HELLO EVERYONE!!!...I hope yall don't forget to do blogs with this GREAT weekend!
so, this week, as usual, we took more AP test...non-calculator multiple choice MONDAY...calculator multiple choice TUESDAY...calculator free responce WEDNESDAY...non-calculator free responce THURSDAY!...&& FRIDAY..that was SENIOR DAY..& it was really fun =)
so, once again i will go over what i know. i don't have many questions because B-Rob offers to help at lunch every day for free responce and the multiple choice we've been working on for corrections!...so here's what i know and that you should know in order to pass this AP Exam.
RELATED RATES:
Steps:
1. Identify all variables and equations
2. Identify what you are looking for
3. Make a sketch and label
4. Write an equation(s) involving your variables (only have 1 unknown)
5. Take the derivative with respect to TIME!
6. Substitute in the Derivative and solve
limits:
Rule #1 - When the degree (exponent) of the bottom is GREATER than the degree of the top, the limit is Zero.
Rule #2 - When the degree (exponent) of the bottom is SMALLER than the degree of the top, the limit is infinity. (positive or negative)
Rule #3 - When the degrees are equal, the limit is the coeffecients.
linierazation:
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
implicit derivatives:
First Derivative:
1. take the derivative of both sides
2. everytime you take the derivative of y note it with dy/dx or y^1
3. solve for dy/dx
Second Derivative:
first you find the first derivative and solve it for dy/dx by using the steps for the first derivative steps.
you then take the second derivative of the solved equation. Plugging in d^2y/d^2x everytime you take the derivative of y again. and where you have dy/dx you plug in your solved equation for that.
once you have everything plugged in and ready to go you then solve for d^2y/d^2x
Intermediate Value Theorem:
1. if f is continuous on [a,b] and k is any number between f(a)and f(b), then there is at least 1 number c when f(c)=k.
* basically you cannot skip any y value
HOW TO FIND THE EQUATION OF A TANGENT LINE: [I always miss this one or do it wrong for some reason…so I went look up the directions of how to solve them.]
1. take f1(x)
2. plug x in to find your slope m
3. plug x into f(x)to get y
4. using m and (x,y) plug it into the equation (y-y1)=m(x-x1).
well, good luck and remember that the AP is on May 5th!
I hope everyone has a great weekend .. SeniorDay-Movies-Sleepover-Prom-PromMania-Sleepover-Hooters-FRIENDS =) ..
~ElliE~
Saturday, April 24, 2010
Wednesday, April 21, 2010
Post 35
Ok I'll go over a few things I've learned this week about the free response sections of the ap (although today is Wednesday and the blogs were due Sunday).
Some things I learned include how to find bounds on an integral of area or volume when the bounds are not already given using the intersection function on your calculator, that a chart (although I hate them) on the free response portion of the ap is the best and easiest problem to have, that there is another way to do trapezoid rheeman's sum when using a chart, that RATE is not a derivative, it is an integral, and that you will always have to find the slope for part a of these problems.
Ok, first things first, how to find bounds when they are not given. Most of the time these problems will be shown on the calculator portion unless the bounds are clearly stated. Before, I just skipped these problems because I didn't know how to find them and when I did figure out how to find them I completley could not decide whether I wanted to use the x point or the y point for my bounds. But now I have all of this straight, so I will share it with you all. Here are the steps
1. Plug your function(s) into y=
For this, you'll either have two equations that will intersect or one equation that will intersect with either the x or y axis. A point of intersection is just what you want
2. After you plugged your function(s) into y=, you can graph them and find your intersection points. To do this hit 2nd, calc, intersect, enter. The function will guide you to locate your first line, then your second line, then your guess. After this it will give you your point. Always use the x point for this unless you are rotating about the y axis. Make sure you find all your points of intersection that connect to either the area or the volume you are rotating for your bounds.
So the chart problems are supposibly the easiest parts on the ap. I still get them wron all the time, ha. Part a will always ask you for a slope (even though it may say something else) always take the slope.
If you forgot (I did) Rheeman sums are really just finding the area of something. Another way to do a trapezoid rule is this:
(b1-b2/2)(h)
This is the first base minus the second base divided by 2 times the height. You'll add these all together and divide by your subinterval to find your answer.
I don't really remember optimization. I'll need help on this considering I'll need to do my prob/stats project on something relating to it.
Thanks.
Some things I learned include how to find bounds on an integral of area or volume when the bounds are not already given using the intersection function on your calculator, that a chart (although I hate them) on the free response portion of the ap is the best and easiest problem to have, that there is another way to do trapezoid rheeman's sum when using a chart, that RATE is not a derivative, it is an integral, and that you will always have to find the slope for part a of these problems.
Ok, first things first, how to find bounds when they are not given. Most of the time these problems will be shown on the calculator portion unless the bounds are clearly stated. Before, I just skipped these problems because I didn't know how to find them and when I did figure out how to find them I completley could not decide whether I wanted to use the x point or the y point for my bounds. But now I have all of this straight, so I will share it with you all. Here are the steps
1. Plug your function(s) into y=
For this, you'll either have two equations that will intersect or one equation that will intersect with either the x or y axis. A point of intersection is just what you want
2. After you plugged your function(s) into y=, you can graph them and find your intersection points. To do this hit 2nd, calc, intersect, enter. The function will guide you to locate your first line, then your second line, then your guess. After this it will give you your point. Always use the x point for this unless you are rotating about the y axis. Make sure you find all your points of intersection that connect to either the area or the volume you are rotating for your bounds.
So the chart problems are supposibly the easiest parts on the ap. I still get them wron all the time, ha. Part a will always ask you for a slope (even though it may say something else) always take the slope.
If you forgot (I did) Rheeman sums are really just finding the area of something. Another way to do a trapezoid rule is this:
(b1-b2/2)(h)
This is the first base minus the second base divided by 2 times the height. You'll add these all together and divide by your subinterval to find your answer.
I don't really remember optimization. I'll need help on this considering I'll need to do my prob/stats project on something relating to it.
Thanks.
Sunday, April 18, 2010
post 35
yeah, so this week we took tests every single day, and had the career day thing friday, and the tests were super hard, but hopefully i learn from my mistakes and do better etc... so anyways, here is my blog
Related Rates:
1: identify all variables in equations
2: identify what you are looking for
3: sketch and label
4: write an equation involving your variables. (you can only have one unknown so a secondary equation may be given)
5: take the derivative with respect to time.
6: substitute derivative and solve.
Example: the variables x and y are functions of t and related by the equation y=2x^3-x+4 when x=2, dy/dt=-1. Find dy/dt when x=2
alright, so you put down the equation, y=2x^3-x+4.
Then you take the derivative of that, so you get dy/dt=6x^2(dx/dt)-(dx/dt)
then you plug in to find that dy/dt=6(2)^2(-1)-(-1)
and that is further simplified to, dy/dt=-23.
Linearization:
f(x)=f(c)+f'(c)(x-c)
example: Approximate the tangent line to y=x^2 at x=1
you find all the different values: dy/dx=2x dy/dx=2 y=(1)^2=1
then you plug into the formula to get: f(x)=1+2(x-1)
example 2: use differentials to approximate: sq root(16.5)
steps:
1: identify an equation--- f(x)=sq root(x)
2:f(x)+f;(x)dx--- sqrt(x)+ (1/(2sqrt(x)))(dx)
3:determine dx-- .5
4:determine x--- 16
5:plug in--- sqrt(16)+(1/2sqrt(16))(.5)= 4.0625
error= .0005
yeah, so um, i guess i could use some help on taking the second implicit derivative of something, i always manage to screw it up somehow..
Related Rates:
1: identify all variables in equations
2: identify what you are looking for
3: sketch and label
4: write an equation involving your variables. (you can only have one unknown so a secondary equation may be given)
5: take the derivative with respect to time.
6: substitute derivative and solve.
Example: the variables x and y are functions of t and related by the equation y=2x^3-x+4 when x=2, dy/dt=-1. Find dy/dt when x=2
alright, so you put down the equation, y=2x^3-x+4.
Then you take the derivative of that, so you get dy/dt=6x^2(dx/dt)-(dx/dt)
then you plug in to find that dy/dt=6(2)^2(-1)-(-1)
and that is further simplified to, dy/dt=-23.
Linearization:
f(x)=f(c)+f'(c)(x-c)
example: Approximate the tangent line to y=x^2 at x=1
you find all the different values: dy/dx=2x dy/dx=2 y=(1)^2=1
then you plug into the formula to get: f(x)=1+2(x-1)
example 2: use differentials to approximate: sq root(16.5)
steps:
1: identify an equation--- f(x)=sq root(x)
2:f(x)+f;(x)dx--- sqrt(x)+ (1/(2sqrt(x)))(dx)
3:determine dx-- .5
4:determine x--- 16
5:plug in--- sqrt(16)+(1/2sqrt(16))(.5)= 4.0625
error= .0005
yeah, so um, i guess i could use some help on taking the second implicit derivative of something, i always manage to screw it up somehow..
Posting...#35
I'm just going to do some example problems for this blog so yall can look at them and if you don't already know they will help you
1. Let f(x)=x^3+9x on [1,3].
(a) What is the average rate of change in f on [1,3]
f(3)-f(1)
Average rate=---------
3-1
=54-10
-----------
2
=22
(b) On what intervals is f ' (x) >0?
f '(x)=3x^2+9
=3(x^2+1)
Since x^2 +1 >0 for all x, f '(x)>0 for all x. Thus f ' (x)>0 on the interval of [1,3].
(c) Find a value c in the interval [1,3] such that the average rate of change in f over the interval [1,3] equals f ' (c).
f ' (c)=3c^2+9
22=3c^2+9
13=3c^2
c^2=13/3
=2.082
And....
Let g(x)=x-sin(pie x) on [0,2].
Find the Critical values of g.
g'(x)=1-pie cos (pie x)
0=1-(pie)cos((pie)x)
(pie)cos((pie)x)=1
cos((pie)x)=1/(pie)
(pie)x=cos^-1(1/pie)
cos^-1(1/pie)
x=--------------
pie
=o.397, 1.603
Can any one help me with Reames an example would help alot
Post 35
This week we took a practice AP test which consisted of four parts. I"m quite amazed that i did better on the free response than the multiple choice. I'll start by going over some easy concepts:
First derivative test:
You have to take the derivative of the function and then set it equal to zero. Then solve for the x intercepts (critical points). After that, set the intercepts up into intervals between negative infinity and infinity. Plug in a number between those intervals to find max or mins, or if the graph is increasing or decreasing.
Second derivative test:
You take the derivative of the function twice and set equal to zero. Solve for the x intercepts once more and set them up into intervals between negative infinity and infinity. Plug in values between those intervals into the second derivative to find if the graph is concave up, concave down, or where there is a point of inflection.
Tangent lines:
You will be given a function and an x intercept. If no y intercept is given, plug the x value into the original function and solve for y. Then take the derivative of the function and plug in the x value to find the slope. Then put everything into point-slope form y-y1=slope(x-x1).
Limit rules:
If the degree on top is smaller than the degree on the bottom, the limit is zero
If the degree on top is bigger than the degree on the bottom, the limit is infinity.
If the degree on top is the same as the degree on the bottom, divide the coefficients and that will be the limit
Things i struggle with:
complicated integrals (such as ones that u have to use trig identities for)
Related Rates
Optimization
I hope you all have a great week :)
First derivative test:
You have to take the derivative of the function and then set it equal to zero. Then solve for the x intercepts (critical points). After that, set the intercepts up into intervals between negative infinity and infinity. Plug in a number between those intervals to find max or mins, or if the graph is increasing or decreasing.
Second derivative test:
You take the derivative of the function twice and set equal to zero. Solve for the x intercepts once more and set them up into intervals between negative infinity and infinity. Plug in values between those intervals into the second derivative to find if the graph is concave up, concave down, or where there is a point of inflection.
Tangent lines:
You will be given a function and an x intercept. If no y intercept is given, plug the x value into the original function and solve for y. Then take the derivative of the function and plug in the x value to find the slope. Then put everything into point-slope form y-y1=slope(x-x1).
Limit rules:
If the degree on top is smaller than the degree on the bottom, the limit is zero
If the degree on top is bigger than the degree on the bottom, the limit is infinity.
If the degree on top is the same as the degree on the bottom, divide the coefficients and that will be the limit
Things i struggle with:
complicated integrals (such as ones that u have to use trig identities for)
Related Rates
Optimization
I hope you all have a great week :)
post 3509284508245
this week in calc we did our first actual AP and got a score from 1-5. I got a 2, & was 11 points from passing. :( hopefully these next few weeks i can improve what i'm lacking in most, which is FREE RESPONSE. i got 0/27 then a 3/27.
OPTIMIZATION!:
1. Identify primary and secondary equations. Primary deals with the variable that is being maximized or minimized. The secondary equation is the other one.
2. Solve the secondary equation for one variable
3. Plug ^that variable back into the primary. If the primary equation only has one variable you can skip this step.
4. Take the derivative of the primary equation after plugging in the variable
5. Set it equal to zero and solve.
6. Plug that variable back into the secondary equation in order to solve for the last missing variable.
ABSOLUTE MAX/MIN!:
1. First derivative test
2. Plug critical values into the origonal function to get y-values
3. Plug endpoints into the origional function to get y-values
4. The highest y-value is the absolute maximum
5. The lowest y-value is the absolute minimum
since i did so horribly on free response, my question is going to be all of it. ha, i didn't know barely any of them! if someone wants to explain parts of it, or at least one problem. that would be very helpful
OPTIMIZATION!:
1. Identify primary and secondary equations. Primary deals with the variable that is being maximized or minimized. The secondary equation is the other one.
2. Solve the secondary equation for one variable
3. Plug ^that variable back into the primary. If the primary equation only has one variable you can skip this step.
4. Take the derivative of the primary equation after plugging in the variable
5. Set it equal to zero and solve.
6. Plug that variable back into the secondary equation in order to solve for the last missing variable.
ABSOLUTE MAX/MIN!:
1. First derivative test
2. Plug critical values into the origonal function to get y-values
3. Plug endpoints into the origional function to get y-values
4. The highest y-value is the absolute maximum
5. The lowest y-value is the absolute minimum
since i did so horribly on free response, my question is going to be all of it. ha, i didn't know barely any of them! if someone wants to explain parts of it, or at least one problem. that would be very helpful
Ash's 35 post
Geez, 35th week of school. This is kinda depressing...
This week was completely intense and it showed me that I'm really not ready to take this test. 4 days of testing and 1 day of career thing...fun.
Anyway, I did learn something this week about calculus: I love my calculator just about more than any other technology...just about.
My mind blanked on how to do a few simple things that I've relied on my calculator for, but during the practice APs I remember? I think it might have been the intimidation factor...
Also, the free response was killer...I think I may need to memorize the wiki problems, but I might need a little more help?
I don't have my APs on me otherwise I'd ask a ton of questions, but I mainly need help on the non-calculator portion. I've been spoiled into relying on my calculator for everything...I need someone's help to relearn how to do things by hand.
Also, I learned that the trig chart NEVER goes away....EVER...
I must pull mine out from the abyss and relearn it
Things to remember on Non-Calc Multiple Choice:
Trig Chart
Derivative formulas
How to solve for x
How to graph
How to find the limit
Things to remember on Calc multiple choice:
Everything
Bring batteries
Always make sure you actually have a calculator...don't walk into a room unprepared..
I hope everyone has a good week! Don't somehow burn yourself on something and don't forget corrections!
[[btw, bold is things I really need help with]]
This week was completely intense and it showed me that I'm really not ready to take this test. 4 days of testing and 1 day of career thing...fun.
Anyway, I did learn something this week about calculus: I love my calculator just about more than any other technology...just about.
My mind blanked on how to do a few simple things that I've relied on my calculator for, but during the practice APs I remember? I think it might have been the intimidation factor...
Also, the free response was killer...I think I may need to memorize the wiki problems, but I might need a little more help?
I don't have my APs on me otherwise I'd ask a ton of questions, but I mainly need help on the non-calculator portion. I've been spoiled into relying on my calculator for everything...I need someone's help to relearn how to do things by hand.
Also, I learned that the trig chart NEVER goes away....EVER...
I must pull mine out from the abyss and relearn it
Things to remember on Non-Calc Multiple Choice:
Trig Chart
Derivative formulas
How to solve for x
How to graph
How to find the limit
Things to remember on Calc multiple choice:
Everything
Bring batteries
Always make sure you actually have a calculator...don't walk into a room unprepared..
I hope everyone has a good week! Don't somehow burn yourself on something and don't forget corrections!
[[btw, bold is things I really need help with]]
Post #35
Review of AP questions..
3. If the integral of f(x) dx on [a,b] = a +2b, then the integral of (f(x)+5) dx on [a,b] =
The mistake I did on this problem was I forgot to integrate the 5 so my answer came out to be a+2b+5 which is an answer choice.
If you integrate the 5 you get a+2b + 5x9
If you plug in your bounds you get a+2b+5b-5a which simplifies to 7b-4a (also an answer choice and the correct answer)
You have to watch out for their tricks.
12. At what point on the graph of y=1/2x^2 is the tangent line parallel to the line 2x-4y=3?
The steps to doing these problems are to
1. Take the derivative of the equation
y' = x
2. Set that equal to the slope of the line ( slope = -a/b)
x= -2/-4 x= 1/2
3. Solve for x
x=1/2
4. Plug back into original equation to find y.
1/2 (1/2)^2
(1/2) (1/4) = 1/8
The point is (1/2,1/8)
5b (short answer). Find the particular solution y=f(x) to the differential equation with the initial condition f(-1) =1.
The equation is dy/dx = 1+y /x
The steps for particular solution are:
1. Separate variables
2. Integrate each side
3. Put +c on x side (left)
4. Do e^ = e^
5. Simplify
6. Solve for c
7. Solve for y
The easiest way to separate variables in this problem is to cross multiply.
dyx = dx (1+y)
dy x / 1+y = dx
Multiply each side by 1/x
dy/ 1+y = dx/x
2. Now integrate each side
The derivative of y is 1, which is what is in the numerator so you know this is going to be natural log
ln|1+y| = ln|x| +c
3. e^ln |1+y| = e^ |x| + e^c
4. 1+y = c|x| (Since you multiply when adding exponents)
5. You are given f(-1) = 1 so you know your point is (-1,1)
Now plug in and solve for c.
1+1 = c |-1|
2 = c (1)
c=2
6. Plug c back into the equation and solve for y.
|1+y| = 2|x|
y = 2|x| -1
Hope this helps with corrections.
I can use some help on related rate questions and questions like 84 when it says if the cross sections of S perpendicular to the x-axis and also remembering being in radians.
3. If the integral of f(x) dx on [a,b] = a +2b, then the integral of (f(x)+5) dx on [a,b] =
The mistake I did on this problem was I forgot to integrate the 5 so my answer came out to be a+2b+5 which is an answer choice.
If you integrate the 5 you get a+2b + 5x9
If you plug in your bounds you get a+2b+5b-5a which simplifies to 7b-4a (also an answer choice and the correct answer)
You have to watch out for their tricks.
12. At what point on the graph of y=1/2x^2 is the tangent line parallel to the line 2x-4y=3?
The steps to doing these problems are to
1. Take the derivative of the equation
y' = x
2. Set that equal to the slope of the line ( slope = -a/b)
x= -2/-4 x= 1/2
3. Solve for x
x=1/2
4. Plug back into original equation to find y.
1/2 (1/2)^2
(1/2) (1/4) = 1/8
The point is (1/2,1/8)
5b (short answer). Find the particular solution y=f(x) to the differential equation with the initial condition f(-1) =1.
The equation is dy/dx = 1+y /x
The steps for particular solution are:
1. Separate variables
2. Integrate each side
3. Put +c on x side (left)
4. Do e^ = e^
5. Simplify
6. Solve for c
7. Solve for y
The easiest way to separate variables in this problem is to cross multiply.
dyx = dx (1+y)
dy x / 1+y = dx
Multiply each side by 1/x
dy/ 1+y = dx/x
2. Now integrate each side
The derivative of y is 1, which is what is in the numerator so you know this is going to be natural log
ln|1+y| = ln|x| +c
3. e^ln |1+y| = e^ |x| + e^c
4. 1+y = c|x| (Since you multiply when adding exponents)
5. You are given f(-1) = 1 so you know your point is (-1,1)
Now plug in and solve for c.
1+1 = c |-1|
2 = c (1)
c=2
6. Plug c back into the equation and solve for y.
|1+y| = 2|x|
y = 2|x| -1
Hope this helps with corrections.
I can use some help on related rate questions and questions like 84 when it says if the cross sections of S perpendicular to the x-axis and also remembering being in radians.
post 35
LINERAZATION
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
LIMIT RULES
The limit rules are:
1. if the highest exponent is the same on the top and bottom then the limit is the top coefficient over the bottom coefficient of the highest exponents.
2. If the highest exponent is on the top then the limit is infinity.
3. But if the highest exponent is on the bottom then the limit is 0.
RELATED RATES:
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
First derivative test:
For the first derivative test, you are solving for max and mins and may be trying to see where the graph is increasing and decreasing. You take the derivative of the function and and set it equal to zero and solve for the x values (critical points). Then you set those points up into intervals between negative infinity and infinity. Then, you plug in numbers between those intervals to see if it is positive or negative.
Second derivative test:
For the second derivative test, you are solving to see whether the graph is concave up, concave down, or where there is a point of inflection in the graph. You take the derivative of the function twice and set it equal to zero and solve for the x values. You set those values up into intervals between negative infinity and infinity. You then plug in numbers between those intervals to see if it is positive or negative. If it is positive, it is concave up. If it is negative it is concave down. Where there is a change in concavity, there is a point of inflection.
I dont remember optimization. help
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
LIMIT RULES
The limit rules are:
1. if the highest exponent is the same on the top and bottom then the limit is the top coefficient over the bottom coefficient of the highest exponents.
2. If the highest exponent is on the top then the limit is infinity.
3. But if the highest exponent is on the bottom then the limit is 0.
RELATED RATES:
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
First derivative test:
For the first derivative test, you are solving for max and mins and may be trying to see where the graph is increasing and decreasing. You take the derivative of the function and and set it equal to zero and solve for the x values (critical points). Then you set those points up into intervals between negative infinity and infinity. Then, you plug in numbers between those intervals to see if it is positive or negative.
Second derivative test:
For the second derivative test, you are solving to see whether the graph is concave up, concave down, or where there is a point of inflection in the graph. You take the derivative of the function twice and set it equal to zero and solve for the x values. You set those values up into intervals between negative infinity and infinity. You then plug in numbers between those intervals to see if it is positive or negative. If it is positive, it is concave up. If it is negative it is concave down. Where there is a change in concavity, there is a point of inflection.
I dont remember optimization. help
Post Number Thirty Five
Well this is really almost it. Unfortunately I still suck at calculus even after a whole year..
Review I guess..
Linearization:
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
Substitution:
1.Find a derivative inside the interval
2. set u = the non-derivative
3. take the derivative of u
4. substitute back in
Related Rates:
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
Limit Rules:
1.if the highest exponent is the same on the top and bottom then the limit is the top coefficient over the bottom coefficient of the highest exponents.
2. If the highest exponent is on the top then the limit is infinity.
3. But if the highest exponent is on the bottom then the limit is 0.
I hope b-rob is still doing the whole lunch thing. I didn’t know about it until now and I could use it.. But anyways maybe as I keep working they I get better.
Review I guess..
Linearization:
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
Substitution:
1.Find a derivative inside the interval
2. set u = the non-derivative
3. take the derivative of u
4. substitute back in
Related Rates:
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
Limit Rules:
1.if the highest exponent is the same on the top and bottom then the limit is the top coefficient over the bottom coefficient of the highest exponents.
2. If the highest exponent is on the top then the limit is infinity.
3. But if the highest exponent is on the bottom then the limit is 0.
I hope b-rob is still doing the whole lunch thing. I didn’t know about it until now and I could use it.. But anyways maybe as I keep working they I get better.
post 35
So this past week we took an AP test to show us what we need to improve on. So now I will review some old things we learned this school year.
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
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\
Also I am going to talk about taking implicit derivatives. 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 my problems I need help doing problems with graphs.
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
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\
Also I am going to talk about taking implicit derivatives. 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 my problems I need help doing problems with graphs.
post 35
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
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
okay so ive been having trouble with trapizoidal and the particle problems....if anyone can explain please do cause im lost....
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
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
okay so ive been having trouble with trapizoidal and the particle problems....if anyone can explain please do cause im lost....
What post is this again?
Anyway, I will now explain a few types of questions that in particular I got wrong on this week's AP, just to help me review my mistakes etc.
Number 83, Calculator Portion.
83. What is the area of the region in the first quadrant enclosed by the graphs of y=cos x, y=x, and the y-axis?
Okay so instinctively when I read this problem, I for some reason jumped to the conclusion that it was a volume problem and it would easy. Okay, I was completely wrong--it's an area problem. Other than that, my only advice here is that we/I need to be very particular in what we do when we read the problems. Identify what you are supposed to do, and double check that. Write it down next to the problem if need be...it might help you remember that "Okay, this is an area problem, not volume, so I won't be multiplying by Pi".
Number 21, Non-Calculator Portion
21. The limit as x goes to 1 of (x)/(ln x) is ?
Okay so this problem, I plugged in for 1 (because that is what you do for definite limits) and I got that it was 1 / 0. Okay so for whatever reason in my mind, I decided that this was a point in time where L'Hospital's rule applied; however, I was very wrong. L'Hospital's rule only applies when it comes out to be 0/0 or infinity/infinity. So, this answer is simply DNE.
Number 22, Non-Calculator Portion
22. What are all values of x for which the function f defined by f(x)=(x^2-3)e^-x is increasing?
Okay so for this problem, I identified "increasing" as to being my clue word. It hinted to me that I needed to take the first derivative, and set equal to 0 and solve for my critical points. Well, the only problem here was that the derivative was slightly...odd. It was different, but the trick is that you want to completely distribute everything in so that you can see what you can factor out. You end up being able to factor out the ugly e^-x, and you are left with a pretty polynomial to factor. The only tips on this problem is to watch your derivative, and remember that for first-derivative test, you have to plug into the first derivative, not the original.
Number 23, Non-Calculator Portion
23. If the region enclosed by the y-axis, the line y=2, and the curve y=sqrt(x) is revolved about the y-axis, the volume of the solid generated is
So the most important mistake that most people will make with this is that they won't realize it is being rotated about the y-axis. The reason this changes things is because you need to solve your equation for x. So, our equation becomes x=y^2. Now it is just a simple integral, however, you have to remember that since it is volume, it becomes squared. So the integral is of y^4, not y^2. Also, I guess it's important to note that the line y=2 is used as your bounds (drawing a picture will help immensely in figuring this out).
Have a great day :-).
Number 83, Calculator Portion.
83. What is the area of the region in the first quadrant enclosed by the graphs of y=cos x, y=x, and the y-axis?
Okay so instinctively when I read this problem, I for some reason jumped to the conclusion that it was a volume problem and it would easy. Okay, I was completely wrong--it's an area problem. Other than that, my only advice here is that we/I need to be very particular in what we do when we read the problems. Identify what you are supposed to do, and double check that. Write it down next to the problem if need be...it might help you remember that "Okay, this is an area problem, not volume, so I won't be multiplying by Pi".
Number 21, Non-Calculator Portion
21. The limit as x goes to 1 of (x)/(ln x) is ?
Okay so this problem, I plugged in for 1 (because that is what you do for definite limits) and I got that it was 1 / 0. Okay so for whatever reason in my mind, I decided that this was a point in time where L'Hospital's rule applied; however, I was very wrong. L'Hospital's rule only applies when it comes out to be 0/0 or infinity/infinity. So, this answer is simply DNE.
Number 22, Non-Calculator Portion
22. What are all values of x for which the function f defined by f(x)=(x^2-3)e^-x is increasing?
Okay so for this problem, I identified "increasing" as to being my clue word. It hinted to me that I needed to take the first derivative, and set equal to 0 and solve for my critical points. Well, the only problem here was that the derivative was slightly...odd. It was different, but the trick is that you want to completely distribute everything in so that you can see what you can factor out. You end up being able to factor out the ugly e^-x, and you are left with a pretty polynomial to factor. The only tips on this problem is to watch your derivative, and remember that for first-derivative test, you have to plug into the first derivative, not the original.
Number 23, Non-Calculator Portion
23. If the region enclosed by the y-axis, the line y=2, and the curve y=sqrt(x) is revolved about the y-axis, the volume of the solid generated is
So the most important mistake that most people will make with this is that they won't realize it is being rotated about the y-axis. The reason this changes things is because you need to solve your equation for x. So, our equation becomes x=y^2. Now it is just a simple integral, however, you have to remember that since it is volume, it becomes squared. So the integral is of y^4, not y^2. Also, I guess it's important to note that the line y=2 is used as your bounds (drawing a picture will help immensely in figuring this out).
Have a great day :-).
post 35
We have been taking ap tests all week long. Here are some problems.
1. d/dx cos^2(x^3)
chain rule!
-2cos(x^3)sin(x^3)(3x^2)
-6x^2cos(x^3)sin(x^3)
2. An equation of the line tangent to the graph of y=cos(2x) at x=pi/4 is
tangent line steps: take derivative, plug in x to get slope, use point slope formula, and if not given y plug x value into original
cos(2x)
-2sin(2x)
-2sin2(pi/4)
-2sin(pi/2)
-2=slope
cos(2(pi/4))
cos(pi/2)=0
y=-2(x-pi/4)
3. Let f be a function defined for all real numbers x. If f^1(x)=(4-x^2)/x2, then f is decreasing on the interval *(4-x^2 is really the absolute value of 4-x^2)
first derivative test!
f^1(x)=(4-x^2)/x2 =0
4-x^2=0
-x$2=4
x=+ or - 2
(-infinity,-2) (-2,2) (2,infinity)
plug in -3=-ve/decreasing
plug in 0=-ve/decreasing
plug in 3=+ve/increasing
So decreasing on the interval (-infinity,2)
4. 0S(pi/4) e^tanx/cos^2x dx is
u=tanx du=sec^2x = 1/cos^2x
0Spi/4 e^u du
e^tanx *(have to integrate from 0 to pi/4)
e^tan(pi/4) - e^tan0
e^1 - e^0
=e-1
5. If f(x)=ln((x^2)-1), then f^1(x) *(x^2-1)is really the absolute value of x^2-1
just take derivative!
f^1(x)=ln(x^2-1)
(1/x^2-1)(2x)
2x/(x^2)-1
I'm not to great with integration!
1. d/dx cos^2(x^3)
chain rule!
-2cos(x^3)sin(x^3)(3x^2)
-6x^2cos(x^3)sin(x^3)
2. An equation of the line tangent to the graph of y=cos(2x) at x=pi/4 is
tangent line steps: take derivative, plug in x to get slope, use point slope formula, and if not given y plug x value into original
cos(2x)
-2sin(2x)
-2sin2(pi/4)
-2sin(pi/2)
-2=slope
cos(2(pi/4))
cos(pi/2)=0
y=-2(x-pi/4)
3. Let f be a function defined for all real numbers x. If f^1(x)=(4-x^2)/x2, then f is decreasing on the interval *(4-x^2 is really the absolute value of 4-x^2)
first derivative test!
f^1(x)=(4-x^2)/x2 =0
4-x^2=0
-x$2=4
x=+ or - 2
(-infinity,-2) (-2,2) (2,infinity)
plug in -3=-ve/decreasing
plug in 0=-ve/decreasing
plug in 3=+ve/increasing
So decreasing on the interval (-infinity,2)
4. 0S(pi/4) e^tanx/cos^2x dx is
u=tanx du=sec^2x = 1/cos^2x
0Spi/4 e^u du
e^tanx *(have to integrate from 0 to pi/4)
e^tan(pi/4) - e^tan0
e^1 - e^0
=e-1
5. If f(x)=ln((x^2)-1), then f^1(x) *(x^2-1)is really the absolute value of x^2-1
just take derivative!
f^1(x)=ln(x^2-1)
(1/x^2-1)(2x)
2x/(x^2)-1
I'm not to great with integration!
Post #35
So I need to improve on this thing called calculus.
EXAMPLE 1:
(aSb = integral from a to be)
If aSb f(x)dx=a+2b, then aSb (f(x)+5)dx=
a+2b-(5x)
*have to integrate 5x from a to b)
5b-5a+a+2b
=7b-4a
EXAMPLE 2:
(1/2)Se^(1/2)dt=
u=t/2 du= 1/2dt
1/2Se^u du
2(1/2)Se^u du
e^u +C
e^t/2 +C
EXAMPLE 3:
At what point on the graph of y=(1/2)x^2 is the tangent line parallel to the line 2x-4y=3?
2x-4y=3
formula: -A/B
-2/-4 = 1/2
x=1/2
(1/2)x^2
(1/2)(1/2)^2 = 1/8
(1/2 , 1/8)
EXAMPLE 4:
Let f be a differentiable function such that f(3)=2 and f^1(3)=5. If the tangent line to the graph of f at x=3 is used to find an approximation to a zero of f, that approximation is?
f(3)=2 (3=x, 2=y)
f^1(3)=5 (<--5 slope)
y-2=5(x-3)
y-2=5(3.1-3)
y-2=5(.1)
y=2.5
*It is an approximation so you pick the answer choice closest which is 2.6.
EXAMPLE 5:
If x^2+y^2=25,what is the value of d^2y/dx^2 at the point (4,3)?
x^2+y^2=25
2x+2y(dy/dx)=0
dy/dx= -2x/2y
(2y)(-2)-(-2x)(2dy/dx)/(2y)^2
*plug in point for dy/dx
dy/dx=-2(4)/2(3) = -4/3
2(3)(-2)-(-2(4))(2(-4/3))/2(3)^2
-12-(-8)(-8/3)/2(9)
=-25/27
I need help with:
OPTIMIZATION
ANGLE OF ELEVATION
EXAMPLE 1:
(aSb = integral from a to be)
If aSb f(x)dx=a+2b, then aSb (f(x)+5)dx=
a+2b-(5x)
*have to integrate 5x from a to b)
5b-5a+a+2b
=7b-4a
EXAMPLE 2:
(1/2)Se^(1/2)dt=
u=t/2 du= 1/2dt
1/2Se^u du
2(1/2)Se^u du
e^u +C
e^t/2 +C
EXAMPLE 3:
At what point on the graph of y=(1/2)x^2 is the tangent line parallel to the line 2x-4y=3?
2x-4y=3
formula: -A/B
-2/-4 = 1/2
x=1/2
(1/2)x^2
(1/2)(1/2)^2 = 1/8
(1/2 , 1/8)
EXAMPLE 4:
Let f be a differentiable function such that f(3)=2 and f^1(3)=5. If the tangent line to the graph of f at x=3 is used to find an approximation to a zero of f, that approximation is?
f(3)=2 (3=x, 2=y)
f^1(3)=5 (<--5 slope)
y-2=5(x-3)
y-2=5(3.1-3)
y-2=5(.1)
y=2.5
*It is an approximation so you pick the answer choice closest which is 2.6.
EXAMPLE 5:
If x^2+y^2=25,what is the value of d^2y/dx^2 at the point (4,3)?
x^2+y^2=25
2x+2y(dy/dx)=0
dy/dx= -2x/2y
(2y)(-2)-(-2x)(2dy/dx)/(2y)^2
*plug in point for dy/dx
dy/dx=-2(4)/2(3) = -4/3
2(3)(-2)-(-2(4))(2(-4/3))/2(3)^2
-12-(-8)(-8/3)/2(9)
=-25/27
I need help with:
OPTIMIZATION
ANGLE OF ELEVATION
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