So how’s everyone’s scores coming along on the AP tests…hopefully a little better than mine, but I can’t complain too much! Anywayyssss…
Let’s see what I remember shall we.. if you take the derivative of f^2(x) you treat it as if it was like sine. You do 2f(x) times f’(x) times 1. So it’s almost as if you have (f(x))^2 and you use the chain rule. Bring the 2 to the front, subtract 1, times by the derivative of the inside. And since the derivative of f(x) is f’(x) times the derivative of x, which is 1. These rules seem to help me A LOT!
Lets say you have:
x^2-2 DIVIDED BY 4-x^2 AS THE LIMIT APPROACHES 2
soo..
for this you end up getting two divided by zero. For finite limits though, if you get a number over zero, it’s always infinity. Therefore the answer is INFINITY!
Remember:
If the top exponent is greater than the bottom it’s the limit as it approaches infinity, and if the top exponent is less than the bottom it’s the limit as it approaches zero!
DON’T GET THEM CONFUSED LIKE I DID!
Here’s a trick:
Remember if you’re divided a number by another number and you have a bigger number on top it’s usually not zero. Also, if you’re dividing a number by another number and you have a smaller number on top it’s usually not a big number. Therefore, when the bigger number is on top since it’s a bigger number than zero it’s INFINITY and when the bigger number is on the bottom that means that it’s being lessened so it’s ZERO!
I guess we need to start studing for the next AP Exam huh!...[well I do anyways]
GOOD LUCK!
~ElliE~
Saturday, January 16, 2010
numero 22
THIS WEEK IN CALC WE JUST WENT OVER ALL OF OUR A.P. STUFF, SAME AS THE REST OF THE THIRD 9 WEEKS, AND SAME THING WE WILL BE DOING FOR THE REST OF THE YEAR.
formulas for derivatives
d/dx c=0 (c is a #)
d/dx cu=cu' (c is #)
d/dx cx=c (c is a #)
d/dx u+v=u'+v'
d/dx uv=uv'+vu'
d/dx u/v=(vu'-uv')/v^2
d/dx sinx=cosx(x')
d/dx cosx=-sinx(x')
d/dx tanx=sec^2x(x')
d/dx secx=secxtanx(x')
d/dx cscx=-cscxtanx(x')
d/dx cotx=-csc^2x(x')
d/dx lnu= 1/u(u')
d/dx e^u=e^u(u')
***just remember that integration is opposite for all these derivative formulas. ***
Optimization can be used for finding the maximum/minimum amount of area of something. Steps in order to optimize anything:
1. Identify primary and secondary equations. Primary deals with the variable that is being maximized or minimized. The secondary equation is usually the other equation that ties in all the information given in the problem.
2. Solve the secondary equation for one variable and then plug that variable back into the primary. If the primary equation only have one variable you can skip this step.
3. Take the derivative of the primary equation after plugging in the variable, set it equal to zero, and then solve for the variable.
4. Plug that variable back into the secondary equation in order to solve for the last missing variable. Check endpoint if necessary to find the maximum or minimum answers
I'M NOT DOING SO GREAT ON AP RIGHT NOW, BUT HOPEFULLY ALL THE CORRECTIONS AND STUFF WILL HELP ME OUT. Can someone explain to me linearization. it has completely left my brain
formulas for derivatives
d/dx c=0 (c is a #)
d/dx cu=cu' (c is #)
d/dx cx=c (c is a #)
d/dx u+v=u'+v'
d/dx uv=uv'+vu'
d/dx u/v=(vu'-uv')/v^2
d/dx sinx=cosx(x')
d/dx cosx=-sinx(x')
d/dx tanx=sec^2x(x')
d/dx secx=secxtanx(x')
d/dx cscx=-cscxtanx(x')
d/dx cotx=-csc^2x(x')
d/dx lnu= 1/u(u')
d/dx e^u=e^u(u')
***just remember that integration is opposite for all these derivative formulas. ***
Optimization can be used for finding the maximum/minimum amount of area of something. Steps in order to optimize anything:
1. Identify primary and secondary equations. Primary deals with the variable that is being maximized or minimized. The secondary equation is usually the other equation that ties in all the information given in the problem.
2. Solve the secondary equation for one variable and then plug that variable back into the primary. If the primary equation only have one variable you can skip this step.
3. Take the derivative of the primary equation after plugging in the variable, set it equal to zero, and then solve for the variable.
4. Plug that variable back into the secondary equation in order to solve for the last missing variable. Check endpoint if necessary to find the maximum or minimum answers
I'M NOT DOING SO GREAT ON AP RIGHT NOW, BUT HOPEFULLY ALL THE CORRECTIONS AND STUFF WILL HELP ME OUT. Can someone explain to me linearization. it has completely left my brain
Friday, January 15, 2010
Post 1/15/2010..#22
More AP practice questions work for you??? Okay, that is what I shall do then.
EXAMPLE 1:Given the equation y = (x-3)/(2-5x). Find dy/dx.
Quotient Rule (you could technically do product, but I prefer things set in stone (plug into a formula nd all))
dy/dx = ((2-5x)(1) -(-5)(x-3))/(2-5x)^2
= (2-5x + 5x -15)/(2-5x)^2
= - 13/(2-5x)^2
What I did: I took the derivative of the function using the quotient rule**be sure not to mix up NEGATIVES!!
EXAMPLE 2: What is the maximum value for the following: f(x) = xe^-x
Take derivative using product rule (multiplying two things).
x(-e^-x) + e^-x(1)remember that the der. e is e^w/e times the der. of the exponent
Simplify:
-xe^-x + e^-x
You can factor out an e^-x
(-x+1)e^-x
To find the critical values (possible maxima) set (-x+1) equal to zero and solve for x.
This yields x = 1. Now plug in that one to e^-x.
This then gives you 1/e====>your maximum value.
EXAMPLE 3: The table shows the speed of an object in feet per second, during a 3 second period.
time(sec)-----0--1--2--3
speed(ft/sec)-30-22-12-0
Estimate the distance traveled using the trapezoid method.
All you have to do for this particular problem is find delta x then plug into the formula: delta x /2 [f(first one) +2f(next)...f(last one)]
b-a/n = delta x
3-0/3 = 1
1/2[30 + 2(22) + 2(12) + 0]
1/2[20 + 44 + 24]
=49
EXAMPLE 4: Which best describes the behavior of the function y = arctan(1/lnx) at x = t?
A. It has a jump discontinuity
B. It has an infinite discontinuity.
C. It has a removable discontinuity
D. It is both continuous and differentiable
E. It is continuous but not differentiable.
Now keep in mind that this is in the calculator portion...so if they give you a function and you have your calculuator...GRAPH it! All the question is asking you is what's happening at x = 1???
Well when looking at the graph you can clearly see that it is not continuous....(*whispers*there's a jump). So you rule out D and E. It's not a removable because there's no hole...but it clearly jumps!!! SO the answer is A!!!!!
~MalPal
EXAMPLE 1:Given the equation y = (x-3)/(2-5x). Find dy/dx.
Quotient Rule (you could technically do product, but I prefer things set in stone (plug into a formula nd all))
dy/dx = ((2-5x)(1) -(-5)(x-3))/(2-5x)^2
= (2-5x + 5x -15)/(2-5x)^2
= - 13/(2-5x)^2
What I did: I took the derivative of the function using the quotient rule**be sure not to mix up NEGATIVES!!
EXAMPLE 2: What is the maximum value for the following: f(x) = xe^-x
Take derivative using product rule (multiplying two things).
x(-e^-x) + e^-x(1)remember that the der. e is e^w/e times the der. of the exponent
Simplify:
-xe^-x + e^-x
You can factor out an e^-x
(-x+1)e^-x
To find the critical values (possible maxima) set (-x+1) equal to zero and solve for x.
This yields x = 1. Now plug in that one to e^-x.
This then gives you 1/e====>your maximum value.
EXAMPLE 3: The table shows the speed of an object in feet per second, during a 3 second period.
time(sec)-----0--1--2--3
speed(ft/sec)-30-22-12-0
Estimate the distance traveled using the trapezoid method.
All you have to do for this particular problem is find delta x then plug into the formula: delta x /2 [f(first one) +2f(next)...f(last one)]
b-a/n = delta x
3-0/3 = 1
1/2[30 + 2(22) + 2(12) + 0]
1/2[20 + 44 + 24]
=49
EXAMPLE 4: Which best describes the behavior of the function y = arctan(1/lnx) at x = t?
A. It has a jump discontinuity
B. It has an infinite discontinuity.
C. It has a removable discontinuity
D. It is both continuous and differentiable
E. It is continuous but not differentiable.
Now keep in mind that this is in the calculator portion...so if they give you a function and you have your calculuator...GRAPH it! All the question is asking you is what's happening at x = 1???
Well when looking at the graph you can clearly see that it is not continuous....(*whispers*there's a jump). So you rule out D and E. It's not a removable because there's no hole...but it clearly jumps!!! SO the answer is A!!!!!
~MalPal
Monday, January 11, 2010
Post #21
Well we have now gone straight ap practice tests. This is making me nervous! I'm forgetting some easy stuff and making stupid mistakes. There are a few things though that I am actually remembering and learning while correcting these tests:
1. The derivative of e^u is simply (e^u)u^1. I don't know why I always forget that.
2. Average value means take the integral.
Example: The average value of f(x)= -1/x^2 on [1/2, 1].
1/b-a 1S1/2 (-1/x^2)
1/(1-1/2)S x^-2
2[(-x^-1)/-1] = 2/x still have to integrate from 1/2 to 1
(2/1)-(2/(1/2) = -2
3. Approximate=linearization
Example: If f(1)=2 and f^1(1)=5, use the equation of the line tangent to the graph of f at x=1 to approximate f(1.2).
All you do is put it in to point slope form and solve
y-2=5(x-1)
y-2=5(1.2-1)
y-2=5(.2)
y=1+2= 3
4. Make sure to read the problem carefully because half the time you can eliminate at least two answer choices.
5. And finally use your calculator for the calculator section! It sounds dumb, but we all don't. Always try to graph the equation given or use the table and your calculator can take derivatives and integrals.
My question is about this formula H^1(x)=1/f^1(H(x)). I put a star by this formula in my notes, but I can't tell what kind of problems it is used for. Can anyone help with that?
1. The derivative of e^u is simply (e^u)u^1. I don't know why I always forget that.
2. Average value means take the integral.
Example: The average value of f(x)= -1/x^2 on [1/2, 1].
1/b-a 1S1/2 (-1/x^2)
1/(1-1/2)S x^-2
2[(-x^-1)/-1] = 2/x still have to integrate from 1/2 to 1
(2/1)-(2/(1/2) = -2
3. Approximate=linearization
Example: If f(1)=2 and f^1(1)=5, use the equation of the line tangent to the graph of f at x=1 to approximate f(1.2).
All you do is put it in to point slope form and solve
y-2=5(x-1)
y-2=5(1.2-1)
y-2=5(.2)
y=1+2= 3
4. Make sure to read the problem carefully because half the time you can eliminate at least two answer choices.
5. And finally use your calculator for the calculator section! It sounds dumb, but we all don't. Always try to graph the equation given or use the table and your calculator can take derivatives and integrals.
My question is about this formula H^1(x)=1/f^1(H(x)). I put a star by this formula in my notes, but I can't tell what kind of problems it is used for. Can anyone help with that?
21
Optimization
1 Identify primary and secondary equations your primary is the one your or maximizing or minimizing and your secondary is the other equation
2. Solve for your secondary variable and plug into your primary equation if your primary only has on variable this isn’t necessary
3. Plug into secondary equation to find the other value check your end points if necessary
Related Rates:
1. Identify all variables and equations
2. Identify what you are looking for
3. Make a sketch and label
4. Write an equation involving your variables
5. Take the derivative with respect to time
6. Substitute in derivative and solve.
for some things im still having trouble with are rolles theorm and mean value theroem. Can somebody help
1 Identify primary and secondary equations your primary is the one your or maximizing or minimizing and your secondary is the other equation
2. Solve for your secondary variable and plug into your primary equation if your primary only has on variable this isn’t necessary
3. Plug into secondary equation to find the other value check your end points if necessary
Related Rates:
1. Identify all variables and equations
2. Identify what you are looking for
3. Make a sketch and label
4. Write an equation involving your variables
5. Take the derivative with respect to time
6. Substitute in derivative and solve.
for some things im still having trouble with are rolles theorm and mean value theroem. Can somebody help
Sunday, January 10, 2010
twenty-one
no school tomorrow! yessssss :) haha, so this week mon-fri, we did the same thing. every day. review ap test questions, took a couple practice ap tests in 5th hour, went over them. pretty much everything we've learned this year, we went over. haha
Related Rates:
1. Identify all variables and equations
2. Identify what you are looking for
3. Make a sketch and label
4. Write an equation involving your variables
5. Take the derivative with respect to time
6. Substitute in derivative and solve.
Example:
Air is being pumped into a spherical balloon at a rate of 4.5 cubic ft/min. Find the rate of change of the radius when the radius is 2 ft.
1. r=2ft; dv/dt = 4.5 ft^3/min
2. NEED TO FIND dr/dt.
3. Volume of sphere: v=4/3 pi r^3
4. derivative: dv/dt = 4 pi r^2 dr/dt
5. Plug in: 4.5 = 4pi(2)^2 dr/dt
6. 4.5=16 pi dr/dt
dr/dt = 9/32 pi ft/min
i finally am starting to understand all the lram rram tram mram stuff, but i couldn't explain it. i have to see it. AP tests seem to have a lot of repeating questions. i get e integration too!
u is always whatever e is raised to(the exponent)
du is always the derivative.
simply use substitution and solve.
i also understand disks and washers.. but unfortunately i did forget the formulas. haha
what i mainly have problems with is word problems.. which is pretty much everything! so, idk what to do. i guess i just need practice. someone please refresh my memory with disk and washers area and volume formulas for both!
Related Rates:
1. Identify all variables and equations
2. Identify what you are looking for
3. Make a sketch and label
4. Write an equation involving your variables
5. Take the derivative with respect to time
6. Substitute in derivative and solve.
Example:
Air is being pumped into a spherical balloon at a rate of 4.5 cubic ft/min. Find the rate of change of the radius when the radius is 2 ft.
1. r=2ft; dv/dt = 4.5 ft^3/min
2. NEED TO FIND dr/dt.
3. Volume of sphere: v=4/3 pi r^3
4. derivative: dv/dt = 4 pi r^2 dr/dt
5. Plug in: 4.5 = 4pi(2)^2 dr/dt
6. 4.5=16 pi dr/dt
dr/dt = 9/32 pi ft/min
i finally am starting to understand all the lram rram tram mram stuff, but i couldn't explain it. i have to see it. AP tests seem to have a lot of repeating questions. i get e integration too!
u is always whatever e is raised to(the exponent)
du is always the derivative.
simply use substitution and solve.
i also understand disks and washers.. but unfortunately i did forget the formulas. haha
what i mainly have problems with is word problems.. which is pretty much everything! so, idk what to do. i guess i just need practice. someone please refresh my memory with disk and washers area and volume formulas for both!
Ash's 21st Post
So, this blog is not going to explain much, if anything, because, I'll be honest, there isn't much that I understand. I think I'm getting more confused as the days pass... These AP tests are killing me..I'm doing horribly on these. I don't know why! Maybe it's the fact that it's supposed to be beastly hard and my mind is thinking that I can't do it (subconsciously...wow, shows my self esteem level, eh?).
I think I may have asked this before, but Is there a method to taking the AP tests the way there are methods to take the ACT and whatnot?
I'm getting confused on even the limits! After a little bit of derivatives, I finally see that I can just take the coefficients and divide them, but I'm making simple things way too complicated and complicated things...even more so than they already are!
I second Steph's blog..I also need help with the keywords and what triggers what. I've always had that problem in Math.
I look at my notebook, and think something looks simple, but after I look away and try to do an example problem, I FAIL!! I try to learn this information, but I'm pretty sure, as Mrs. Robinson has suggested once before, that I'm not studying correctly...well, not correctly, but you know what I mean. For formulas, Riemann Sums, for example, what is your method of memorizing them? I think I need to try different methods of studying, but I want to see what works for others and see if maybe I can combine their methods to study more efficiently.
On the AP tests, can someone explain to me a cone problem? One of them is number 17, but I couldn't read the problem, so I've only got a diagram. But I've never understood cones.
Oh, another thing I don't understand is tables! I cannot read them!! I don't know if I see a table and freak out or what. I get confused with the tables like this
x | f(x) | f'(x) | g
and there's a bunch of numbers under here..
I can never tell what they want...does anyone have a method of reading tables? That can also come in handy for the Science portion of the ACT!
Speaking of which! I cannot find this online, but I may not have searched the right things. Does anyone know what score you need in math to get out of a college course? Just like if you get a 30 in English, you get to skip English 101 and 102 or something like that. My scores in math this time leave much to be desired. They keep decreasing, does anyone know any good websites that can help or give problems?
Sorry for all of the questions :)
I think I may have asked this before, but Is there a method to taking the AP tests the way there are methods to take the ACT and whatnot?
I'm getting confused on even the limits! After a little bit of derivatives, I finally see that I can just take the coefficients and divide them, but I'm making simple things way too complicated and complicated things...even more so than they already are!
I second Steph's blog..I also need help with the keywords and what triggers what. I've always had that problem in Math.
I look at my notebook, and think something looks simple, but after I look away and try to do an example problem, I FAIL!! I try to learn this information, but I'm pretty sure, as Mrs. Robinson has suggested once before, that I'm not studying correctly...well, not correctly, but you know what I mean. For formulas, Riemann Sums, for example, what is your method of memorizing them? I think I need to try different methods of studying, but I want to see what works for others and see if maybe I can combine their methods to study more efficiently.
On the AP tests, can someone explain to me a cone problem? One of them is number 17, but I couldn't read the problem, so I've only got a diagram. But I've never understood cones.
Oh, another thing I don't understand is tables! I cannot read them!! I don't know if I see a table and freak out or what. I get confused with the tables like this
x | f(x) | f'(x) | g
and there's a bunch of numbers under here..
I can never tell what they want...does anyone have a method of reading tables? That can also come in handy for the Science portion of the ACT!
Speaking of which! I cannot find this online, but I may not have searched the right things. Does anyone know what score you need in math to get out of a college course? Just like if you get a 30 in English, you get to skip English 101 and 102 or something like that. My scores in math this time leave much to be desired. They keep decreasing, does anyone know any good websites that can help or give problems?
Sorry for all of the questions :)
Week of January 4
Well since I have a couple practice exams from first hour, and one from fifth hour, and the exam review we did in class last week, I guess I'll do some questions from them.
Water is poured at a constant rate into a conical reservior. If the depth of the water, h, is graphed as a function of time, the graph is:
A) decreasing
B) constant
C) linear
D) concave up
E) concave down
You know the question is talking about the first derivate, because it says a 'constant rate'. From that, you can eliminate that the graph is: C) linear, D) concave up, and E) concave down. You then use logic to realize that if the width of the cone keeps getting bigger or smaller, the height would not be a constant, so the A) constant would be elimated. This would leave you with A) decreasing.
If F(3) = 8 and F '(3) = -4 then F(3.02) is approximately:
A) -8.08
B) 7.92
C) 7.98
D) 8.02
E) 8.03
You can use y-intercept form to solve this problem.
The problem gives you that x = 3, y = 8, and m = -4
So you plug it into the y-intercept form and get:
y - 8 = -4 (x - 3)
Equals: y - 8 = -4x + 12
So you then plug the x they want you to find, 3.02, into the problem and get that:
y = 7.92
So the answer is B) 7.92.
Ahh, well school is cancelled Monday, so I guess I will see you all on Tuesday!!!
Water is poured at a constant rate into a conical reservior. If the depth of the water, h, is graphed as a function of time, the graph is:
A) decreasing
B) constant
C) linear
D) concave up
E) concave down
You know the question is talking about the first derivate, because it says a 'constant rate'. From that, you can eliminate that the graph is: C) linear, D) concave up, and E) concave down. You then use logic to realize that if the width of the cone keeps getting bigger or smaller, the height would not be a constant, so the A) constant would be elimated. This would leave you with A) decreasing.
If F(3) = 8 and F '(3) = -4 then F(3.02) is approximately:
A) -8.08
B) 7.92
C) 7.98
D) 8.02
E) 8.03
You can use y-intercept form to solve this problem.
The problem gives you that x = 3, y = 8, and m = -4
So you plug it into the y-intercept form and get:
y - 8 = -4 (x - 3)
Equals: y - 8 = -4x + 12
So you then plug the x they want you to find, 3.02, into the problem and get that:
y = 7.92
So the answer is B) 7.92.
Ahh, well school is cancelled Monday, so I guess I will see you all on Tuesday!!!
post 21
okay, well umm, this last week we practiced ap tests and although it makes me sad to say, i think i did betteron the no calculator portion of the test than i did on the calculator allowed portion. but now is not the time for that, so i am just going to get this blog over with. for this blog i am going to write about riemann summs.
The first formula you need to know is x=(b-a)/n [a,b] with n subintervals. You will need to know this because each of the next formulas require that you know what x is.
LRAM- left hand approximation. (this puts the rectangles used to find the area on the left side of the curve) x[f(a)+f(a+x)+...f(b)]
RRAM- right hand approximation. (this puts the rectangles used to find the area on the right side of the curve) x[f(a+x)+...f(b)]
MRAM- approximation from the middle. (this puts the rectangles right on top of the curve, so that the curve goes through the middle of each one) x[f(mid)+f(mid)+...]
Trapezoidal- this does not use squares, instead it uses trapezoids to eliminate most of the empty space inside the curve, and I think this is the most accurate. x/2[f(a)+2f(a+x)+2f(a+2x)+...f(b)]
okay, well i know how to do all the stuff i mentioned above except for mram, so even though i have the formula i am not exactly sure how to do it. any help would be appreciated
The first formula you need to know is x=(b-a)/n [a,b] with n subintervals. You will need to know this because each of the next formulas require that you know what x is.
LRAM- left hand approximation. (this puts the rectangles used to find the area on the left side of the curve) x[f(a)+f(a+x)+...f(b)]
RRAM- right hand approximation. (this puts the rectangles used to find the area on the right side of the curve) x[f(a+x)+...f(b)]
MRAM- approximation from the middle. (this puts the rectangles right on top of the curve, so that the curve goes through the middle of each one) x[f(mid)+f(mid)+...]
Trapezoidal- this does not use squares, instead it uses trapezoids to eliminate most of the empty space inside the curve, and I think this is the most accurate. x/2[f(a)+2f(a+x)+2f(a+2x)+...f(b)]
okay, well i know how to do all the stuff i mentioned above except for mram, so even though i have the formula i am not exactly sure how to do it. any help would be appreciated
Post #21
Hello, and i'd like to wish everyone a wonderful day tomorrow being that we do not have school. YES! Finally, our day of for inclimate weather! :)
So, i'm not really comfortable with any of the AP things we have been going over lately..so instead of lying about this, i'm going to try and get some help, because with my scores..it's much needed.
If anyone can:
*help me remember which words in the question tell me to do what Calculus operation
Like: telling between when to take an integral for velocty and speed and when to take the derivative of something.
*Related rate problems
I've always had trouble with these, I can't quite grasp the steps and mix it in with the problem.
*Substitution
This is something i didn't understand from the beginning. I don't understand how you plug in the integral and equation..because i always end up with taking the integral of the original equation..help?
*Linearization
I'm not quite sure when to do it, i know i'm supposed to do linearization when i see approximate..but i'm not sure how to do it. Is it just finding a slope?
I know this isn't one of my best blogs, but honestly..i'm doing this for my help, these are the topics i need buku help with..and when i got my scores back it made me realize that i'm not comfortable with any of my calculus topics..
Hopefully someone out there has some good hints and clues for me..because there's a short in my brain when i'm reading my notes and reviewing the problems, i just can't find the key words and make hints of what to do!
So, i'm not really comfortable with any of the AP things we have been going over lately..so instead of lying about this, i'm going to try and get some help, because with my scores..it's much needed.
If anyone can:
*help me remember which words in the question tell me to do what Calculus operation
Like: telling between when to take an integral for velocty and speed and when to take the derivative of something.
*Related rate problems
I've always had trouble with these, I can't quite grasp the steps and mix it in with the problem.
*Substitution
This is something i didn't understand from the beginning. I don't understand how you plug in the integral and equation..because i always end up with taking the integral of the original equation..help?
*Linearization
I'm not quite sure when to do it, i know i'm supposed to do linearization when i see approximate..but i'm not sure how to do it. Is it just finding a slope?
I know this isn't one of my best blogs, but honestly..i'm doing this for my help, these are the topics i need buku help with..and when i got my scores back it made me realize that i'm not comfortable with any of my calculus topics..
Hopefully someone out there has some good hints and clues for me..because there's a short in my brain when i'm reading my notes and reviewing the problems, i just can't find the key words and make hints of what to do!
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