So, the twelfth week of school and calculus is over, but we didn't really go over any new topics this week. We reviewed and studied for the tests on wednesday and thursday, and i understand related rates and implicit derivatives and second implicit derivatives, but some of the stuff on those tests were super hard! but a lot of it was my fault because i did not study good, next time, i'll remember to study rolles theorem, mean value theorem, and all those geometric formulas that come in handy for related rates. anyway, i supposed i should do an example problem or two, to show what i do understand. here we go...
first, i'll say some stuff about limits that i failed to know before we took the tests.
1. if the top and bottom exponents are the same, the answer is the top coefficient over the bottom coefficient.
2. if the top exponent is bigger than the bottom exponent, the answer is infinity.
3. if the top is less than the bottom, it goes to 0.
Next, i just found a problem that i knew how to do in some old packet that we had, so i will post this as an example.
EXAMPLE: Assume that x and y are both differentiable functions of t. Find dy/dt when x=64 and dx/dt=5 for the equation y=sqrt(x)
first, you take the derivative of the equation, so you get dy/dt=1/2(x)^-1/2(dx/dt)
then, you just plug in the given values, so you get dy/dt=1/2(64)^-1/2(5)
and when you solve, you get dy/dt=5/16.
now, for what i don't understand, it is linearization, big surprize... i know you have to draw the little picture, and take the derivative of the formula, and plug in and everything, but i just don't know how to do the stuff...
Monday, November 9, 2009
Post #12
Alright, another week of Calculus down. I studied, failed, and learned this week. I do understand most of the related rates problems and more of limits, but I am still having trouble with things we learned last week.
I really understand problems like these so, let me do one:
The radius, r, of a circle is increasing at a rate of 2 centimeters per minute. Find the rate of change of area, A, when the radius is 10.
1. It helps me to copy down what is given and what I'm trying to exactly find:
A=(pi)r^2
dr/dt= 2
r= 10
dA/dt= ?
2. You then take the derivative of the formula, and plug in your given:
dA/dt= 2r(dr/dt)(pi)
dA/dt= (2)(10)(2)pi
dA/dt= 40pi
Also, we went over limits this week which I'm understanding better.
Rules for Limits:
1. if the degree of top equals thedegree of bottom, the answer is the top coefficient over bottom coefficient
2. if top degree is bigger than bottom degree, the answer is positive or negative infinity
2. if top degree is less than bottom degree, the answer is 0
B-rob also stressed to us that problems when you have to plug in and the denominatior equals zero then you have to either factor and cancel OR plug into your calculator.
For what I don't understand:
angle of elevation and linearization
I really understand problems like these so, let me do one:
The radius, r, of a circle is increasing at a rate of 2 centimeters per minute. Find the rate of change of area, A, when the radius is 10.
1. It helps me to copy down what is given and what I'm trying to exactly find:
A=(pi)r^2
dr/dt= 2
r= 10
dA/dt= ?
2. You then take the derivative of the formula, and plug in your given:
dA/dt= 2r(dr/dt)(pi)
dA/dt= (2)(10)(2)pi
dA/dt= 40pi
Also, we went over limits this week which I'm understanding better.
Rules for Limits:
1. if the degree of top equals thedegree of bottom, the answer is the top coefficient over bottom coefficient
2. if top degree is bigger than bottom degree, the answer is positive or negative infinity
2. if top degree is less than bottom degree, the answer is 0
B-rob also stressed to us that problems when you have to plug in and the denominatior equals zero then you have to either factor and cancel OR plug into your calculator.
For what I don't understand:
angle of elevation and linearization
WEEK TWELVE
So this week in Calculus we reviewed Monday and Tuesday for the multiple-choice test on Wednesday and the free response test on Thursday. On Friday, we reviewed limits because a majority of the class did not do well on that.
Limits:
Infinite limits:
1) if degree of top = degree of bottom:: the answer is the top coefficient over bottom coefficient.
2) top degree > bottom degree = +- infinity
3) top degree < bottom degree = 0
Finite Limits:
Plug in x
Examples:
limit x à infinity x / (x^3 + 9) = 0
limit x à 3 x / (x^3 + 9) = 3 / 36 = 1 / 12
Something I still do not completely understand is related rates. I get how to do them, just don’t understand some of the properties.
Example: Let A be the area of a circle of radius r that is changing with respect to time. If dr/dt is constant, is dA/dt contant? Explain.
Limits:
Infinite limits:
1) if degree of top = degree of bottom:: the answer is the top coefficient over bottom coefficient.
2) top degree > bottom degree = +- infinity
3) top degree < bottom degree = 0
Finite Limits:
Plug in x
Examples:
limit x à infinity x / (x^3 + 9) = 0
limit x à 3 x / (x^3 + 9) = 3 / 36 = 1 / 12
Something I still do not completely understand is related rates. I get how to do them, just don’t understand some of the properties.
Example: Let A be the area of a circle of radius r that is changing with respect to time. If dr/dt is constant, is dA/dt contant? Explain.
Post 12
So last week we reviewed and took two tests. Wednesday we took the first part of the test, which was the multiple choice part, and Thursday we took the free response part of the test. I completley failed the multiple choice portion of the test Wednesday. Thursday it felt like I did better. On number one I had no clue what I was doing because it was the one kind of problem I didn't know how to do. It was a problem with related rates, I think, and air planes. Supposedly the answer to the question was either 500 or -500. I ended up with 499.25? I have no clue how, but it's pretty close to the real answer. So on the multiple choice portion of the test, many people (including myself) did horribly on limits. I have no clue why I did horribly on limits because I know how to work them. I just blanked out. Anyway, because of our failure on limits, Mrs. Robinson gave us a packet that concentrated only on limits. She also gave us notes once again on how to find and figure out limits.
The steps she gave us are as follows:
1. if when plugged in, the denominatior of the limit equals zero
a. factor and cancel
b. plug into your calculator
Many people do not know how to use their calculator to find a limit, so I'll give the steps
1. plug limit into y=
2. second table
3. plug numbers to the left and right of what x is approaching
Ex: lim x->0(sin^8x/x^8)
You would first plug this limit into y=, then you would hit second table. Once you get into your table, you plug in values to the left and right of zero (-.1, -.01, -.001, .001, .01, .1). When these values are plugged in, both sides give you 0.9. This shows you that the limit x->0(sin^8/x^8) approaches 1.
I still don't know how to do angle of elevation problems or problems with air planes.
The steps she gave us are as follows:
1. if when plugged in, the denominatior of the limit equals zero
a. factor and cancel
b. plug into your calculator
Many people do not know how to use their calculator to find a limit, so I'll give the steps
1. plug limit into y=
2. second table
3. plug numbers to the left and right of what x is approaching
Ex: lim x->0(sin^8x/x^8)
You would first plug this limit into y=, then you would hit second table. Once you get into your table, you plug in values to the left and right of zero (-.1, -.01, -.001, .001, .01, .1). When these values are plugged in, both sides give you 0.9. This shows you that the limit x->0(sin^8/x^8) approaches 1.
I still don't know how to do angle of elevation problems or problems with air planes.
post 12
So Monday and Tuesday we reviewed stuff we didn't understand so we could be ready for the tests on wednesday and thursday. i stilll have trouble w/ most of it. our testw as on everything we did this whole y ear. i understand some of it now.
im not really good at lookin at a graph and finding its horizontal tangents and whether or not it is concave up or down. and i dont know what its derivative is supposed to look like and i'd like some hlep w/ this :)
Another thing I still can't seem to understand is optimization and angles of elevation. I don't know what my problem is becuase i dont know it from teh beginning. i just get so confused in what im trying to do. i know the steps well enough its just actually doin the steps that gets me.
So after the tests we found out that as a whole we suck at limits.
limit rules:
1. If the degree of the top equals the degree of the bottom the answer is the top coefficient over the bottom coefficient.
For example, lim x>infinity x^5/4x^5 + 2 = 1/4
2. If the degree of the top is greater than the degree of the bottom, the answer is + or - infinity
Ex: lim x>infinity x^5 -3x^1/ x + 1 = infinity
3. If the degree of the top i less than the degree of the bottom, the answer is 0.
Ex: lim x>infinity x/x^8 + 2 = 0
the rules are pretty easy.
Linearization is something i dont really think i know too well. so if anyone wants to help w/ all this i'd appreciate it ;)
im not really good at lookin at a graph and finding its horizontal tangents and whether or not it is concave up or down. and i dont know what its derivative is supposed to look like and i'd like some hlep w/ this :)
Another thing I still can't seem to understand is optimization and angles of elevation. I don't know what my problem is becuase i dont know it from teh beginning. i just get so confused in what im trying to do. i know the steps well enough its just actually doin the steps that gets me.
So after the tests we found out that as a whole we suck at limits.
limit rules:
1. If the degree of the top equals the degree of the bottom the answer is the top coefficient over the bottom coefficient.
For example, lim x>infinity x^5/4x^5 + 2 = 1/4
2. If the degree of the top is greater than the degree of the bottom, the answer is + or - infinity
Ex: lim x>infinity x^5 -3x^1/ x + 1 = infinity
3. If the degree of the top i less than the degree of the bottom, the answer is 0.
Ex: lim x>infinity x/x^8 + 2 = 0
the rules are pretty easy.
Linearization is something i dont really think i know too well. so if anyone wants to help w/ all this i'd appreciate it ;)
Sunday, November 8, 2009
Post #12
So this week in calculus, we spent most of the week preparing for our tests on Wednesday and Thursday, and on Friday we reviewed limits.
Lim as x ->-10 x+10/ x^2 – 100
If you plug in 10 into the bottom you get zero so you have to rely on other methods.
In this case, the bottom can factor out to (x-10)(x+10), cancelling out the (x+10)s and leaving you with 1/ (x-10).
You now can plug in your -10 and get -1/20 as your limit.
Example 2: Find the limit:
Lim x-> 0 sin^5x/x^5
Since you cannot factor anything out, you have to plug into the table function of your calculator.
Hit y= and plug in your equation, don’t forget to put parentheses then hit 2nd graph.
Once in your table function, you have to enter numbers to the right of zero and to the left of zero.
0-.1= .9917
0-.01= .99992
0-.001=1
0+. 001=1
0+. 01= .99992
0+. 1= .9917
The limit as x approaches 0 is 1.
To find the limit as x approaches infinity, you use your limit rules, which are:
1. If the degree at the top is greater than the degree at the bottom than the limit is + or – infinity.
Lim x-> infinity 4x^2+3/ 5x+9
Limit= infinity
2. If the degree at the top is less than the degree at the bottom, then the limit is zero.
Limit = 0
3. If the degree at the top is equal to the degree at the bottom, then you divide the coefficients to find your limit.
Lim x-> infinity 3x^2+8/ x^2+ 7
Lim= 3
I seem to still be having trouble with graphs such as when giving the graph of f’(x) and you have to find where f’’(x) is concave up or down or where f(x) is increasing or decreasing. I understand what words go with each graph except I don’t understand how to find what they are asking for. Also, if the question asks to find where f(x) is concave up or down, do you find where f’’(x) is concave up or down because concave up and down goes with the second derivative? Or do you find where the original graph is concave up or down?
Help would greatly be appreciated since this shows up on every test.
Posting...#12
So last week was an average calculus week. Monday we reviewed for out big test on Wednesday and Thursday. Tuesday we reviewed on our test that was Wednesday and Thursday. Wednesday we took a multiple choice test that I think I did alright on but I could of did bad or I could of did really good. Thursday now I didn’t do good at all on our free response test I don’t know what happen I just didn’t know the stuff. Friday we got a limit packet I think because people did bad on the limit parts of the test and I had like 3 question but I left mine at school but at least I’m almost finished it.
So some limit rules since we are baddies are:
If the degree of the top equals the degree of the bottom the answer is the top coefficient over the bottom coefficient.For example, lim x>infinity x^5/4x^5 + 2 = 1/4 If the degree of the top is greater than the degree of the bottom, the answer is + or - infinityEx: lim x>infinity x^5 -3x^1/ x + 1 = infinity
If the degree of the top i less than the degree of the bottom, the answer is 0.Ex: lim x>infinity x/x^8 + 2 = 0
So since I can’t ask my few limit questions I have I’ll on how to do angle of elevations since I’m a BK(bad kid) at them and I ask this question every week it’s just I can’t learn how to do them so help.
So some limit rules since we are baddies are:
If the degree of the top equals the degree of the bottom the answer is the top coefficient over the bottom coefficient.For example, lim x>infinity x^5/4x^5 + 2 = 1/4 If the degree of the top is greater than the degree of the bottom, the answer is + or - infinityEx: lim x>infinity x^5 -3x^1/ x + 1 = infinity
If the degree of the top i less than the degree of the bottom, the answer is 0.Ex: lim x>infinity x/x^8 + 2 = 0
So since I can’t ask my few limit questions I have I’ll on how to do angle of elevations since I’m a BK(bad kid) at them and I ask this question every week it’s just I can’t learn how to do them so help.
12th post
This week in calculus, we took a test on everything we have learned so far this year. I definitely know how to do related rates, limits, and first derivative and second derivative test. The only thing i have trouble with when it comes to first derivative and second derivative tests is when i have to deal with tangent functions. Here is an example of related rates.
Lets say they want to know the rate of change of the volume of a sphere:
the radius is 5, and the rate of change of the sphere is 10. Find Dv/Dt.
the formula for the volume of a sphere is:
V= 1/3(pi)(r^3)
so from here you can plug in the given:
V=1/3(pi)(5^3)(10)
now take the derivative of the function:
dV/dt=(pi)(5^2)(10)
now you can simplify and solve for dV/dt:
dV/dt= 250(pi)
Now lets look at an example of limits:
Find the limit as x goes to infinity
x^3-8
-------
2x^3-4
in this problem we can use our limit rules. Since the degree on the top is the same as the degree on the bottom, the answer will be the coefficients:
so the answer is x-> infinity is 1/2
Some of the things i still do not understand is angle of elevation, and optimization. When dealing with optimization, once i figure out what my formulas are, i can work the problem from there, but i still have trouble figuring out the formulas and drawing a picture to find the formulas. When dealing with angle of elevation, i do not even understand where to start. I do not know how it is different from related rates and i need somebody to explain exactly what i'm looking for and what i need to work the problem. If anyone can help me out i would really appreciate it. Thanks :)
post number 12
This week we took a test and studied alot.
One thing that i understand in calculus is related rates.
First you read your problem and dertimne what you are givin. Be sure to watch out if they give you something like ten miles/Hr, that you put that as a rate (dr/dt). Then you sketch and label your problem. Usually means you draw a traiangle. After that you set up your equation and take the derivative. when taking the derivative you have to be sure to take it with respect to time. When you do this you plug in your givins and solve for your unknown. Be sure when solving you only have one unknown. After you plug in your left with a simpel equation that can be easily solved.
One thing that i dont understand to much in calc is things such as tangent lines and limits when looking at the graph. I dont know but i guess i just didn't pick up these lessons. Tangent lines just mess with me. I know you take the derivative but once i do that i get lost and confused when plugging into slope or which ever formula it is.
One thing that i understand in calculus is related rates.
First you read your problem and dertimne what you are givin. Be sure to watch out if they give you something like ten miles/Hr, that you put that as a rate (dr/dt). Then you sketch and label your problem. Usually means you draw a traiangle. After that you set up your equation and take the derivative. when taking the derivative you have to be sure to take it with respect to time. When you do this you plug in your givins and solve for your unknown. Be sure when solving you only have one unknown. After you plug in your left with a simpel equation that can be easily solved.
One thing that i dont understand to much in calc is things such as tangent lines and limits when looking at the graph. I dont know but i guess i just didn't pick up these lessons. Tangent lines just mess with me. I know you take the derivative but once i do that i get lost and confused when plugging into slope or which ever formula it is.
post 12
Twelfth week of calculus. Monday-Study Tuesday-Study Wednesday-Test Thursday-Test Friday-LEARN!
Our test was on everything we did this whole year. I actually understand a lot of things now that we reviewed for everything for two days. The things I didn't understand on the test were just the formulas. I knew how to work all the related rates and angle of elevation problems but I couldn't remember any of the formulas.But I finally did learn them, so here they are.
CONE- V=3/4(pi)(h)(r^2)
CUBE- V=E^3
RECTANGLE- A=1/2(B)(H)
SQUARE- A=L x W P=2(L+W)
SPHER- V=1/3(pi)(r^3)
CIRCLE- A=2(pi)(r)
how to find a tangent line:
1. Take derivative
2. Plug x in to derivative to find slope
3. Plug slope into original to find y-intercepts
4. plug in slope and point into the equation (y-y1)=m(x-x1).
I don't really understand linearization. Well, i really just dont remember it. Because we didn't have that much practice, but hopefully we'll be going over it next week. If anyone did understand it please refresh my memory.
I don't really understand linearization. Well, i really just dont remember it. Because we didn't have that much practice, but hopefully we'll be going over it next week. If anyone did understand it please refresh my memory.
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