degree is greater than the bottom degree the answer is plus or minus infinity
Ex: lim x>infinity x^6 + 4x^4/x = infinity
If the top degree is less than the bottom degree the answer is zero
Ex: lim x>infinity x + 4/x^2 = 0
Reminders:
If the bottom is 0 don’t assume the limit does not exist right away, first factor and cancel or use your calculator.
Limits have to match from left and right.
a/b
So on to what we learned this week. We learned about integration. Integrations is the area under a curve. We learned about rieman sums which is an approximation of area using rectangles or trapezoids.
LRAM- left hand approximation
x[f(a) + f(a + x) + … f(b – x)]
RRAM – right hand approximation
x[f(a + x) + … + f(b)]
MRAM
x[f(mid) + f (mid) + …]
Trapezoidal
x/2[f(a) + 2f(a + x) + 2f(a + 2x) + … + f(b)]
I understand this stuff except for MRAM. Also for trapezoidal I get kind of mixed up in the formula so if anyone can help me with this please explain differently..
We then learned that there are two types of integration: indefinite and definite
The symbol for these looks sort of like a S, so that is what I will use here.
S=opposite of a derivative
Indefinite: answer is an equation, all the same properties of derivatives apply, pull a number out, treat separate terms separately
Ex: S x^3 dx would equal (1/4)x^4 + C
Definite: answer is a number, you are given the interval and must plug in for that interval
I understand the concept of integration, but it isn’t clicking in my head yet when trying to do it. I know the rules and how to do it, I’m just not getting it. Hopefully it will come. The one thing I do know is you always put + C after indefinite integrals J_
This week we had a test on limits, easy right? Yep I thought so too, but I guess I thought wrong. I psych myself out way too much. So let’s go over limit rules.
Infinite limits:
If the top degree equals the bottom degree the answer is the top coefficient over the bottom coefficient.
Ex: lim x>infinity x^2/3x^2 = 1/3
If the top degree is greater than the bottom degree the answer is plus or minus infinity
Ex: lim x>infinity x^6 + 4x^4/x = infinity
If the top degree is less than the bottom degree the answer is zero
Ex: lim x>infinity x + 4/x^2 = 0
Reminders:
If the bottom is 0 don’t assume the limit does not exist right away, first factor and cancel or use your calculator.
Limits have to match from left and right.
a/b
So on to what we learned this week. We learned about integration. Integrations is the area under a curve. We learned about rieman sums which is an approximation of area using rectangles or trapezoids.
LRAM- left hand approximation
x[f(a) + f(a + x) + … f(b – x)]
RRAM – right hand approximation
x[f(a + x) + … + f(b)]
MRAM
x[f(mid) + f (mid) + …]
Trapezoidal
x/2[f(a) + 2f(a + x) + 2f(a + 2x) + … + f(b)]
I understand this stuff except for MRAM. Also for trapezoidal I get kind of mixed up in the formula so if anyone can help me with this please explain differently..
We then learned that there are two types of integration: indefinite and definite
The symbol for these looks sort of like a S, so that is what I will use here.
S=opposite of a derivative
Indefinite: answer is an equation, all the same properties of derivatives apply, pull a number out, treat separate terms separately
Ex: S x^3 dx would equal (1/4)x^4 + C
Definite: answer is a number, you are given the interval and must plug in for that interval
I understand the concept of integration, but it isn’t clicking in my head yet when trying to do it. I know the rules and how to do it, I’m just not getting it. Hopefully it will come. The one thing I do know is you always put + C after indefinite integrals J_
Post 13
Monday - no school
Tuesday - B-rob wasn't there; studied for limits test
Wednesday - took limits test and started on integration
Integration is the area under a curve. We learned four different ways of approximating this area. These four ways included four riemann sums--approximations of area using rectangles or trapezoids. These four ways are:
LRAM - left hand approximations
x[ f(a) + f(a +x) + ... f(b) ]
RRAM - right hand approximations
x[ f(a + x) + ... f(b) ]
MRAM - midpoint approximations
x[ f(mid) + f(mid) + ... ]
Trapezoid - approximations using a trapezoid
x/2[ f(a) + 2f(a + x) + 2f(a + 2x) + ... f(b) ]
MRAM and Trapezoid are the most precise of the approximations, though MRAM is a pain. These are very easy to do. To begin, one must find x. To find x = b-a/n, then plug into equations.
Ex: find the area of x - 3 on the interval [0,2] n = 4
1. x = 2-0/4 = 1/2
2. 1/2[ f(0) + f(1/2) + f(1) + f(3/2) ]
*Remember, the number of terms should match up with your n
3. 1/2[ -3 - (5/2) - 2 - (3/2) ] = 1/2 - 9 = -9/2
*The answer may come out negative, but remember it's area that you're finding, and area cannot be negative, so your answer will become positive.
Wednesday - I don't remember?
Thursday - instead of approximating areas, we learned to find the direct answers
There are two types of integration: indefinite and definite. Indefinite integration is when the answer comes out to be an equation, and definite integration is when the answer comes out to be a number.
An integral is basically a backwards derivative. For integration, all the same properties of derivitives apply, except you do the opposite. Instead of subtracting 1 from the exponent and multiplying it to the term, you will add 1 to the exponent and divide the term by it.
For definite integrals: integral a b = F(b) - F(a) = #
I pretty much understand indefinite integrals, but I'm having trouble on the homework on the ones with definite integrals. I'm getting mixed up when there is a product or quotent rule. Also, I remember in class we learned something about splitting the integral up and adding the two split up parts together? When do we do that?
Tuesday - B-rob wasn't there; studied for limits test
Wednesday - took limits test and started on integration
Integration is the area under a curve. We learned four different ways of approximating this area. These four ways included four riemann sums--approximations of area using rectangles or trapezoids. These four ways are:
LRAM - left hand approximations
x[ f(a) + f(a +x) + ... f(b) ]
RRAM - right hand approximations
x[ f(a + x) + ... f(b) ]
MRAM - midpoint approximations
x[ f(mid) + f(mid) + ... ]
Trapezoid - approximations using a trapezoid
x/2[ f(a) + 2f(a + x) + 2f(a + 2x) + ... f(b) ]
MRAM and Trapezoid are the most precise of the approximations, though MRAM is a pain. These are very easy to do. To begin, one must find x. To find x = b-a/n, then plug into equations.
Ex: find the area of x - 3 on the interval [0,2] n = 4
1. x = 2-0/4 = 1/2
2. 1/2[ f(0) + f(1/2) + f(1) + f(3/2) ]
*Remember, the number of terms should match up with your n
3. 1/2[ -3 - (5/2) - 2 - (3/2) ] = 1/2 - 9 = -9/2
*The answer may come out negative, but remember it's area that you're finding, and area cannot be negative, so your answer will become positive.
Wednesday - I don't remember?
Thursday - instead of approximating areas, we learned to find the direct answers
There are two types of integration: indefinite and definite. Indefinite integration is when the answer comes out to be an equation, and definite integration is when the answer comes out to be a number.
An integral is basically a backwards derivative. For integration, all the same properties of derivitives apply, except you do the opposite. Instead of subtracting 1 from the exponent and multiplying it to the term, you will add 1 to the exponent and divide the term by it.
For definite integrals: integral a b = F(b) - F(a) = #
I pretty much understand indefinite integrals, but I'm having trouble on the homework on the ones with definite integrals. I'm getting mixed up when there is a product or quotent rule. Also, I remember in class we learned something about splitting the integral up and adding the two split up parts together? When do we do that?
Post #13
Calculus - Week 13
Okay, first off. I'm royally p*$$ed as the time of this post. My calculator, yet again, is broken... Anyway..
As for what we did / learned this week.
We took a quiz on limits - easy.
We learned LRAM, RRAM, MRAM, and Trapezoidal calculations of approximating area under a curve.
We then learned integrals, which is not an approximation but an exact answer.
Two types include indefinite and definite integrals.
Indefinite will give you an equation. Definite will give you a number.
For integration you basically do the opposite of a derivative. For polynomials, you multiply each term by the reciprocal of its current exponent + 1 and add 1 to the exponent. For anything else you simply go to your list...for example the integral of cos is just sin (the derivative of sin is cos). Another example is (1/x) will be ln(x).
For definite integrals you have the same thing except it now gives you an interval that it wants the area under. So the number on top of the integral symbol is b and the number on the bottom is a. So once integrating, you will do F(b) - F(a) where F(x) denotes the integrated f(x).
It's all pretty simple...will just take us some time to get used to it.
Anyway, going back to being p/oed.
Okay, first off. I'm royally p*$$ed as the time of this post. My calculator, yet again, is broken... Anyway..
As for what we did / learned this week.
We took a quiz on limits - easy.
We learned LRAM, RRAM, MRAM, and Trapezoidal calculations of approximating area under a curve.
We then learned integrals, which is not an approximation but an exact answer.
Two types include indefinite and definite integrals.
Indefinite will give you an equation. Definite will give you a number.
For integration you basically do the opposite of a derivative. For polynomials, you multiply each term by the reciprocal of its current exponent + 1 and add 1 to the exponent. For anything else you simply go to your list...for example the integral of cos is just sin (the derivative of sin is cos). Another example is (1/x) will be ln(x).
For definite integrals you have the same thing except it now gives you an interval that it wants the area under. So the number on top of the integral symbol is b and the number on the bottom is a. So once integrating, you will do F(b) - F(a) where F(x) denotes the integrated f(x).
It's all pretty simple...will just take us some time to get used to it.
Anyway, going back to being p/oed.
Post #13
This week was a lot of learning. We reviewed and took a quiz on limits and learned about Integration. I am having trouble with this stuff; I guess it just hasn't clicked yet.
We learned how to find the area under a curve and how to approximate this by using LRAM, RRAM, MRAM, or Trapezoidal. The two I'm most comfortable with is LRAM and RRAM.
Lets start with LRAM first:
f(x)=x^2-3 [1,4] n=3
First find your delta x:
(4-1)/3= 3/3= 1
Plug into formula next:
delta x=[f(a)+f(a+delta x)+...f(b+delta x)]
1[f(1)+f(2)+f(3)]
Use original equation to figure out f(1)..f(3):
1^2-3=3 2^2-3=1 3^2-3=6
1[-2+1+6]= 1[5]= 5
Next is RRAM:
f(x)=x^2-3 [1,4] n=3
Use the same steps, except the formula is delta x=[f(a+delta x)+...+f(b)]
delta x=1
1[f(2)+f(3)+f(4)]
1[1+6+13]= 1[20]= 20
I don't understand how to do MRAM. So, can someone explain that to me please?
Next we have this weird looking S, which means the opposite of a derivative. There are two types: indefinite and definite. Indefinites answer will be an EQUATION and a definites answer will be a NUMBER. It seems like a simple concept, but for some reason I confuse myself. One thing I do know is to ALWAYS put +C behind the equation for indefinite. I really need some extra practice with these kinds of problems.
We learned how to find the area under a curve and how to approximate this by using LRAM, RRAM, MRAM, or Trapezoidal. The two I'm most comfortable with is LRAM and RRAM.
Lets start with LRAM first:
f(x)=x^2-3 [1,4] n=3
First find your delta x:
(4-1)/3= 3/3= 1
Plug into formula next:
delta x=[f(a)+f(a+delta x)+...f(b+delta x)]
1[f(1)+f(2)+f(3)]
Use original equation to figure out f(1)..f(3):
1^2-3=3 2^2-3=1 3^2-3=6
1[-2+1+6]= 1[5]= 5
Next is RRAM:
f(x)=x^2-3 [1,4] n=3
Use the same steps, except the formula is delta x=[f(a+delta x)+...+f(b)]
delta x=1
1[f(2)+f(3)+f(4)]
1[1+6+13]= 1[20]= 20
I don't understand how to do MRAM. So, can someone explain that to me please?
Next we have this weird looking S, which means the opposite of a derivative. There are two types: indefinite and definite. Indefinites answer will be an EQUATION and a definites answer will be a NUMBER. It seems like a simple concept, but for some reason I confuse myself. One thing I do know is to ALWAYS put +C behind the equation for indefinite. I really need some extra practice with these kinds of problems.
post 13
This week we reviewed limits and took a test on them on Wednesday.... after reviewing limits, we then learned how to do integration. The first part of integration that we learned was:
LRAM- this is when they want left hand approximation
x[f(a)+ f(a+x)...f(b-x0)]
RRAM- when you want right hand approximation
x[f(a+x)+...+f(b)]
MRAM- this is when you use the midpoint formula
x[f(mid) + f(mid)+...]
Trapezoidle- the best graph approximation will always be trapezoidal
x/2[f(a)+2f(a+x)+2f(a+2x)+...f(b)]
These formulas are primarily used when finding the area under a curve.
The next concept we learned was definite and indefinite integration.
When working indefinite integrals, your answer will always be an equation.
When working backwards always remember the formula to find what the derivative was:
sx^n= x^n-1/n+1... and always remember to add a +C at the end to account for any number that may be there.
Definite integrals will always give you a number for your answer. Denfinite integrals are used to sometimes find the area. the formula is bSa f(x)dx=F(b)-F(a)=#
One of the things i do not understand about integration is the absolute value problem that was worked on the board. If anyone can help me out i would appreciate it. Some of the other things i am still not good at and having trouble with are angle of elevation, and linearization.
Oh and for those of you who are still having problems with related rates, here is a quick reminder:
1. Identify all variables
2. Identify equations that will be used
3. Take the derivative of the formula
4. Plug in all numbers
5. Solve for the unknown
Hope this helps!
LRAM- this is when they want left hand approximation
x[f(a)+ f(a+x)...f(b-x0)]
RRAM- when you want right hand approximation
x[f(a+x)+...+f(b)]
MRAM- this is when you use the midpoint formula
x[f(mid) + f(mid)+...]
Trapezoidle- the best graph approximation will always be trapezoidal
x/2[f(a)+2f(a+x)+2f(a+2x)+...f(b)]
These formulas are primarily used when finding the area under a curve.
The next concept we learned was definite and indefinite integration.
When working indefinite integrals, your answer will always be an equation.
When working backwards always remember the formula to find what the derivative was:
sx^n= x^n-1/n+1... and always remember to add a +C at the end to account for any number that may be there.
Definite integrals will always give you a number for your answer. Denfinite integrals are used to sometimes find the area. the formula is bSa f(x)dx=F(b)-F(a)=#
One of the things i do not understand about integration is the absolute value problem that was worked on the board. If anyone can help me out i would appreciate it. Some of the other things i am still not good at and having trouble with are angle of elevation, and linearization.
Oh and for those of you who are still having problems with related rates, here is a quick reminder:
1. Identify all variables
2. Identify equations that will be used
3. Take the derivative of the formula
4. Plug in all numbers
5. Solve for the unknown
Hope this helps!
13!
During week thirteen we learned some new concepts and also reviewed some old stuff.
the new thing that we learned included integration and reiman sums.
we revied limits which was taught at the end of last year/beginning of this year.
REIMAN SUMS:
-an approximation of area using rectangles and trapezoids.
LRAM- left hand approximationx
[f(a) + f(a + x) + … f(b – x)]
RRAM – right hand approximationx
[f(a + x) + … + f(b)]
MRAMx
[f(mid) + f (mid) + …]
Trapezoidal
x/2[f(a) + 2f(a + x) + 2f(a + 2x) + … + f(b)]
INTEGRATION: basically a derivative.. backwards, but there are no quotient rules, product rules, etc.
There are two types of intergation. These two types of integration are indefinite and definite.
Indefinite: the result of indefinite integration is an equation.
Definite: the result of definite integration is a number.
as far as limits, that stuff is relatively easy, i just needed a refresher.
Infinite limits:
top degree=the bottom degree
-the answer is the top coefficient over the bottom coefficient.
Ex: lim as x approaches infinity of 3x-1/2x+3
lim=3/2
top degree>bottom degree
-the answer is infinity
top degree
as far as what i dont understan.. everything is relatively easy, it will just take alot of practice to get used to integration. this may be a dumb question/statement but i don't really know where the numbers ontop of the S come from for integration and why sometimes they change within the same problem.
the new thing that we learned included integration and reiman sums.
we revied limits which was taught at the end of last year/beginning of this year.
REIMAN SUMS:
-an approximation of area using rectangles and trapezoids.
LRAM- left hand approximationx
[f(a) + f(a + x) + … f(b – x)]
RRAM – right hand approximationx
[f(a + x) + … + f(b)]
MRAMx
[f(mid) + f (mid) + …]
Trapezoidal
x/2[f(a) + 2f(a + x) + 2f(a + 2x) + … + f(b)]
INTEGRATION: basically a derivative.. backwards, but there are no quotient rules, product rules, etc.
There are two types of intergation. These two types of integration are indefinite and definite.
Indefinite: the result of indefinite integration is an equation.
Definite: the result of definite integration is a number.
as far as limits, that stuff is relatively easy, i just needed a refresher.
Infinite limits:
top degree=the bottom degree
-the answer is the top coefficient over the bottom coefficient.
Ex: lim as x approaches infinity of 3x-1/2x+3
lim=3/2
top degree>bottom degree
-the answer is infinity
top degree
as far as what i dont understan.. everything is relatively easy, it will just take alot of practice to get used to integration. this may be a dumb question/statement but i don't really know where the numbers ontop of the S come from for integration and why sometimes they change within the same problem.
Post # 13
So, now that I'm having the greatest day ever because I just realized that next week is Thanksgiving Break...I'll come do my blog.
This week was really good in calculus, except for the fact that i don't really grasp the LRAM, RRAM, and MIDPT. stuff..it's kinda weird in my head..so if anyone can clarify?
But i'll talk about integrals, which is like the opposite of a derivative. In a derivatve where you multiply the exponent to the coefficient, with integrals, you divide and add one to the exponent.
There are two types:
1. Indefinate - only an equation with the intergral symbol; you simply take the derivative backwards, you divide the exponent from the coefficient and add one to the exponent. Also, you MUST place a + c at the end of the equation. Why, you may ask, because you don't know if the beginning equation had a constant at the end of it, so you MUST mark it, or it will be COMPLETELY WRONG. YOU WILL ALWAYS GET AN EQUATION ANSWER.
Now, let's do an example problem:
S x^2 + 4x + 9
S (x/2)^3 + (2x)^2 + 9x + C
2. Definate Integrals - almost the same thing except there will be a number at the top and bottom of the integral symbol; the end of your equation will be marked with a dx. You treat it the same as an indefinate integral except you plug in the top number, or the b, into the derivative and then you have to SUBTRACT the plugged in bottom number, a, from the derivative plugged in. FOR THESE TYPES OF EQUATIONS, YOU WILL GET A NUMBER ANSWER, NOT AN EQUATION!
These are pretty simple, but just alot to remember because you have to remember not to take the derivative and to do the opposite. The thing i need help with would be the LRAM AND RRAM stuff..i never understand what to take from it..so can anyone help?
This week was really good in calculus, except for the fact that i don't really grasp the LRAM, RRAM, and MIDPT. stuff..it's kinda weird in my head..so if anyone can clarify?
But i'll talk about integrals, which is like the opposite of a derivative. In a derivatve where you multiply the exponent to the coefficient, with integrals, you divide and add one to the exponent.
There are two types:
1. Indefinate - only an equation with the intergral symbol; you simply take the derivative backwards, you divide the exponent from the coefficient and add one to the exponent. Also, you MUST place a + c at the end of the equation. Why, you may ask, because you don't know if the beginning equation had a constant at the end of it, so you MUST mark it, or it will be COMPLETELY WRONG. YOU WILL ALWAYS GET AN EQUATION ANSWER.
Now, let's do an example problem:
S x^2 + 4x + 9
S (x/2)^3 + (2x)^2 + 9x + C
2. Definate Integrals - almost the same thing except there will be a number at the top and bottom of the integral symbol; the end of your equation will be marked with a dx. You treat it the same as an indefinate integral except you plug in the top number, or the b, into the derivative and then you have to SUBTRACT the plugged in bottom number, a, from the derivative plugged in. FOR THESE TYPES OF EQUATIONS, YOU WILL GET A NUMBER ANSWER, NOT AN EQUATION!
These are pretty simple, but just alot to remember because you have to remember not to take the derivative and to do the opposite. The thing i need help with would be the LRAM AND RRAM stuff..i never understand what to take from it..so can anyone help?
Post #13
One more week until we have a break!!! This past week in calculus we continued to learn and review limits. We had a test on limits wednesday and after many packets i'm sure almost everyone knew what they were doing, but if not here's a few rules to refresh your memory:
LIMITS:
When you have a zero [any number] in the denominator, you try to fator and cancle but if taht diesn't work then you use the table function in your calculator with whatever number minus .1...minus .01... minus .001... plus .001... plus .01... and plus .1 NOW, you look at what the first three and bottom three are approaching in the middle of them, and that's your answer
also, if you have as the limit goes to infinity look at the degree of the top and the degree of the bottom, if the top and bottom or equal then you just take there coeffients and simplify, if the top degree is greater than the bottom degree then it will be plus or minus infinity, and if the top degree is less than the bottom degree then it will be zero.
Next we learned about Integration..there is four steps of integration in order to find whats close:
LRAM - left hand approximation
the formula is x[f(a)+f(a+x)+...f(b-x)]
RRAM - right hand apporximation
the formula is x[f(a+x)=...f(b)]
MRAM - the middle
the formula is x[f(mid)+f(mid)+...]
Trapezoidle -
the formula is (x/2)[f(a)+2f(a+x)+2f(a+2x)+...f(b)]
Remember that in any graph the best approximation is trapizodial.
We then learned about the two types of integhration .. indefinite and definite ..
Remember that indefinite answers will be an equation while definite answers witll be a number.
The equation for indefinite is [{x^(n-1)}/{n+1}]+c
so if you have x^3dx then the answer will be [(1)/(4)]x^4 + c
The equation for definite is f(b)-f(a)
so if you have 0 to 3 x^@dx, the answer will be (1/3)(3)^3 - (1/3)(0)^3 = 9-0 = 9
I'm pretty sure i understood limits very well as well as integration, but for some reason i just cannot grasp the concept of indefinite and definite integration.
~Ellie
LIMITS:
When you have a zero [any number] in the denominator, you try to fator and cancle but if taht diesn't work then you use the table function in your calculator with whatever number minus .1...minus .01... minus .001... plus .001... plus .01... and plus .1 NOW, you look at what the first three and bottom three are approaching in the middle of them, and that's your answer
also, if you have as the limit goes to infinity look at the degree of the top and the degree of the bottom, if the top and bottom or equal then you just take there coeffients and simplify, if the top degree is greater than the bottom degree then it will be plus or minus infinity, and if the top degree is less than the bottom degree then it will be zero.
Next we learned about Integration..there is four steps of integration in order to find whats close:
LRAM - left hand approximation
the formula is x[f(a)+f(a+x)+...f(b-x)]
RRAM - right hand apporximation
the formula is x[f(a+x)=...f(b)]
MRAM - the middle
the formula is x[f(mid)+f(mid)+...]
Trapezoidle -
the formula is (x/2)[f(a)+2f(a+x)+2f(a+2x)+...f(b)]
Remember that in any graph the best approximation is trapizodial.
We then learned about the two types of integhration .. indefinite and definite ..
Remember that indefinite answers will be an equation while definite answers witll be a number.
The equation for indefinite is [{x^(n-1)}/{n+1}]+c
so if you have x^3dx then the answer will be [(1)/(4)]x^4 + c
The equation for definite is f(b)-f(a)
so if you have 0 to 3 x^@dx, the answer will be (1/3)(3)^3 - (1/3)(0)^3 = 9-0 = 9
I'm pretty sure i understood limits very well as well as integration, but for some reason i just cannot grasp the concept of indefinite and definite integration.
~Ellie
13th post
so this week we learned about indefinite integrals and definite integrals its pretty much like derivatives, so:
indefinite integrals:
an indefinite integral is just the symbol with an equation behind it. taking an integral is just like taking the derivative backwards. if you are given x^3, instead of subtracting one from the exponent, you would add and instead of multiplying the coefficient by the exponent, you would divide. and no matter what problem it is, you must mark the end of your problem with + C. the reason you do this is because c counts as any possible constant that could be there. your answer for indefinite integrals will always be an equation!
for example:
S x^2 + 4x + 9
S x^3/3 + 4x^2/2 + 9x + C
definite integrals:
a definite integral is pretty much the same except there will be two numbers found at the top and bottom of your integral symbol (the long skinny "s") and after your equation.. it will be marked with dx, but you pretty much ignore that. you treat it the same, you take the derivative backwards.. the only thing is that you pllug in your top number "b" into the derivative and your bottom number "a" into the derivative. once you do that.. you subtract f'(b) - f'(a), then you have your answer. you do not have to mark this answer with + C. your answer for definite integrals will always be a number.
reminder:
Rules for Limits:
1. if the degree of top equals thedegree 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
2. if top degree is less than bottom degree, the answer is 0
related rates!!!:
The steps for related rates are:
1. Pick out all variables
2. Pick out all equations
3. Pick out what you are looking for
4. Sketch a graph and label
5. Create an equation with your variables
6. Take the derivative respecting time
7. Substitute back into the derivative
8. Solve
linearization:
The steps for solving linearization problems are:
1. Pick out the equation
2. f(x)+f`(x)dx
3. Figure out your dx
4. Figure out your x
5. Plug in everything you get
alright so im kind of understanding this indefinite and definite integrals business. im still having difficulties with linearization a little i think im getting better but i could use some examples.
indefinite integrals:
an indefinite integral is just the symbol with an equation behind it. taking an integral is just like taking the derivative backwards. if you are given x^3, instead of subtracting one from the exponent, you would add and instead of multiplying the coefficient by the exponent, you would divide. and no matter what problem it is, you must mark the end of your problem with + C. the reason you do this is because c counts as any possible constant that could be there. your answer for indefinite integrals will always be an equation!
for example:
S x^2 + 4x + 9
S x^3/3 + 4x^2/2 + 9x + C
definite integrals:
a definite integral is pretty much the same except there will be two numbers found at the top and bottom of your integral symbol (the long skinny "s") and after your equation.. it will be marked with dx, but you pretty much ignore that. you treat it the same, you take the derivative backwards.. the only thing is that you pllug in your top number "b" into the derivative and your bottom number "a" into the derivative. once you do that.. you subtract f'(b) - f'(a), then you have your answer. you do not have to mark this answer with + C. your answer for definite integrals will always be a number.
reminder:
Rules for Limits:
1. if the degree of top equals thedegree 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
2. if top degree is less than bottom degree, the answer is 0
related rates!!!:
The steps for related rates are:
1. Pick out all variables
2. Pick out all equations
3. Pick out what you are looking for
4. Sketch a graph and label
5. Create an equation with your variables
6. Take the derivative respecting time
7. Substitute back into the derivative
8. Solve
linearization:
The steps for solving linearization problems are:
1. Pick out the equation
2. f(x)+f`(x)dx
3. Figure out your dx
4. Figure out your x
5. Plug in everything you get
alright so im kind of understanding this indefinite and definite integrals business. im still having difficulties with linearization a little i think im getting better but i could use some examples.
week 13
ok, so it's week 13 already and progress reports are about to come out for the 2nd nine weeks. school year is flying by!
this week we went over limits because we had a quiz on wednesday, it was scheduled for tuesday but b rob was absent. towards the end of the week, we learned something new! integrals!
the symbol for an integral is a long skinny capital S. there are 2 different kinds of integrals, definite and indefinite.
INDEFINITE INTEGRALS:
an indefinite integral is just the symbol with an equation behind it. taking an integral is just like taking the derivative backwards. if you are given x^3, instead of subtracting one from the exponent.. you would add and instead of multiplying the coefficient by the exponent.. you would divide. and no matter what problem it is, you must mark the end of your problem with + C. it will be counted completely wrong if you dont! THE REASON YOU DO THIS IS BECAUSE C COUNTS AS ANY POSSIBLE CONSTANT THAT COULD BE THERE. your answer for indefinite integrals will always be an equation!
for example:
S x^2 + 4x + 9
S x^3/3 + 4x^2/2 + 9x + C
DEFINITE INTEGRALS:
a definite integral is pretty much the same except there will be two numbers found at the top and bottom of your integral symbol (the long skinny "s") and after your equation.. it will be marked with dx, but you pretty much ignore that. you treat it the same, you take the derivative backwards.. the only thing is that you pllug in your top number "b" into the derivative and your bottom number "a" into the derivative. once you do that.. you subtract f'(b) - f'(a), then you have your answer. you do not have to mark this answer with + C. your answer for definite integrals will always be a number!
for example:
3
S x^2 dx
0
= x^3/3 ..... (3)^3/3 - (0)^3/3 ...... = 9-0
= 9
just remember that these problems do get harder. some involve functions like sine and cosine. same rules apply though, just remember your derivative rules for those types of functions.
i need help with 35 and 37 on the back page...(293) of the homework, its pictures! what do i do?
this week we went over limits because we had a quiz on wednesday, it was scheduled for tuesday but b rob was absent. towards the end of the week, we learned something new! integrals!
the symbol for an integral is a long skinny capital S. there are 2 different kinds of integrals, definite and indefinite.
INDEFINITE INTEGRALS:
an indefinite integral is just the symbol with an equation behind it. taking an integral is just like taking the derivative backwards. if you are given x^3, instead of subtracting one from the exponent.. you would add and instead of multiplying the coefficient by the exponent.. you would divide. and no matter what problem it is, you must mark the end of your problem with + C. it will be counted completely wrong if you dont! THE REASON YOU DO THIS IS BECAUSE C COUNTS AS ANY POSSIBLE CONSTANT THAT COULD BE THERE. your answer for indefinite integrals will always be an equation!
for example:
S x^2 + 4x + 9
S x^3/3 + 4x^2/2 + 9x + C
DEFINITE INTEGRALS:
a definite integral is pretty much the same except there will be two numbers found at the top and bottom of your integral symbol (the long skinny "s") and after your equation.. it will be marked with dx, but you pretty much ignore that. you treat it the same, you take the derivative backwards.. the only thing is that you pllug in your top number "b" into the derivative and your bottom number "a" into the derivative. once you do that.. you subtract f'(b) - f'(a), then you have your answer. you do not have to mark this answer with + C. your answer for definite integrals will always be a number!
for example:
3
S x^2 dx
0
= x^3/3 ..... (3)^3/3 - (0)^3/3 ...... = 9-0
= 9
just remember that these problems do get harder. some involve functions like sine and cosine. same rules apply though, just remember your derivative rules for those types of functions.
i need help with 35 and 37 on the back page...(293) of the homework, its pictures! what do i do?
Subscribe to:
Posts (Atom)