Well, this past week in calculus was .. just two days. Monday we reviewed some more things that we did not understand throughout the year. I can honestly say that I am BEGINNING to understand the stuff that Mrs. Robinson taught us about the picture of a graph, so I think that’s what I’m going to explain.
• If you have f(x):
o Use a solid line to divide where it’s increasing or decreasing: if it’s increasing then the first derivative is positive, and if it’s decreasing then the first derivative is negative.
o Use a dotted line to separate concave up and concave down: if it’s concave up then the second derivative is positive, and if it’s concave down then the second derivative is negative.
o Put a large dot on MAX and MIN: MAXs and MINs mean that it’s a horizontal tangent and the first derivative is zero.
o Put an open circle on points of inflection: points of inflection is where the second derivative is zero.
• If you have f’(x):
o Use a solid line where it’s increasing or decreasing: if it’s increasing then the second derivative is positive, and if it’s decreasing then the second derivative is negative.
o Use a dotted line where it’s above and below the axis: when it’s above then the original is increasing and when it’s below then the original is decreasing.
o Use solid dots on MAX’s and MIN’s then the second derivative is equal to zero and it’s a point of inflection on the derivative.
o Then put open circles on the x intercepts: when the MAX is above the axis then it’s below for the second derivative, and when the MIN is below the axis then it’s above for the second derivative.
So I know that I understand the concept perfectly fine and it’s really simple, it’s just that looking at a graph confuses me so I would need more practice to look at the graph and figure out what is what. I didn’t do so well on my exam so it looks like I’ll be reviewing a lot over the break so I can catch up. THANK GOD IT’S THE HOLIDAYS.
~ElliE~
Friday, December 18, 2009
Tuesday, December 15, 2009
post seventeen
Ok so this week we were preparing for log one and for our exam.
Some things that i understand in calculus include:
Tangent Line:
1. If you are only given a x value, plug into your equation to find the y value.
2. take the derivative of the equation
3. Plug in your x value into the derivative to find the slope
4. Put in point slope
The formula for the volume of disks is S (top)^2 - (bottom)^2 dx
The formula for the area of washers is S (top) - (bottom)
STEPS:
1. Draw the graphs of the equations
2. Subtract top graph's equation by the bottom graph's equation(in disks each equation would be squared)
3. Set equations equal and solve for x to find bounds
4. Plug in the bounds and the outcome of step 2
5. Integrate
These are fairly simple processes and equations to work with. For what i dont know is linierization. I dont even have notes on this
Some things that i understand in calculus include:
Tangent Line:
1. If you are only given a x value, plug into your equation to find the y value.
2. take the derivative of the equation
3. Plug in your x value into the derivative to find the slope
4. Put in point slope
The formula for the volume of disks is S (top)^2 - (bottom)^2 dx
The formula for the area of washers is S (top) - (bottom)
STEPS:
1. Draw the graphs of the equations
2. Subtract top graph's equation by the bottom graph's equation(in disks each equation would be squared)
3. Set equations equal and solve for x to find bounds
4. Plug in the bounds and the outcome of step 2
5. Integrate
These are fairly simple processes and equations to work with. For what i dont know is linierization. I dont even have notes on this
Sunday, December 13, 2009
Post #17
Last week in calculus was review week. We didn't learn anything new so I'm going to review some of the old stuff because it will be on our exam.
Tangent Line:
1. If you are only given a x value, plug into your equation to find the y value.
2. take the derivative of the equation
3. Plug in your x value into the derivative to find the slope
4. Put in point slope
EXAMPLE: y=3x^3 + 4x^2 +5 at x=1
1. 3(1)^3+4(1)^2+5 = 12
2. 9x^2 + 8x
3. 9(1)^2 + 8(1) = 17
4. y-12 = 17 (x-1)
Normal line is the same except you take the negative reciprocal of the slope to plug in.
y-12 = -1/17 (x-1)
Rolle's Theorem:
In order to use Rolle's theorem, the fuction must be continuous and differentiable on the interval given and f(a) = f(b)
Example:
-x^2+14x on [0,14]
The function is continuous and differentiable
f(0) = -(0)^2 + 14(0) = 0
f(14) = -(14)^2 + 14(14) = 0
Therefore Rolle's applies
Now take the derivative: -2x+14
and set equal to zero: -2x+14 = 0
solve for x: x=7 or c=7
Mean value theorem:
The function again has to be continuous and differentiable at the interval given and you plug into f(b) - f(a) / b-a
Example: x^2 on [1,7]
The function is continuous and differentiable
First take the derivative: 2x
Then plug into the formula: f(7) - f(1) / 7-1
14-2/ 7-1 = 12/6 which equals 2.
I am still having troubles with graphs and going from one graph to another. I don't understand how to find increasing/decreasing, concave up/down, maxs, mins, and points of inflection. Help please!
17th post
alright, we just reviewed this week, so i am going to go over integration. Integration is pretty much the opposite of a derivative.
there are two types of integration, definite integration and indefinite integration.
indefinite integration is easy, you just take the equation, and do this to it, lets say the equation is x^3+2x^2
to take the integral of something, first you have to raise the exponent by one, so lets take the first part of the problem, x^3, and do that to it. it becomes x^4. Now, you just multiple the whole thing by the inverse of the new exponent. so it would become (1/4)x^4.
then you do the same thing to the second part of the problem. 2x^2 becomes 2x^3, then you have to multiply, and it becomes (2/3)x^3
then you put the two parts back together like normal, and it becomes (1/4)x^4+(2/3)x^3.
but we are not done yet!! for indefinite integrals, you always have to add C at the end of the equation, so your final answer becomes: (1/4)x^4+(2/3)x^3+C
and for definite integrals, you just do the same thing, but for definite integrals, they give you a point, and you just plug in the x value from the point, and you do not add the C at the end!!! so lets just use the example problem from indefinite integration, but they'll also give you a point, let's say (2,2) for example. You would integrate it normally, and get (1/4)x^4+(2/3)x^3, then you plug in your x value, which is 2. so then you get (1/4)(2)^4+(2/3)(2)^3, and after that you can just put it into your calculator, which I do not have with me, and you'll just get some number as your answer. So for definite integration, you'll get a a number answer, and for indefinite integration, you'll get an equation.
there are two types of integration, definite integration and indefinite integration.
indefinite integration is easy, you just take the equation, and do this to it, lets say the equation is x^3+2x^2
to take the integral of something, first you have to raise the exponent by one, so lets take the first part of the problem, x^3, and do that to it. it becomes x^4. Now, you just multiple the whole thing by the inverse of the new exponent. so it would become (1/4)x^4.
then you do the same thing to the second part of the problem. 2x^2 becomes 2x^3, then you have to multiply, and it becomes (2/3)x^3
then you put the two parts back together like normal, and it becomes (1/4)x^4+(2/3)x^3.
but we are not done yet!! for indefinite integrals, you always have to add C at the end of the equation, so your final answer becomes: (1/4)x^4+(2/3)x^3+C
and for definite integrals, you just do the same thing, but for definite integrals, they give you a point, and you just plug in the x value from the point, and you do not add the C at the end!!! so lets just use the example problem from indefinite integration, but they'll also give you a point, let's say (2,2) for example. You would integrate it normally, and get (1/4)x^4+(2/3)x^3, then you plug in your x value, which is 2. so then you get (1/4)(2)^4+(2/3)(2)^3, and after that you can just put it into your calculator, which I do not have with me, and you'll just get some number as your answer. So for definite integration, you'll get a a number answer, and for indefinite integration, you'll get an equation.
post #17
Okay so almost everything we did this week was basically in preparation for the exam on tuesdayyy, so i'm just going to go over some things from the past as a review.
Some things that we just learned that are extremely easy form me are finding the volume and area of disks and washers. atleast i know i'll pass some portion of the exams, assuming ofcourse, that they're even on there.
Area between curves:
formula: bSa top equation-bottom equation
example: Find the area of the region enclosed by y=-4x^2+41x+94 and y=x-2 between x=1 and x=7.
1. draw a picture
2. subtract the two equations: (-4x^2+41x+94-(x-2)-don't square, it's not volumee!
(-4x^2+41x+94-(-x+2)
3.combine like terms and integrate: S(-4x^2+41x+94-(-x+2))
(-4/3)x^3+20x^2+96x
4. plug in 1 and 6 and subtract: (3584/3)-(344/3)= 1080
FORMULA FOR DISKS: pi S[r(x)]^2dx, this formula is almost exactly the same as when you are asked to find the area of a disk, however, while finding the area you do not square the radius (the given equation).
FORMULA FOR WASHERS: pi S[(top)^2-(bottom)^2]dx
TANGENT LINES!
Steps:
Take the first der...f ‘(x).
Plug in x to find your slope: m.
Plug x into original functiong..f(x) to find y (if not already given).
Using your slope..m and (x,y), plug it into slope-intercept form: (y-y1) = m(x-x1)
okayy something i don't understandd! umm i really don't get how to integrate a fraction, which would be integration by substitution i'm assumine. also some things from the past that would be nice to go over is ofcourse optimization, related rates type stuff. other than that, i think i'm okayy, maybe trapezoidal rule but that's about it! help would be WONDERFULL!
Some things that we just learned that are extremely easy form me are finding the volume and area of disks and washers. atleast i know i'll pass some portion of the exams, assuming ofcourse, that they're even on there.
Area between curves:
formula: bSa top equation-bottom equation
example: Find the area of the region enclosed by y=-4x^2+41x+94 and y=x-2 between x=1 and x=7.
1. draw a picture
2. subtract the two equations: (-4x^2+41x+94-(x-2)-don't square, it's not volumee!
(-4x^2+41x+94-(-x+2)
3.combine like terms and integrate: S(-4x^2+41x+94-(-x+2))
(-4/3)x^3+20x^2+96x
4. plug in 1 and 6 and subtract: (3584/3)-(344/3)= 1080
FORMULA FOR DISKS: pi S[r(x)]^2dx, this formula is almost exactly the same as when you are asked to find the area of a disk, however, while finding the area you do not square the radius (the given equation).
FORMULA FOR WASHERS: pi S[(top)^2-(bottom)^2]dx
TANGENT LINES!
Steps:
Take the first der...f ‘(x).
Plug in x to find your slope: m.
Plug x into original functiong..f(x) to find y (if not already given).
Using your slope..m and (x,y), plug it into slope-intercept form: (y-y1) = m(x-x1)
okayy something i don't understandd! umm i really don't get how to integrate a fraction, which would be integration by substitution i'm assumine. also some things from the past that would be nice to go over is ofcourse optimization, related rates type stuff. other than that, i think i'm okayy, maybe trapezoidal rule but that's about it! help would be WONDERFULL!
Ash's 17th Blog
Wow
Only one more week until we can finally be freed from school for two entire weeks! I'm so excited!!
This week we spent...doesn't stuff? I don't actually remember...so let's skip to Friday
Friday: Beast packet :D
Saturday: ACT + Subway (omg $5 6inch..YUM!) + PJs + Walmart (1.5 hours, with Mal's mom to get EIGHT things...) + New Orleans Hamburger (YUM! SEAFOOD!) + Movies (standing outside trying to decide what movie to watch, New Moon or the Blindside...for...thirty...minutes) + NEW MOON! FINALLY! + Calculus packet again
Sunday: Calculus packet + English packet + Tic Tac Toe + Calculus packet..againn
So..yea...that's all I have to say :)
Hmm...what do I know that I can explain?
Umm...Riemann Sums?
I'm pretty sure that's way over-done, but I have no idea what else to explain! =/
Delta = b-a/n (top number above S minus the bottom number below S divided by the number of intervals)
LRAM: delta(f(a)+f(a+delta)+f(a+2delta)...+f(b-delta))
RRAM: delta(f(a+delta)+f(a+2delta)+...f(b))
MRAM: delta(f(mid)+f(mid)...)
Trapezoidal: delta(x)/2 [ f(a) + 2f(a + delta x) + 2f(a + 2deltax)
+ f(b))
Just don't forget to not go over the amount of intervals you have (n).
Now, I've got major questions
1. Trapezoidal graphs...where can I get the software?
2. What's deltax? is that the same thing as delta..???
3. GRAPHING! AHHH!!
4. Also, for limits, are they asking for the open circle or the closed circle? I always get confused! And each time I change it, I get it wrong..Help!!!
5. Another thing for limits, what do you do if it's over zero? I remember something about it...did she or Alex go over that? Anyway, I can't find those notes and I didn't get it before...=/
Okay, this is my extreme problem:
I've never been able to remember the steps for math
I can do it if I know what I'm doing (makes total sense right?)
But I can't remember the steps
Does ANYONE have a way to do remember?? =/
Only one more week until we can finally be freed from school for two entire weeks! I'm so excited!!
This week we spent...doesn't stuff? I don't actually remember...so let's skip to Friday
Friday: Beast packet :D
Saturday: ACT + Subway (omg $5 6inch..YUM!) + PJs + Walmart (1.5 hours, with Mal's mom to get EIGHT things...) + New Orleans Hamburger (YUM! SEAFOOD!) + Movies (standing outside trying to decide what movie to watch, New Moon or the Blindside...for...thirty...minutes) + NEW MOON! FINALLY! + Calculus packet again
Sunday: Calculus packet + English packet + Tic Tac Toe + Calculus packet..againn
So..yea...that's all I have to say :)
Hmm...what do I know that I can explain?
Umm...Riemann Sums?
I'm pretty sure that's way over-done, but I have no idea what else to explain! =/
Delta = b-a/n (top number above S minus the bottom number below S divided by the number of intervals)
LRAM: delta(f(a)+f(a+delta)+f(a+2delta)...+f(b-delta))
RRAM: delta(f(a+delta)+f(a+2delta)+...f(b))
MRAM: delta(f(mid)+f(mid)...)
Trapezoidal: delta(x)/2 [ f(a) + 2f(a + delta x) + 2f(a + 2deltax)
+ f(b))
Just don't forget to not go over the amount of intervals you have (n).
Now, I've got major questions
1. Trapezoidal graphs...where can I get the software?
2. What's deltax? is that the same thing as delta..???
3. GRAPHING! AHHH!!
4. Also, for limits, are they asking for the open circle or the closed circle? I always get confused! And each time I change it, I get it wrong..Help!!!
5. Another thing for limits, what do you do if it's over zero? I remember something about it...did she or Alex go over that? Anyway, I can't find those notes and I didn't get it before...=/
Okay, this is my extreme problem:
I've never been able to remember the steps for math
I can do it if I know what I'm doing (makes total sense right?)
But I can't remember the steps
Does ANYONE have a way to do remember?? =/
Posting...#17
Last week in calculus we did not learn anything new. We took a practic ap monday and wensday then we went over it for tuesday and thursday.
Rolle's Theorem: Let f be continuous on a closed interval [a,b] and differentiable on the open interval (a,b) then there exsists a number c in (a,b) such that
f(b)-f(a)
f ' (c)=----------
b-a
Exampmle: F(x)=x^4 -2x^2 on [-2,2] find c f '(c)=0
Check if its continuous : YES
Check if its differentiable : YES
f(-2)=(-2)^4-2(-2)^2=8
F(2)=(2)^4-2(2)^2=8
f(-2)=f(2)
4x^3-4x=0
4x(x^2-1)=0
X=0,1,-1
Awnser C= -1,0,1
And thats it for roles theroem its pretty easy but for what im having trouble with is even easier so it would be great to get help.
I am confused on Critical Points so can some one explain how to do that for me.
Rolle's Theorem: Let f be continuous on a closed interval [a,b] and differentiable on the open interval (a,b) then there exsists a number c in (a,b) such that
f(b)-f(a)
f ' (c)=----------
b-a
Exampmle: F(x)=x^4 -2x^2 on [-2,2] find c f '(c)=0
Check if its continuous : YES
Check if its differentiable : YES
f(-2)=(-2)^4-2(-2)^2=8
F(2)=(2)^4-2(2)^2=8
f(-2)=f(2)
4x^3-4x=0
4x(x^2-1)=0
X=0,1,-1
Awnser C= -1,0,1
And thats it for roles theroem its pretty easy but for what im having trouble with is even easier so it would be great to get help.
I am confused on Critical Points so can some one explain how to do that for me.
Post Number Seventeen
this week we reviewed for our exam, so i'll just go over stuff that i know and do not know as my blog.
Let's start this off on a good note and explain something i DO understand. Average value :)
The formula for average value is 1/b-a(definite integral)f(a)
For example, Find the average value of f(x) = x^2 on [0,5]
1/5-0 S from 0 to 5 x^2 dx
= 1/5 [1/3x^3]
=1/5[125/3 - 0] = 25/3
Now for something i still don't understand and never did: SUBSTITUTION.
help please?
I also understand rolles and mean value theorems, you just follow them and plug in :)
This may be sad, but i am having trouble finding critical numbers and absolute extrema, those kinds of problems. I need a slight review on this if anyone cares.
I know how to do RRAM and LRAM, but am still having trouble with MRAM and mostly TRAM..
One more thing i understand is volume by disks, so i'll explain that quickly.
Volume by disks is used only with solid objects. The formula used is pi definite integral [R(x)]^2 dx
This is pretty easy, everything is basically given to you, the only place i usually mess up is in the algebra itself.
Well i'm stressing about the exam but hey, what can i do?
Goodnight Everyone :)
Let's start this off on a good note and explain something i DO understand. Average value :)
The formula for average value is 1/b-a(definite integral)f(a)
For example, Find the average value of f(x) = x^2 on [0,5]
1/5-0 S from 0 to 5 x^2 dx
= 1/5 [1/3x^3]
=1/5[125/3 - 0] = 25/3
Now for something i still don't understand and never did: SUBSTITUTION.
help please?
I also understand rolles and mean value theorems, you just follow them and plug in :)
This may be sad, but i am having trouble finding critical numbers and absolute extrema, those kinds of problems. I need a slight review on this if anyone cares.
I know how to do RRAM and LRAM, but am still having trouble with MRAM and mostly TRAM..
One more thing i understand is volume by disks, so i'll explain that quickly.
Volume by disks is used only with solid objects. The formula used is pi definite integral [R(x)]^2 dx
This is pretty easy, everything is basically given to you, the only place i usually mess up is in the algebra itself.
Well i'm stressing about the exam but hey, what can i do?
Goodnight Everyone :)
Post #17
Calculus Week #17
Anyway so we are supposed to explain a concept and ask a question or two about another. Well to be quite honest, this week I actually learned something I should have known a long time ago.
Trapezoidal!
The formula is really simple and easy to remember...anyway, it's
Let x = (b-a)/n
x/2 [f(a) + 2f(a+x) + 2f(a+2x) ... f(b)]
This beats trying to find the actual area under the curve and wondering which one of the answer choices might be close. Lol...
Anyway, let's see...what else can I explain.
Okay a crash course to help you all on the take-home test:
If it asks for a tangent line, take the derivative, plug in x, get your slope, use that in point-slope form with the point you had with that slope.
If it asks for critical values...set derivative equal to 0 and solve for x.
Make sure to test end points.
Relative maximums or minimums are when you have multiple maxs and mins. The relative ones are the ones that are not absolute. The ones that are absolute are the ones that have either the highest y value or lowest y value.
Remember, if you are allowed in calculator, you can check your integration.
fnInt((equation),x,a,b) for f(x) = (equation) on the interval [a,b].
Or just plug it into y= on the plot then go to SECOND, CALC and go to the last option. Then type in a press ENTER then press b and press ENTER. Voila, a graphed picture of the area under your curve.
For the area between two curves, you do the top equation minus bottom equation. If you have no a,b to plug in, set the two equations equal and solve. Use that as your lower and upper bounds.
For volume by discs of a solid rotated about an axis, do the definite integral from a to b of the equation squared...times pi. Pretty simple.
For volume by washers, it's the same thing as area between two curves...except this time you square both equations as well, as in the above example.
Average value is 1/(b-a) times the definite integral from a to b of the function.
Average rate of change is a slope which is a derivative.
Derivative of position is velocity, derivative of velocity is acceleration.
So first derivative is velocity, second derivative is acceleration.
Let's see what else I can cover...
If they give you acceleration and you take the integral to find the velocity equation, don't forget you need to solve for +c. Usually they will give you some information such as v(1) = 1 would be used to plug in and solve for c.
Anyway, I think I gave a pretty decent crash course for the take home test. Don't forget to use your calculator...you can check almost all of the problems in your calculator so you might as well do it!
Anyway so we are supposed to explain a concept and ask a question or two about another. Well to be quite honest, this week I actually learned something I should have known a long time ago.
Trapezoidal!
The formula is really simple and easy to remember...anyway, it's
Let x = (b-a)/n
x/2 [f(a) + 2f(a+x) + 2f(a+2x) ... f(b)]
This beats trying to find the actual area under the curve and wondering which one of the answer choices might be close. Lol...
Anyway, let's see...what else can I explain.
Okay a crash course to help you all on the take-home test:
If it asks for a tangent line, take the derivative, plug in x, get your slope, use that in point-slope form with the point you had with that slope.
If it asks for critical values...set derivative equal to 0 and solve for x.
Make sure to test end points.
Relative maximums or minimums are when you have multiple maxs and mins. The relative ones are the ones that are not absolute. The ones that are absolute are the ones that have either the highest y value or lowest y value.
Remember, if you are allowed in calculator, you can check your integration.
fnInt((equation),x,a,b) for f(x) = (equation) on the interval [a,b].
Or just plug it into y= on the plot then go to SECOND, CALC and go to the last option. Then type in a press ENTER then press b and press ENTER. Voila, a graphed picture of the area under your curve.
For the area between two curves, you do the top equation minus bottom equation. If you have no a,b to plug in, set the two equations equal and solve. Use that as your lower and upper bounds.
For volume by discs of a solid rotated about an axis, do the definite integral from a to b of the equation squared...times pi. Pretty simple.
For volume by washers, it's the same thing as area between two curves...except this time you square both equations as well, as in the above example.
Average value is 1/(b-a) times the definite integral from a to b of the function.
Average rate of change is a slope which is a derivative.
Derivative of position is velocity, derivative of velocity is acceleration.
So first derivative is velocity, second derivative is acceleration.
Let's see what else I can cover...
If they give you acceleration and you take the integral to find the velocity equation, don't forget you need to solve for +c. Usually they will give you some information such as v(1) = 1 would be used to plug in and solve for c.
Anyway, I think I gave a pretty decent crash course for the take home test. Don't forget to use your calculator...you can check almost all of the problems in your calculator so you might as well do it!
Post 17
Ok, so last week we pretty much just reviewed for the upcoming exams. Mrs. Robinson gave us two practice APs, which I failed. They mostly went over things we learned recently like integrals. The first one was much much better than the second one, which is pretty pathetic considering the second one was calculator allowed. We were also given the take home portion of the exam. This portion is only twenty multiple choice questions and is also calculator allowed. The one we will take on exam did will include multiple choice and essay, both of which calculators will not be allowed. Today I was going through the take home portion and I realized I forgot a lot of things we did in the beginning of the year. I thought I knew how to do some problems, but my answers were coming out wrong.
Anyway, two weeks ago we learned how to find volume of a curve. This is done by rotating the curve about an axis or line to make it a solid object. We did this by using both disks and washers. Disks are used when there is only one equation and the object rotated turns out to be a solid. When there are two equations, washers are used since there will be a hole in the figure.
The formula for disks is:
pi S a-b [R(x)]^2 dx
This is basically a definite integral where you plug in the function you are given and square it. Many people forget to square it. This completley changes everything, so don't forget.
The formula for washers is:
pi S a-b top^2-bottom^2 dx
This is also a definite integral where you have to square the functions. I like these because they're very similar to area under a curve, and I also liked those. The only difference is to square both functions given and put a pi on the outside of the integral.
Most of the things we learned recently I'm fine with, but I came to realize I'm confused with a lot of old things. I get average value, average rate of change, and other various things with those words arranged in different ways. I never really caught onto average rate of change, and I still don't know what it is. I also get confused with linerizaton. When I look at the questions, I never know what to do with them. Can someone explain?
Anyway, two weeks ago we learned how to find volume of a curve. This is done by rotating the curve about an axis or line to make it a solid object. We did this by using both disks and washers. Disks are used when there is only one equation and the object rotated turns out to be a solid. When there are two equations, washers are used since there will be a hole in the figure.
The formula for disks is:
pi S a-b [R(x)]^2 dx
This is basically a definite integral where you plug in the function you are given and square it. Many people forget to square it. This completley changes everything, so don't forget.
The formula for washers is:
pi S a-b top^2-bottom^2 dx
This is also a definite integral where you have to square the functions. I like these because they're very similar to area under a curve, and I also liked those. The only difference is to square both functions given and put a pi on the outside of the integral.
Most of the things we learned recently I'm fine with, but I came to realize I'm confused with a lot of old things. I get average value, average rate of change, and other various things with those words arranged in different ways. I never really caught onto average rate of change, and I still don't know what it is. I also get confused with linerizaton. When I look at the questions, I never know what to do with them. Can someone explain?
post 17
This week was mainly a review week in order to get ready for the upcoming midterm exam. We reviewed everything that we learned throughout this year.
Another thing I am going to talk about since I understand is linearization. The steps for working linearization problems are:
1. Identify the equation
2. Use the formula f(x)+f ' (x)dx
3. Determine your dx in the problem
4. Then determine your x in the problem
5. Plug in everything you get
6. Solve the equation
Also I will talk about integration. Integration is used to find the area under a curve. The Riemann sum approximates the area using the rectangles or trapezoids. The Riemanns Sums are:
LRAM-Left hand approximation=delta x[f(a)+f(a+delta x)+...f(b-delta x)]
RRAM-Right hand approximation=delta x[f(a+delta x)+...f(b)]
MRAM-Middle approximation=delta x[f(mid)+f(mid)+...]
Trapezoidal-delta x/2[f(a)+2f(a+delta x)+2f(a+2 delta x)+...f(b)]
*delta x=b-a/number of subintervals
Related rates are one thing I understood fromt the start so I am going to talk about them next. The steps for related rates are:
1. Identify all of the variables and equations
2. Identify the things that you are looking for
3. Sketch a graph and then label that graph
4. Create and write an equation using all of the variables
5. Take the derivative of this equation with respect to time
6. Substitute everything back in
7. Solve the equation
Also I am going to talk about taking implicit derivatives since I understand how to take them. The steps for taking implicit derivatives are:
1. Take the derivative of both sides like you would normally do
2. Everytime the derivative of y is taken it needs to be notated with either y ' or dy/dx
3. Solve for dy/dx or y ' as if you are solving for x.
For things I am not good with is the trapezoidal rule. I get LRAM, RRAM, and MRAM but I just can not get trapezoidal. So if someone can help me before the test that would be great.
Another thing I am going to talk about since I understand is linearization. The steps for working linearization problems are:
1. Identify the equation
2. Use the formula f(x)+f ' (x)dx
3. Determine your dx in the problem
4. Then determine your x in the problem
5. Plug in everything you get
6. Solve the equation
Also I will talk about integration. Integration is used to find the area under a curve. The Riemann sum approximates the area using the rectangles or trapezoids. The Riemanns Sums are:
LRAM-Left hand approximation=delta x[f(a)+f(a+delta x)+...f(b-delta x)]
RRAM-Right hand approximation=delta x[f(a+delta x)+...f(b)]
MRAM-Middle approximation=delta x[f(mid)+f(mid)+...]
Trapezoidal-delta x/2[f(a)+2f(a+delta x)+2f(a+2 delta x)+...f(b)]
*delta x=b-a/number of subintervals
Related rates are one thing I understood fromt the start so I am going to talk about them next. The steps for related rates are:
1. Identify all of the variables and equations
2. Identify the things that you are looking for
3. Sketch a graph and then label that graph
4. Create and write an equation using all of the variables
5. Take the derivative of this equation with respect to time
6. Substitute everything back in
7. Solve the equation
Also I am going to talk about taking implicit derivatives since I understand how to take them. The steps for taking implicit derivatives are:
1. Take the derivative of both sides like you would normally do
2. Everytime the derivative of y is taken it needs to be notated with either y ' or dy/dx
3. Solve for dy/dx or y ' as if you are solving for x.
For things I am not good with is the trapezoidal rule. I get LRAM, RRAM, and MRAM but I just can not get trapezoidal. So if someone can help me before the test that would be great.
post 17
we reviewed mostly this past week so ima show you the formulas andd steps for washers and disks.
The formula for the volume of disks is S (top)^2 - (bottom)^2 dx
The formula for the area of washers is S (top) - (bottom)
STEPS:
1. Draw the graphs of the equations
2. Subtract top graph's equation by the bottom graph's equation(in disks each equation would be squared)
3. Set equations equal and solve for x to find bounds
4. Plug in the bounds and the outcome of step 2
5. Integrate
volume by disks:
the formula is pi times the integral of the [function given] squared times dx. so just solve it by taking the integral of it and then pluging in the numbers they give you. just like before you'll have two numbers so whatever the answer is for the top one will be first and then you subtract the answer you get for the bottom one. then graph
volume by washers:
the formla is pie times the integral of the [top function] squared minus the [bottom function] squared times dx. so to do this, if you don't have the in between number you have to set the functions equal, but if you do, then it's worked the same way as above. square the formula's that were given and simplify. then take the integral of it and plug in the numbers they give you or you found by setting the formulas equal to each other and then solve like any other one by subracting them. then graph.
im not too good at related rates, optimization, and angles of elevevation still.
The formula for the volume of disks is S (top)^2 - (bottom)^2 dx
The formula for the area of washers is S (top) - (bottom)
STEPS:
1. Draw the graphs of the equations
2. Subtract top graph's equation by the bottom graph's equation(in disks each equation would be squared)
3. Set equations equal and solve for x to find bounds
4. Plug in the bounds and the outcome of step 2
5. Integrate
volume by disks:
the formula is pi times the integral of the [function given] squared times dx. so just solve it by taking the integral of it and then pluging in the numbers they give you. just like before you'll have two numbers so whatever the answer is for the top one will be first and then you subtract the answer you get for the bottom one. then graph
volume by washers:
the formla is pie times the integral of the [top function] squared minus the [bottom function] squared times dx. so to do this, if you don't have the in between number you have to set the functions equal, but if you do, then it's worked the same way as above. square the formula's that were given and simplify. then take the integral of it and plug in the numbers they give you or you found by setting the formulas equal to each other and then solve like any other one by subracting them. then graph.
im not too good at related rates, optimization, and angles of elevevation still.
post 17
alright so this week in calc we reviewed, reviewed, reviewed, and took two practice ap tests, which i actually did pretty good on ! :-)
calc exam is tuesday, and i just found out that the packet b rob gave us on friday is due the day of the exam! i thought it was our packet for over the christmas holidays! ahhhhh, i need to get started on that. haha. okay well, i'll explain average rate of change.
you are given a problem y=3t+2 and a time iterval from [0,2] seconds. well, your formula is:
f(b)-f(a)/b-a. 0=a,2=b. so, first you must plug in 0 and 2 to find out what your plug ins will be. so you would get [2,8]. then plug it into your formula, 6/2 = 3. so your answer would be three. don't forget to put your units behind it though, whatever that may be.
i'll also, tell you limit rules for finding infinity because i will never ever ever forget these.
if a limit is approaching infinity:
1. top degree < bottom degree it's equal to 0
2. top degree > bottom degree it's equal to +/- infinity. ****
3. top degree = bottom degree divide leading coefficients.
****find out if it's positive or negative infinity by graphing it in your calculator.
also, i'll throw in that volume for disks and washers is squared, while area is not. although it is the exact same formula. figured i would just say that since a lot of people seem to forget those formulas, when really you only have to remember one.
what i dont get still, is LRAM, MRAM, RRAM, TRAM.
also i kinda forgot how to find the equation of a tangent line. don't make funny of me, but i can't remember to save my life and it's driving me crazy cuz i know how simple it is. also, if anyone wants to go over optimization just to refresh my memory, and all the related rates stuff would be nice too :-)
calc exam is tuesday, and i just found out that the packet b rob gave us on friday is due the day of the exam! i thought it was our packet for over the christmas holidays! ahhhhh, i need to get started on that. haha. okay well, i'll explain average rate of change.
you are given a problem y=3t+2 and a time iterval from [0,2] seconds. well, your formula is:
f(b)-f(a)/b-a. 0=a,2=b. so, first you must plug in 0 and 2 to find out what your plug ins will be. so you would get [2,8]. then plug it into your formula, 6/2 = 3. so your answer would be three. don't forget to put your units behind it though, whatever that may be.
i'll also, tell you limit rules for finding infinity because i will never ever ever forget these.
if a limit is approaching infinity:
1. top degree < bottom degree it's equal to 0
2. top degree > bottom degree it's equal to +/- infinity. ****
3. top degree = bottom degree divide leading coefficients.
****find out if it's positive or negative infinity by graphing it in your calculator.
also, i'll throw in that volume for disks and washers is squared, while area is not. although it is the exact same formula. figured i would just say that since a lot of people seem to forget those formulas, when really you only have to remember one.
what i dont get still, is LRAM, MRAM, RRAM, TRAM.
also i kinda forgot how to find the equation of a tangent line. don't make funny of me, but i can't remember to save my life and it's driving me crazy cuz i know how simple it is. also, if anyone wants to go over optimization just to refresh my memory, and all the related rates stuff would be nice too :-)
WEEK 17
Since we took the two practice AP exams this week, I think I'll just explain some problems off of them.
Number 5: If y = e^2x + tan(2x), then y'(pi) =
first take the derivative: y' = e^2x (2) + sec^2(2x)(2)
Then you plug in pi: 2e^2pi + 2sec(2pi)^2 = 2e^2pi + 2
Number 16: If y = sin^2(5x), dy/dx =
dy/dx = 2 (sin 5x)(cos 5x)(5) <--- take care of exponent, take derivative of sin, take derivative of the inside (5x).
dy/dx = 5 (sin(2 * 5x)) <--- 2sinxcosx = sin2x
dy/dx = 5sin(10x)
Number 18:
remember that f(x) = x
f'(x) = v(x)
f''(x) = a(x)
Number 20: If f(1) = 2 and f'(1) = 5, use the equation of the line tangent to the graph of f at x=1 to approximate f(1.2).
Given: x = 1
y = 2
m = 5
find: x = 1.2
y - 2 = 5 (x - 1)
y - 2 = 5 (1.2 - 1)
y = 3
Number 27:
Always remember that AVERAGE VALUE = INTEGRATION
1/ b-a
Number 28: y = x^2 - 3x. Find y'(1).
y'(x) = 2x - 3
y'(1) = 2(1) - 3
y'(1) = -1
y'(1) = 1
Number 11:
remember that average rate of change = f(b) - f(a) / b - a
Number 5: If y = e^2x + tan(2x), then y'(pi) =
first take the derivative: y' = e^2x (2) + sec^2(2x)(2)
Then you plug in pi: 2e^2pi + 2sec(2pi)^2 = 2e^2pi + 2
Number 16: If y = sin^2(5x), dy/dx =
dy/dx = 2 (sin 5x)(cos 5x)(5) <--- take care of exponent, take derivative of sin, take derivative of the inside (5x).
dy/dx = 5 (sin(2 * 5x)) <--- 2sinxcosx = sin2x
dy/dx = 5sin(10x)
Number 18:
remember that f(x) = x
f'(x) = v(x)
f''(x) = a(x)
Number 20: If f(1) = 2 and f'(1) = 5, use the equation of the line tangent to the graph of f at x=1 to approximate f(1.2).
Given: x = 1
y = 2
m = 5
find: x = 1.2
y - 2 = 5 (x - 1)
y - 2 = 5 (1.2 - 1)
y = 3
Number 27:
Always remember that AVERAGE VALUE = INTEGRATION
1/ b-a
Number 28: y = x^2 - 3x. Find y'(1).
y'(x) = 2x - 3
y'(1) = 2(1) - 3
y'(1) = -1
y'(1) = 1
Number 11:
remember that average rate of change = f(b) - f(a) / b - a
Post #17
So after 11 make-up posts I ran out of stuff to say, so i'm going back to the beginning of the year and talking about some stuff.
Complex Derivatives y=ln(e^x) (Chain Rule)
First off one should should identify the steps of your problem. In this case they would be:
1. Natural Log
2. e^x
you problem should be (1/(e^x)).(e^x)'
then you find the derivative of e^x which is e^x . x' (x'=1)
so your final problem should be (1/(e^x)).(e^x)
After this you have to simplify algebraically, giving you (e^x)/(e^x) ,which equals 1.
First Derivative Test:
1. Take the derivative of the original problem.
2. Set the first derivative equal to Zero.
3. Solve for x.
4. Create intervals for x. i.e. (-∞, 1) (1, 4) (4, ∞)
5. Pick a number in the intervals then plug that number in the first derivative for x.
6. Solve. For positive numbers, the graph of the derivative is above the x-axis. For negative numbers, the graph of the derivative is below the x-axis. The numbers for x are your points of inflection. (Points of Inflection are only if there is a shift in the graph!!!)
Second Derivative Test:
1. Take the derivative of the first derivative.
2. Set the second derivative equal to Zero.
3. Solve for x.
4. Create intervals for x. i.e. (-∞, 1) (1, 4) (4, ∞)
5. Pick a number in the intervals then plug that number in the second derivative for x.
6. Solve. For positive numbers, the graph of the derivative is above the x-axis. For negative numbers, the graph of the derivative is below the x-axis. The numbers for x are your points of inflection. (Points of Inflection are only if there is a shift in the graph!!!)
Complex Derivatives y=ln(e^x) (Chain Rule)
First off one should should identify the steps of your problem. In this case they would be:
1. Natural Log
2. e^x
you problem should be (1/(e^x)).(e^x)'
then you find the derivative of e^x which is e^x . x' (x'=1)
so your final problem should be (1/(e^x)).(e^x)
After this you have to simplify algebraically, giving you (e^x)/(e^x) ,which equals 1.
First Derivative Test:
1. Take the derivative of the original problem.
2. Set the first derivative equal to Zero.
3. Solve for x.
4. Create intervals for x. i.e. (-∞, 1) (1, 4) (4, ∞)
5. Pick a number in the intervals then plug that number in the first derivative for x.
6. Solve. For positive numbers, the graph of the derivative is above the x-axis. For negative numbers, the graph of the derivative is below the x-axis. The numbers for x are your points of inflection. (Points of Inflection are only if there is a shift in the graph!!!)
Second Derivative Test:
1. Take the derivative of the first derivative.
2. Set the second derivative equal to Zero.
3. Solve for x.
4. Create intervals for x. i.e. (-∞, 1) (1, 4) (4, ∞)
5. Pick a number in the intervals then plug that number in the second derivative for x.
6. Solve. For positive numbers, the graph of the derivative is above the x-axis. For negative numbers, the graph of the derivative is below the x-axis. The numbers for x are your points of inflection. (Points of Inflection are only if there is a shift in the graph!!!)
Post #17
Alrighty then, lets get started.
Today during our study group, I learned average value thanks to Kaitlyn! So I guess I will explain an example of that since I finally get it.
EX: Find the average value of the function f(x)=12x-6x^2 over the interval -5 less than or equal to x less than or greater than 5.
use the formula: 1/b-a S equation given, then solve the definite integral (*S is integral)
1/5-(-5)= 1/10
1/10 (integral form -5 to 5) 12x-6x^2
1/10(12x-2x^3)
1/10(12(5)-2(5)^3)-1/10(12(-5)-2(-5)^3)
1/10(-190)-1/10(190)
-19-(-19)= -38
I do have quite a few questions though..
1. If I get (-1/64)sin(1/8) and I'm suppose to set it equal to 0 then solve..How do I solve that?
2. (like #11 on the packet) For S(3t/t^2+2)^3 I used substitution to solve. For my final answer I got -3/2(t^2+2)^2+c but all the answers have -3/4(t^2+2)^2+c
3. (like #3 on the packet) It says use a graphing utility to graph the function f(x)=12/6-x and locate the absolute extrema of the function on the interval [0,6). If you plug that into your calculator the graph is an asymptote, and I tried taking the derivative and plugging that in too but it is also an asymptote. So how do you get a max and min?
Today during our study group, I learned average value thanks to Kaitlyn! So I guess I will explain an example of that since I finally get it.
EX: Find the average value of the function f(x)=12x-6x^2 over the interval -5 less than or equal to x less than or greater than 5.
use the formula: 1/b-a S equation given, then solve the definite integral (*S is integral)
1/5-(-5)= 1/10
1/10 (integral form -5 to 5) 12x-6x^2
1/10(12x-2x^3)
1/10(12(5)-2(5)^3)-1/10(12(-5)-2(-5)^3)
1/10(-190)-1/10(190)
-19-(-19)= -38
I do have quite a few questions though..
1. If I get (-1/64)sin(1/8) and I'm suppose to set it equal to 0 then solve..How do I solve that?
2. (like #11 on the packet) For S(3t/t^2+2)^3 I used substitution to solve. For my final answer I got -3/2(t^2+2)^2+c but all the answers have -3/4(t^2+2)^2+c
3. (like #3 on the packet) It says use a graphing utility to graph the function f(x)=12/6-x and locate the absolute extrema of the function on the interval [0,6). If you plug that into your calculator the graph is an asymptote, and I tried taking the derivative and plugging that in too but it is also an asymptote. So how do you get a max and min?
Post #17
so..i've been semi-lost this week..so i'm kicking it back, old school.
So, product rule and quotient rule.
First, product rule is done when you need to take the derivative of things being mulitplied. So, it would be something like x(x^2)
So, the first thing you do, is copy the first term times the derivative of the second term plus the copied second term times the derivative of the first term. Product rule is very simple, you just have to remember the rules of simplificiation. Remember to simplify correctly and if you don't remember what to do next on the simplification process, it probably means you're done simplifying.
Now, lets talk about quotient rule..you know you must use quotient rule because you have a fraction.
The rules for quotient rule is copy the bottom times derivative of the top, minus copy the top times the derivative of the bottom. Like product rule, you need to simplify jsut the same. You need to distribute the things necessary and solve it to the simplest terms possible.
The thing i don't understand is the substitution stuff..i always get the same answer after i do the substitution and don't understand the integral.
So, product rule and quotient rule.
First, product rule is done when you need to take the derivative of things being mulitplied. So, it would be something like x(x^2)
So, the first thing you do, is copy the first term times the derivative of the second term plus the copied second term times the derivative of the first term. Product rule is very simple, you just have to remember the rules of simplificiation. Remember to simplify correctly and if you don't remember what to do next on the simplification process, it probably means you're done simplifying.
Now, lets talk about quotient rule..you know you must use quotient rule because you have a fraction.
The rules for quotient rule is copy the bottom times derivative of the top, minus copy the top times the derivative of the bottom. Like product rule, you need to simplify jsut the same. You need to distribute the things necessary and solve it to the simplest terms possible.
The thing i don't understand is the substitution stuff..i always get the same answer after i do the substitution and don't understand the integral.
17th post
The formula for the volume of disks is S (top)^2 - (bottom)^2 dx
The formula for the area of washers is S (top) - (bottom)
The steps are:
1. Draw the graphs of the equations
2. Subtract top graph's equation by the bottom graph's equation(in disks each equation would be squared)
3. Set equations equal and solve for x to find bounds
4. Plug in the bounds and the outcome of step 2
5. Integrate
volume by disks:
the formula is pi times the integral of the [function given] squared times dx. so just solve it by taking the integral of it and then pluging in the numbers they give you. just like before you'll have two numbers so whatever the answer is for the top one will be first and then you subtract the answer you get for the bottom one. then graph
volume by washers:
the formla is pie times the integral of the [top function] squared minus the [bottom function] squared times dx. so to do this, if you don't have the in between number you have to set the functions equal, but if you do, then it's worked the same way as above. square the formula's that were given and simplify. then take the integral of it and plug in the numbers they give you or you found by setting the formulas equal to each other and then solve like any other one by subracting them. then graph.
LRAM is left hand approximation and the formula is:
delta x [f(a) + f( delta x +a) .... + f( delta x - b)]
Say you are asked to calculate the left Riemann Sum for -4x -5 on the interval [-3, -1] divided into 2 subintervals.
delta x would equal: -1+3 /2 = 2/2 = 1
1[ f(-3) + f(-3 +1)]
1[ f( -3) + f(-2)]
then plug into your equation
RRAM is right hand approximation and the formula is:
delta x [ f(a + delta x) + .... + f(b)]
so using the same example:
1[ f( -2) + f(-1)] and then plug into your equation
MRAM is to calculate the middle and the formula is:
delta x [ f(mid) + f(mid) + .... ]
To find midpoints, you would add the two numbers together then divide by two
In this problem the numbers would be: -3 , -2, -1
-3 + -2/ 2 = -5/2 and -2 + -1 / 2 = -3/2
so 1[f(-5/2) + f(-3/2)] and the plug in
Trapezoidal is different because instead of multiplying by delta x, you multiply by delta x/2 and you also have on more term then your number of subintervals.
The formula is : delta x/2 [f(a) + 2f(a + delta x) + 2f(a+ 2 delta x) + ....f(b)]
For this problem: 1/2 [ f(-3) + 2 f(-2) + f( -1)] and then plug in.
Substitution takes the place of the derivative rules for problems such as product rule and quotient rule. 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-1dx
u=2x-1 du=2
rewrite the function as:
1/2{ e^u du, therefore
1/2e^2x-1+C will be the final answer.
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
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
2. if top degree is less than bottom degree, the answer is 0
To find area between curves:
The formula you use is b(int)a (top eq.) - (bottom eq.).
If a and b is not already given to you, then you much set the equations equal to each other and solve.
You find which equation is top/bottom by graphing both and simply looking to see which one is on top.
If the area is on the y-axis, then the a and b values need to be set as y-values, and the equations must be solved for x.
The formula for the area of washers is S (top) - (bottom)
The steps are:
1. Draw the graphs of the equations
2. Subtract top graph's equation by the bottom graph's equation(in disks each equation would be squared)
3. Set equations equal and solve for x to find bounds
4. Plug in the bounds and the outcome of step 2
5. Integrate
volume by disks:
the formula is pi times the integral of the [function given] squared times dx. so just solve it by taking the integral of it and then pluging in the numbers they give you. just like before you'll have two numbers so whatever the answer is for the top one will be first and then you subtract the answer you get for the bottom one. then graph
volume by washers:
the formla is pie times the integral of the [top function] squared minus the [bottom function] squared times dx. so to do this, if you don't have the in between number you have to set the functions equal, but if you do, then it's worked the same way as above. square the formula's that were given and simplify. then take the integral of it and plug in the numbers they give you or you found by setting the formulas equal to each other and then solve like any other one by subracting them. then graph.
LRAM is left hand approximation and the formula is:
delta x [f(a) + f( delta x +a) .... + f( delta x - b)]
Say you are asked to calculate the left Riemann Sum for -4x -5 on the interval [-3, -1] divided into 2 subintervals.
delta x would equal: -1+3 /2 = 2/2 = 1
1[ f(-3) + f(-3 +1)]
1[ f( -3) + f(-2)]
then plug into your equation
RRAM is right hand approximation and the formula is:
delta x [ f(a + delta x) + .... + f(b)]
so using the same example:
1[ f( -2) + f(-1)] and then plug into your equation
MRAM is to calculate the middle and the formula is:
delta x [ f(mid) + f(mid) + .... ]
To find midpoints, you would add the two numbers together then divide by two
In this problem the numbers would be: -3 , -2, -1
-3 + -2/ 2 = -5/2 and -2 + -1 / 2 = -3/2
so 1[f(-5/2) + f(-3/2)] and the plug in
Trapezoidal is different because instead of multiplying by delta x, you multiply by delta x/2 and you also have on more term then your number of subintervals.
The formula is : delta x/2 [f(a) + 2f(a + delta x) + 2f(a+ 2 delta x) + ....f(b)]
For this problem: 1/2 [ f(-3) + 2 f(-2) + f( -1)] and then plug in.
Substitution takes the place of the derivative rules for problems such as product rule and quotient rule. 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-1dx
u=2x-1 du=2
rewrite the function as:
1/2{ e^u du, therefore
1/2e^2x-1+C will be the final answer.
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
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
2. if top degree is less than bottom degree, the answer is 0
To find area between curves:
The formula you use is b(int)a (top eq.) - (bottom eq.).
If a and b is not already given to you, then you much set the equations equal to each other and solve.
You find which equation is top/bottom by graphing both and simply looking to see which one is on top.
If the area is on the y-axis, then the a and b values need to be set as y-values, and the equations must be solved for x.
Post....whatever!
So, Let's go with implicit differentiation...just solve for dy / dx.
Example 1: Use implicit differentiation to find the derivative dy / dx where y x + sin y = 1
Solution to Example 1:
Use the sum rule of differentiation to the whole term on the left of the given equation.
d [xy] / dx + d [siny] / dx = d[1]/dx .
Differentiate each term above using product rule to d [xy] / dx and the cain rule to d [siny] / dx.
x dy / dx + y + (dy / dx) cos(y) = 0 .
Note that in calculating d [siny] / dx, we used the chain rule since y is itself a function of x and sin (y) is a function of a function.
Solve for dy/dx to obtain.
dy / dx = -y / (x + cos y)
Example 2: Use implicit differentiation to find the derivative dy / dx where y 4 + x y 2 + x = 3
Solution to Example 2:
Use the differentiation of a sum formula to left side of the given equation.
d[y 4] / dx + d[x y 2] / dx + d[x] / dx = d[3] / dx
Differentiate each term above using power rule, product rule and chain rule.
4y 3 dy / dx + (1) y 2 + x 2y dy / dx + 1 = 0
Solve for dy/dx.
dy/dx = (-1 - y 2) / (4y 3 + 2xy)
Example 3: Find all points on the graph of the equation
x 2 + y 2 = 4
where the tangent lines are parallel to the line x + y = 2
Solution to Example 3:
Rewrite the given line x + y = 2 in slope intercept form: y = -x + 2 and identify the slope as m = -1. The tangent lines are parallel to this line and therefore their slope are equal to -1. The slope of tangent lines at a point can be found by implicity differentiation of x 2 + y 2 = 4
2x + 2y dy/dx = 0
Let P(a , b) be the point of tangency. At point P the slope is -1. Substituting x by a, y by b and dy/dx by -1 in the above equation, we obtain
2a + 2b (-1) = 0
Point P(a , b) is on the graph of x 2 + y 2 = 4, hence
a 2 + b 2 = 4
Solve the system of equations: 2a - 2b = 0 and a 2 + b 2 = 4 to obtain two points
(-sqrt(2) , -sqrt(2)) and (sqrt(2) , sqrt(2))
I DO NOT UNDERSTAND LINEARIZATION????????!!!?!!!!!!! HELLLLPPPPP...i need somebody...>HELLLPPPP....not just anybody.....HELLLLPPPP....i need someone.......anddddd sommmeetthinnng...idk
BYE BYE ADIOS AMIGOS!!! HASTA MANANA!!!
Example 1: Use implicit differentiation to find the derivative dy / dx where y x + sin y = 1
Solution to Example 1:
Use the sum rule of differentiation to the whole term on the left of the given equation.
d [xy] / dx + d [siny] / dx = d[1]/dx .
Differentiate each term above using product rule to d [xy] / dx and the cain rule to d [siny] / dx.
x dy / dx + y + (dy / dx) cos(y) = 0 .
Note that in calculating d [siny] / dx, we used the chain rule since y is itself a function of x and sin (y) is a function of a function.
Solve for dy/dx to obtain.
dy / dx = -y / (x + cos y)
Example 2: Use implicit differentiation to find the derivative dy / dx where y 4 + x y 2 + x = 3
Solution to Example 2:
Use the differentiation of a sum formula to left side of the given equation.
d[y 4] / dx + d[x y 2] / dx + d[x] / dx = d[3] / dx
Differentiate each term above using power rule, product rule and chain rule.
4y 3 dy / dx + (1) y 2 + x 2y dy / dx + 1 = 0
Solve for dy/dx.
dy/dx = (-1 - y 2) / (4y 3 + 2xy)
Example 3: Find all points on the graph of the equation
x 2 + y 2 = 4
where the tangent lines are parallel to the line x + y = 2
Solution to Example 3:
Rewrite the given line x + y = 2 in slope intercept form: y = -x + 2 and identify the slope as m = -1. The tangent lines are parallel to this line and therefore their slope are equal to -1. The slope of tangent lines at a point can be found by implicity differentiation of x 2 + y 2 = 4
2x + 2y dy/dx = 0
Let P(a , b) be the point of tangency. At point P the slope is -1. Substituting x by a, y by b and dy/dx by -1 in the above equation, we obtain
2a + 2b (-1) = 0
Point P(a , b) is on the graph of x 2 + y 2 = 4, hence
a 2 + b 2 = 4
Solve the system of equations: 2a - 2b = 0 and a 2 + b 2 = 4 to obtain two points
(-sqrt(2) , -sqrt(2)) and (sqrt(2) , sqrt(2))
I DO NOT UNDERSTAND LINEARIZATION????????!!!?!!!!!!! HELLLLPPPPP...i need somebody...>HELLLPPPP....not just anybody.....HELLLLPPPP....i need someone.......anddddd sommmeetthinnng...idk
BYE BYE ADIOS AMIGOS!!! HASTA MANANA!!!
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