Ok, so last week we pretty much just reviewed for the upcoming exams. Mrs. Robinson gave us two practice APs, which I failed. They mostly went over things we learned recently like integrals. The first one was much much better than the second one, which is pretty pathetic considering the second one was calculator allowed. We were also given the take home portion of the exam. This portion is only twenty multiple choice questions and is also calculator allowed. The one we will take on exam did will include multiple choice and essay, both of which calculators will not be allowed. Today I was going through the take home portion and I realized I forgot a lot of things we did in the beginning of the year. I thought I knew how to do some problems, but my answers were coming out wrong.
Anyway, two weeks ago we learned how to find volume of a curve. This is done by rotating the curve about an axis or line to make it a solid object. We did this by using both disks and washers. Disks are used when there is only one equation and the object rotated turns out to be a solid. When there are two equations, washers are used since there will be a hole in the figure.
The formula for disks is:
pi S a-b [R(x)]^2 dx
This is basically a definite integral where you plug in the function you are given and square it. Many people forget to square it. This completley changes everything, so don't forget.
The formula for washers is:
pi S a-b top^2-bottom^2 dx
This is also a definite integral where you have to square the functions. I like these because they're very similar to area under a curve, and I also liked those. The only difference is to square both functions given and put a pi on the outside of the integral.
Most of the things we learned recently I'm fine with, but I came to realize I'm confused with a lot of old things. I get average value, average rate of change, and other various things with those words arranged in different ways. I never really caught onto average rate of change, and I still don't know what it is. I also get confused with linerizaton. When I look at the questions, I never know what to do with them. Can someone explain?
Sunday, December 13, 2009
post 17
This week was mainly a review week in order to get ready for the upcoming midterm exam. We reviewed everything that we learned throughout this year.
Another thing I am going to talk about since I understand is linearization. The steps for working linearization problems are:
1. Identify the equation
2. Use the formula f(x)+f ' (x)dx
3. Determine your dx in the problem
4. Then determine your x in the problem
5. Plug in everything you get
6. Solve the equation
Also I will talk about integration. Integration is used to find the area under a curve. The Riemann sum approximates the area using the rectangles or trapezoids. The Riemanns Sums are:
LRAM-Left hand approximation=delta x[f(a)+f(a+delta x)+...f(b-delta x)]
RRAM-Right hand approximation=delta x[f(a+delta x)+...f(b)]
MRAM-Middle approximation=delta x[f(mid)+f(mid)+...]
Trapezoidal-delta x/2[f(a)+2f(a+delta x)+2f(a+2 delta x)+...f(b)]
*delta x=b-a/number of subintervals
Related rates are one thing I understood fromt the start so I am going to talk about them next. The steps for related rates are:
1. Identify all of the variables and equations
2. Identify the things that you are looking for
3. Sketch a graph and then label that graph
4. Create and write an equation using all of the variables
5. Take the derivative of this equation with respect to time
6. Substitute everything back in
7. Solve the equation
Also I am going to talk about taking implicit derivatives since I understand how to take them. The steps for taking implicit derivatives are:
1. Take the derivative of both sides like you would normally do
2. Everytime the derivative of y is taken it needs to be notated with either y ' or dy/dx
3. Solve for dy/dx or y ' as if you are solving for x.
For things I am not good with is the trapezoidal rule. I get LRAM, RRAM, and MRAM but I just can not get trapezoidal. So if someone can help me before the test that would be great.
Another thing I am going to talk about since I understand is linearization. The steps for working linearization problems are:
1. Identify the equation
2. Use the formula f(x)+f ' (x)dx
3. Determine your dx in the problem
4. Then determine your x in the problem
5. Plug in everything you get
6. Solve the equation
Also I will talk about integration. Integration is used to find the area under a curve. The Riemann sum approximates the area using the rectangles or trapezoids. The Riemanns Sums are:
LRAM-Left hand approximation=delta x[f(a)+f(a+delta x)+...f(b-delta x)]
RRAM-Right hand approximation=delta x[f(a+delta x)+...f(b)]
MRAM-Middle approximation=delta x[f(mid)+f(mid)+...]
Trapezoidal-delta x/2[f(a)+2f(a+delta x)+2f(a+2 delta x)+...f(b)]
*delta x=b-a/number of subintervals
Related rates are one thing I understood fromt the start so I am going to talk about them next. The steps for related rates are:
1. Identify all of the variables and equations
2. Identify the things that you are looking for
3. Sketch a graph and then label that graph
4. Create and write an equation using all of the variables
5. Take the derivative of this equation with respect to time
6. Substitute everything back in
7. Solve the equation
Also I am going to talk about taking implicit derivatives since I understand how to take them. The steps for taking implicit derivatives are:
1. Take the derivative of both sides like you would normally do
2. Everytime the derivative of y is taken it needs to be notated with either y ' or dy/dx
3. Solve for dy/dx or y ' as if you are solving for x.
For things I am not good with is the trapezoidal rule. I get LRAM, RRAM, and MRAM but I just can not get trapezoidal. So if someone can help me before the test that would be great.
post 17
we reviewed mostly this past week so ima show you the formulas andd steps for washers and disks.
The formula for the volume of disks is S (top)^2 - (bottom)^2 dx
The formula for the area of washers is S (top) - (bottom)
STEPS:
1. Draw the graphs of the equations
2. Subtract top graph's equation by the bottom graph's equation(in disks each equation would be squared)
3. Set equations equal and solve for x to find bounds
4. Plug in the bounds and the outcome of step 2
5. Integrate
volume by disks:
the formula is pi times the integral of the [function given] squared times dx. so just solve it by taking the integral of it and then pluging in the numbers they give you. just like before you'll have two numbers so whatever the answer is for the top one will be first and then you subtract the answer you get for the bottom one. then graph
volume by washers:
the formla is pie times the integral of the [top function] squared minus the [bottom function] squared times dx. so to do this, if you don't have the in between number you have to set the functions equal, but if you do, then it's worked the same way as above. square the formula's that were given and simplify. then take the integral of it and plug in the numbers they give you or you found by setting the formulas equal to each other and then solve like any other one by subracting them. then graph.
im not too good at related rates, optimization, and angles of elevevation still.
The formula for the volume of disks is S (top)^2 - (bottom)^2 dx
The formula for the area of washers is S (top) - (bottom)
STEPS:
1. Draw the graphs of the equations
2. Subtract top graph's equation by the bottom graph's equation(in disks each equation would be squared)
3. Set equations equal and solve for x to find bounds
4. Plug in the bounds and the outcome of step 2
5. Integrate
volume by disks:
the formula is pi times the integral of the [function given] squared times dx. so just solve it by taking the integral of it and then pluging in the numbers they give you. just like before you'll have two numbers so whatever the answer is for the top one will be first and then you subtract the answer you get for the bottom one. then graph
volume by washers:
the formla is pie times the integral of the [top function] squared minus the [bottom function] squared times dx. so to do this, if you don't have the in between number you have to set the functions equal, but if you do, then it's worked the same way as above. square the formula's that were given and simplify. then take the integral of it and plug in the numbers they give you or you found by setting the formulas equal to each other and then solve like any other one by subracting them. then graph.
im not too good at related rates, optimization, and angles of elevevation still.
post 17
alright so this week in calc we reviewed, reviewed, reviewed, and took two practice ap tests, which i actually did pretty good on ! :-)
calc exam is tuesday, and i just found out that the packet b rob gave us on friday is due the day of the exam! i thought it was our packet for over the christmas holidays! ahhhhh, i need to get started on that. haha. okay well, i'll explain average rate of change.
you are given a problem y=3t+2 and a time iterval from [0,2] seconds. well, your formula is:
f(b)-f(a)/b-a. 0=a,2=b. so, first you must plug in 0 and 2 to find out what your plug ins will be. so you would get [2,8]. then plug it into your formula, 6/2 = 3. so your answer would be three. don't forget to put your units behind it though, whatever that may be.
i'll also, tell you limit rules for finding infinity because i will never ever ever forget these.
if a limit is approaching infinity:
1. top degree < bottom degree it's equal to 0
2. top degree > bottom degree it's equal to +/- infinity. ****
3. top degree = bottom degree divide leading coefficients.
****find out if it's positive or negative infinity by graphing it in your calculator.
also, i'll throw in that volume for disks and washers is squared, while area is not. although it is the exact same formula. figured i would just say that since a lot of people seem to forget those formulas, when really you only have to remember one.
what i dont get still, is LRAM, MRAM, RRAM, TRAM.
also i kinda forgot how to find the equation of a tangent line. don't make funny of me, but i can't remember to save my life and it's driving me crazy cuz i know how simple it is. also, if anyone wants to go over optimization just to refresh my memory, and all the related rates stuff would be nice too :-)
calc exam is tuesday, and i just found out that the packet b rob gave us on friday is due the day of the exam! i thought it was our packet for over the christmas holidays! ahhhhh, i need to get started on that. haha. okay well, i'll explain average rate of change.
you are given a problem y=3t+2 and a time iterval from [0,2] seconds. well, your formula is:
f(b)-f(a)/b-a. 0=a,2=b. so, first you must plug in 0 and 2 to find out what your plug ins will be. so you would get [2,8]. then plug it into your formula, 6/2 = 3. so your answer would be three. don't forget to put your units behind it though, whatever that may be.
i'll also, tell you limit rules for finding infinity because i will never ever ever forget these.
if a limit is approaching infinity:
1. top degree < bottom degree it's equal to 0
2. top degree > bottom degree it's equal to +/- infinity. ****
3. top degree = bottom degree divide leading coefficients.
****find out if it's positive or negative infinity by graphing it in your calculator.
also, i'll throw in that volume for disks and washers is squared, while area is not. although it is the exact same formula. figured i would just say that since a lot of people seem to forget those formulas, when really you only have to remember one.
what i dont get still, is LRAM, MRAM, RRAM, TRAM.
also i kinda forgot how to find the equation of a tangent line. don't make funny of me, but i can't remember to save my life and it's driving me crazy cuz i know how simple it is. also, if anyone wants to go over optimization just to refresh my memory, and all the related rates stuff would be nice too :-)
WEEK 17
Since we took the two practice AP exams this week, I think I'll just explain some problems off of them.
Number 5: If y = e^2x + tan(2x), then y'(pi) =
first take the derivative: y' = e^2x (2) + sec^2(2x)(2)
Then you plug in pi: 2e^2pi + 2sec(2pi)^2 = 2e^2pi + 2
Number 16: If y = sin^2(5x), dy/dx =
dy/dx = 2 (sin 5x)(cos 5x)(5) <--- take care of exponent, take derivative of sin, take derivative of the inside (5x).
dy/dx = 5 (sin(2 * 5x)) <--- 2sinxcosx = sin2x
dy/dx = 5sin(10x)
Number 18:
remember that f(x) = x
f'(x) = v(x)
f''(x) = a(x)
Number 20: If f(1) = 2 and f'(1) = 5, use the equation of the line tangent to the graph of f at x=1 to approximate f(1.2).
Given: x = 1
y = 2
m = 5
find: x = 1.2
y - 2 = 5 (x - 1)
y - 2 = 5 (1.2 - 1)
y = 3
Number 27:
Always remember that AVERAGE VALUE = INTEGRATION
1/ b-a
Number 28: y = x^2 - 3x. Find y'(1).
y'(x) = 2x - 3
y'(1) = 2(1) - 3
y'(1) = -1
y'(1) = 1
Number 11:
remember that average rate of change = f(b) - f(a) / b - a
Number 5: If y = e^2x + tan(2x), then y'(pi) =
first take the derivative: y' = e^2x (2) + sec^2(2x)(2)
Then you plug in pi: 2e^2pi + 2sec(2pi)^2 = 2e^2pi + 2
Number 16: If y = sin^2(5x), dy/dx =
dy/dx = 2 (sin 5x)(cos 5x)(5) <--- take care of exponent, take derivative of sin, take derivative of the inside (5x).
dy/dx = 5 (sin(2 * 5x)) <--- 2sinxcosx = sin2x
dy/dx = 5sin(10x)
Number 18:
remember that f(x) = x
f'(x) = v(x)
f''(x) = a(x)
Number 20: If f(1) = 2 and f'(1) = 5, use the equation of the line tangent to the graph of f at x=1 to approximate f(1.2).
Given: x = 1
y = 2
m = 5
find: x = 1.2
y - 2 = 5 (x - 1)
y - 2 = 5 (1.2 - 1)
y = 3
Number 27:
Always remember that AVERAGE VALUE = INTEGRATION
1/ b-a
Number 28: y = x^2 - 3x. Find y'(1).
y'(x) = 2x - 3
y'(1) = 2(1) - 3
y'(1) = -1
y'(1) = 1
Number 11:
remember that average rate of change = f(b) - f(a) / b - a
Post #17
So after 11 make-up posts I ran out of stuff to say, so i'm going back to the beginning of the year and talking about some stuff.
Complex Derivatives y=ln(e^x) (Chain Rule)
First off one should should identify the steps of your problem. In this case they would be:
1. Natural Log
2. e^x
you problem should be (1/(e^x)).(e^x)'
then you find the derivative of e^x which is e^x . x' (x'=1)
so your final problem should be (1/(e^x)).(e^x)
After this you have to simplify algebraically, giving you (e^x)/(e^x) ,which equals 1.
First Derivative Test:
1. Take the derivative of the original problem.
2. Set the first derivative equal to Zero.
3. Solve for x.
4. Create intervals for x. i.e. (-∞, 1) (1, 4) (4, ∞)
5. Pick a number in the intervals then plug that number in the first derivative for x.
6. Solve. For positive numbers, the graph of the derivative is above the x-axis. For negative numbers, the graph of the derivative is below the x-axis. The numbers for x are your points of inflection. (Points of Inflection are only if there is a shift in the graph!!!)
Second Derivative Test:
1. Take the derivative of the first derivative.
2. Set the second derivative equal to Zero.
3. Solve for x.
4. Create intervals for x. i.e. (-∞, 1) (1, 4) (4, ∞)
5. Pick a number in the intervals then plug that number in the second derivative for x.
6. Solve. For positive numbers, the graph of the derivative is above the x-axis. For negative numbers, the graph of the derivative is below the x-axis. The numbers for x are your points of inflection. (Points of Inflection are only if there is a shift in the graph!!!)
Complex Derivatives y=ln(e^x) (Chain Rule)
First off one should should identify the steps of your problem. In this case they would be:
1. Natural Log
2. e^x
you problem should be (1/(e^x)).(e^x)'
then you find the derivative of e^x which is e^x . x' (x'=1)
so your final problem should be (1/(e^x)).(e^x)
After this you have to simplify algebraically, giving you (e^x)/(e^x) ,which equals 1.
First Derivative Test:
1. Take the derivative of the original problem.
2. Set the first derivative equal to Zero.
3. Solve for x.
4. Create intervals for x. i.e. (-∞, 1) (1, 4) (4, ∞)
5. Pick a number in the intervals then plug that number in the first derivative for x.
6. Solve. For positive numbers, the graph of the derivative is above the x-axis. For negative numbers, the graph of the derivative is below the x-axis. The numbers for x are your points of inflection. (Points of Inflection are only if there is a shift in the graph!!!)
Second Derivative Test:
1. Take the derivative of the first derivative.
2. Set the second derivative equal to Zero.
3. Solve for x.
4. Create intervals for x. i.e. (-∞, 1) (1, 4) (4, ∞)
5. Pick a number in the intervals then plug that number in the second derivative for x.
6. Solve. For positive numbers, the graph of the derivative is above the x-axis. For negative numbers, the graph of the derivative is below the x-axis. The numbers for x are your points of inflection. (Points of Inflection are only if there is a shift in the graph!!!)
Post #17
Alrighty then, lets get started.
Today during our study group, I learned average value thanks to Kaitlyn! So I guess I will explain an example of that since I finally get it.
EX: Find the average value of the function f(x)=12x-6x^2 over the interval -5 less than or equal to x less than or greater than 5.
use the formula: 1/b-a S equation given, then solve the definite integral (*S is integral)
1/5-(-5)= 1/10
1/10 (integral form -5 to 5) 12x-6x^2
1/10(12x-2x^3)
1/10(12(5)-2(5)^3)-1/10(12(-5)-2(-5)^3)
1/10(-190)-1/10(190)
-19-(-19)= -38
I do have quite a few questions though..
1. If I get (-1/64)sin(1/8) and I'm suppose to set it equal to 0 then solve..How do I solve that?
2. (like #11 on the packet) For S(3t/t^2+2)^3 I used substitution to solve. For my final answer I got -3/2(t^2+2)^2+c but all the answers have -3/4(t^2+2)^2+c
3. (like #3 on the packet) It says use a graphing utility to graph the function f(x)=12/6-x and locate the absolute extrema of the function on the interval [0,6). If you plug that into your calculator the graph is an asymptote, and I tried taking the derivative and plugging that in too but it is also an asymptote. So how do you get a max and min?
Today during our study group, I learned average value thanks to Kaitlyn! So I guess I will explain an example of that since I finally get it.
EX: Find the average value of the function f(x)=12x-6x^2 over the interval -5 less than or equal to x less than or greater than 5.
use the formula: 1/b-a S equation given, then solve the definite integral (*S is integral)
1/5-(-5)= 1/10
1/10 (integral form -5 to 5) 12x-6x^2
1/10(12x-2x^3)
1/10(12(5)-2(5)^3)-1/10(12(-5)-2(-5)^3)
1/10(-190)-1/10(190)
-19-(-19)= -38
I do have quite a few questions though..
1. If I get (-1/64)sin(1/8) and I'm suppose to set it equal to 0 then solve..How do I solve that?
2. (like #11 on the packet) For S(3t/t^2+2)^3 I used substitution to solve. For my final answer I got -3/2(t^2+2)^2+c but all the answers have -3/4(t^2+2)^2+c
3. (like #3 on the packet) It says use a graphing utility to graph the function f(x)=12/6-x and locate the absolute extrema of the function on the interval [0,6). If you plug that into your calculator the graph is an asymptote, and I tried taking the derivative and plugging that in too but it is also an asymptote. So how do you get a max and min?
Post #17
so..i've been semi-lost this week..so i'm kicking it back, old school.
So, product rule and quotient rule.
First, product rule is done when you need to take the derivative of things being mulitplied. So, it would be something like x(x^2)
So, the first thing you do, is copy the first term times the derivative of the second term plus the copied second term times the derivative of the first term. Product rule is very simple, you just have to remember the rules of simplificiation. Remember to simplify correctly and if you don't remember what to do next on the simplification process, it probably means you're done simplifying.
Now, lets talk about quotient rule..you know you must use quotient rule because you have a fraction.
The rules for quotient rule is copy the bottom times derivative of the top, minus copy the top times the derivative of the bottom. Like product rule, you need to simplify jsut the same. You need to distribute the things necessary and solve it to the simplest terms possible.
The thing i don't understand is the substitution stuff..i always get the same answer after i do the substitution and don't understand the integral.
So, product rule and quotient rule.
First, product rule is done when you need to take the derivative of things being mulitplied. So, it would be something like x(x^2)
So, the first thing you do, is copy the first term times the derivative of the second term plus the copied second term times the derivative of the first term. Product rule is very simple, you just have to remember the rules of simplificiation. Remember to simplify correctly and if you don't remember what to do next on the simplification process, it probably means you're done simplifying.
Now, lets talk about quotient rule..you know you must use quotient rule because you have a fraction.
The rules for quotient rule is copy the bottom times derivative of the top, minus copy the top times the derivative of the bottom. Like product rule, you need to simplify jsut the same. You need to distribute the things necessary and solve it to the simplest terms possible.
The thing i don't understand is the substitution stuff..i always get the same answer after i do the substitution and don't understand the integral.
17th post
The formula for the volume of disks is S (top)^2 - (bottom)^2 dx
The formula for the area of washers is S (top) - (bottom)
The steps are:
1. Draw the graphs of the equations
2. Subtract top graph's equation by the bottom graph's equation(in disks each equation would be squared)
3. Set equations equal and solve for x to find bounds
4. Plug in the bounds and the outcome of step 2
5. Integrate
volume by disks:
the formula is pi times the integral of the [function given] squared times dx. so just solve it by taking the integral of it and then pluging in the numbers they give you. just like before you'll have two numbers so whatever the answer is for the top one will be first and then you subtract the answer you get for the bottom one. then graph
volume by washers:
the formla is pie times the integral of the [top function] squared minus the [bottom function] squared times dx. so to do this, if you don't have the in between number you have to set the functions equal, but if you do, then it's worked the same way as above. square the formula's that were given and simplify. then take the integral of it and plug in the numbers they give you or you found by setting the formulas equal to each other and then solve like any other one by subracting them. then graph.
LRAM is left hand approximation and the formula is:
delta x [f(a) + f( delta x +a) .... + f( delta x - b)]
Say you are asked to calculate the left Riemann Sum for -4x -5 on the interval [-3, -1] divided into 2 subintervals.
delta x would equal: -1+3 /2 = 2/2 = 1
1[ f(-3) + f(-3 +1)]
1[ f( -3) + f(-2)]
then plug into your equation
RRAM is right hand approximation and the formula is:
delta x [ f(a + delta x) + .... + f(b)]
so using the same example:
1[ f( -2) + f(-1)] and then plug into your equation
MRAM is to calculate the middle and the formula is:
delta x [ f(mid) + f(mid) + .... ]
To find midpoints, you would add the two numbers together then divide by two
In this problem the numbers would be: -3 , -2, -1
-3 + -2/ 2 = -5/2 and -2 + -1 / 2 = -3/2
so 1[f(-5/2) + f(-3/2)] and the plug in
Trapezoidal is different because instead of multiplying by delta x, you multiply by delta x/2 and you also have on more term then your number of subintervals.
The formula is : delta x/2 [f(a) + 2f(a + delta x) + 2f(a+ 2 delta x) + ....f(b)]
For this problem: 1/2 [ f(-3) + 2 f(-2) + f( -1)] and then plug in.
Substitution takes the place of the derivative rules for problems such as product rule and quotient rule. The steps to substitution are:
1. Find a derivative inside the interval
2. set u = the non-derivative
3. take the derivative of u
4. substitute back in
e integration:
whatever is raised to the e power will be your u and du will be the derivative of u. For example:
e^2x-1dx
u=2x-1 du=2
rewrite the function as:
1/2{ e^u du, therefore
1/2e^2x-1+C will be the final answer.
related rates:
The steps for related rates are….
1. Pick out all variables
2. Pick out all equations
3. Pick out what you are looking for
4. Sketch a graph and label
5. Create an equation with your variables
6. Take the derivative respecting time
7. Substitute back into the derivative
8. Solve
limits:
Rules for Limits:…
1. if the degree of top equals the degree of bottom, the answer is the top coefficient over bottom coefficient
2. if top degree is bigger than bottom degree, the answer is positive or negative infinity
2. if top degree is less than bottom degree, the answer is 0
To find area between curves:
The formula you use is b(int)a (top eq.) - (bottom eq.).
If a and b is not already given to you, then you much set the equations equal to each other and solve.
You find which equation is top/bottom by graphing both and simply looking to see which one is on top.
If the area is on the y-axis, then the a and b values need to be set as y-values, and the equations must be solved for x.
The formula for the area of washers is S (top) - (bottom)
The steps are:
1. Draw the graphs of the equations
2. Subtract top graph's equation by the bottom graph's equation(in disks each equation would be squared)
3. Set equations equal and solve for x to find bounds
4. Plug in the bounds and the outcome of step 2
5. Integrate
volume by disks:
the formula is pi times the integral of the [function given] squared times dx. so just solve it by taking the integral of it and then pluging in the numbers they give you. just like before you'll have two numbers so whatever the answer is for the top one will be first and then you subtract the answer you get for the bottom one. then graph
volume by washers:
the formla is pie times the integral of the [top function] squared minus the [bottom function] squared times dx. so to do this, if you don't have the in between number you have to set the functions equal, but if you do, then it's worked the same way as above. square the formula's that were given and simplify. then take the integral of it and plug in the numbers they give you or you found by setting the formulas equal to each other and then solve like any other one by subracting them. then graph.
LRAM is left hand approximation and the formula is:
delta x [f(a) + f( delta x +a) .... + f( delta x - b)]
Say you are asked to calculate the left Riemann Sum for -4x -5 on the interval [-3, -1] divided into 2 subintervals.
delta x would equal: -1+3 /2 = 2/2 = 1
1[ f(-3) + f(-3 +1)]
1[ f( -3) + f(-2)]
then plug into your equation
RRAM is right hand approximation and the formula is:
delta x [ f(a + delta x) + .... + f(b)]
so using the same example:
1[ f( -2) + f(-1)] and then plug into your equation
MRAM is to calculate the middle and the formula is:
delta x [ f(mid) + f(mid) + .... ]
To find midpoints, you would add the two numbers together then divide by two
In this problem the numbers would be: -3 , -2, -1
-3 + -2/ 2 = -5/2 and -2 + -1 / 2 = -3/2
so 1[f(-5/2) + f(-3/2)] and the plug in
Trapezoidal is different because instead of multiplying by delta x, you multiply by delta x/2 and you also have on more term then your number of subintervals.
The formula is : delta x/2 [f(a) + 2f(a + delta x) + 2f(a+ 2 delta x) + ....f(b)]
For this problem: 1/2 [ f(-3) + 2 f(-2) + f( -1)] and then plug in.
Substitution takes the place of the derivative rules for problems such as product rule and quotient rule. The steps to substitution are:
1. Find a derivative inside the interval
2. set u = the non-derivative
3. take the derivative of u
4. substitute back in
e integration:
whatever is raised to the e power will be your u and du will be the derivative of u. For example:
e^2x-1dx
u=2x-1 du=2
rewrite the function as:
1/2{ e^u du, therefore
1/2e^2x-1+C will be the final answer.
related rates:
The steps for related rates are….
1. Pick out all variables
2. Pick out all equations
3. Pick out what you are looking for
4. Sketch a graph and label
5. Create an equation with your variables
6. Take the derivative respecting time
7. Substitute back into the derivative
8. Solve
limits:
Rules for Limits:…
1. if the degree of top equals the degree of bottom, the answer is the top coefficient over bottom coefficient
2. if top degree is bigger than bottom degree, the answer is positive or negative infinity
2. if top degree is less than bottom degree, the answer is 0
To find area between curves:
The formula you use is b(int)a (top eq.) - (bottom eq.).
If a and b is not already given to you, then you much set the equations equal to each other and solve.
You find which equation is top/bottom by graphing both and simply looking to see which one is on top.
If the area is on the y-axis, then the a and b values need to be set as y-values, and the equations must be solved for x.
Post....whatever!
So, Let's go with implicit differentiation...just solve for dy / dx.
Example 1: Use implicit differentiation to find the derivative dy / dx where y x + sin y = 1
Solution to Example 1:
Use the sum rule of differentiation to the whole term on the left of the given equation.
d [xy] / dx + d [siny] / dx = d[1]/dx .
Differentiate each term above using product rule to d [xy] / dx and the cain rule to d [siny] / dx.
x dy / dx + y + (dy / dx) cos(y) = 0 .
Note that in calculating d [siny] / dx, we used the chain rule since y is itself a function of x and sin (y) is a function of a function.
Solve for dy/dx to obtain.
dy / dx = -y / (x + cos y)
Example 2: Use implicit differentiation to find the derivative dy / dx where y 4 + x y 2 + x = 3
Solution to Example 2:
Use the differentiation of a sum formula to left side of the given equation.
d[y 4] / dx + d[x y 2] / dx + d[x] / dx = d[3] / dx
Differentiate each term above using power rule, product rule and chain rule.
4y 3 dy / dx + (1) y 2 + x 2y dy / dx + 1 = 0
Solve for dy/dx.
dy/dx = (-1 - y 2) / (4y 3 + 2xy)
Example 3: Find all points on the graph of the equation
x 2 + y 2 = 4
where the tangent lines are parallel to the line x + y = 2
Solution to Example 3:
Rewrite the given line x + y = 2 in slope intercept form: y = -x + 2 and identify the slope as m = -1. The tangent lines are parallel to this line and therefore their slope are equal to -1. The slope of tangent lines at a point can be found by implicity differentiation of x 2 + y 2 = 4
2x + 2y dy/dx = 0
Let P(a , b) be the point of tangency. At point P the slope is -1. Substituting x by a, y by b and dy/dx by -1 in the above equation, we obtain
2a + 2b (-1) = 0
Point P(a , b) is on the graph of x 2 + y 2 = 4, hence
a 2 + b 2 = 4
Solve the system of equations: 2a - 2b = 0 and a 2 + b 2 = 4 to obtain two points
(-sqrt(2) , -sqrt(2)) and (sqrt(2) , sqrt(2))
I DO NOT UNDERSTAND LINEARIZATION????????!!!?!!!!!!! HELLLLPPPPP...i need somebody...>HELLLPPPP....not just anybody.....HELLLLPPPP....i need someone.......anddddd sommmeetthinnng...idk
BYE BYE ADIOS AMIGOS!!! HASTA MANANA!!!
Example 1: Use implicit differentiation to find the derivative dy / dx where y x + sin y = 1
Solution to Example 1:
Use the sum rule of differentiation to the whole term on the left of the given equation.
d [xy] / dx + d [siny] / dx = d[1]/dx .
Differentiate each term above using product rule to d [xy] / dx and the cain rule to d [siny] / dx.
x dy / dx + y + (dy / dx) cos(y) = 0 .
Note that in calculating d [siny] / dx, we used the chain rule since y is itself a function of x and sin (y) is a function of a function.
Solve for dy/dx to obtain.
dy / dx = -y / (x + cos y)
Example 2: Use implicit differentiation to find the derivative dy / dx where y 4 + x y 2 + x = 3
Solution to Example 2:
Use the differentiation of a sum formula to left side of the given equation.
d[y 4] / dx + d[x y 2] / dx + d[x] / dx = d[3] / dx
Differentiate each term above using power rule, product rule and chain rule.
4y 3 dy / dx + (1) y 2 + x 2y dy / dx + 1 = 0
Solve for dy/dx.
dy/dx = (-1 - y 2) / (4y 3 + 2xy)
Example 3: Find all points on the graph of the equation
x 2 + y 2 = 4
where the tangent lines are parallel to the line x + y = 2
Solution to Example 3:
Rewrite the given line x + y = 2 in slope intercept form: y = -x + 2 and identify the slope as m = -1. The tangent lines are parallel to this line and therefore their slope are equal to -1. The slope of tangent lines at a point can be found by implicity differentiation of x 2 + y 2 = 4
2x + 2y dy/dx = 0
Let P(a , b) be the point of tangency. At point P the slope is -1. Substituting x by a, y by b and dy/dx by -1 in the above equation, we obtain
2a + 2b (-1) = 0
Point P(a , b) is on the graph of x 2 + y 2 = 4, hence
a 2 + b 2 = 4
Solve the system of equations: 2a - 2b = 0 and a 2 + b 2 = 4 to obtain two points
(-sqrt(2) , -sqrt(2)) and (sqrt(2) , sqrt(2))
I DO NOT UNDERSTAND LINEARIZATION????????!!!?!!!!!!! HELLLLPPPPP...i need somebody...>HELLLPPPP....not just anybody.....HELLLLPPPP....i need someone.......anddddd sommmeetthinnng...idk
BYE BYE ADIOS AMIGOS!!! HASTA MANANA!!!
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