Average Speed
First of all, remember that a slope is the y value, or dy, of a derivative.
Example:
A ball is flung from a little child. It's path is projected as y=4.9t2m in "t" seconds. What is the average speed of the ball from 0 to 3 seconds?
1. Set up equations and intervals: (f(b)-f(a))/(b-a) 4.9t^2 [0,3]
2. Plug in a and b values for t: f(b)=4.9(3)2=44.1 f(a)=4.9(0)2=0
3. Plug into main equation and solve: (44.1-0)/(3-0)=14.7m/s
Average speed is used for many different things, from finding the speed at which a cannonball was launched out of a cannon from how fast a cheetah runs in a straight line trying to catch it's prey. The concept behind average speed is a fairly simple concept that many people understand right away. You're basically finding the slope of the equation using calculus and algebra. If I ask someone what the average speed of a ball from [3,4] if it's path was graphed as y=x.
y=(4) y=(3) (4-3)/(4-3)=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.
Saturday, December 26, 2009
Post #19
Week 19
Well so my Christmas was pretty good, hope everyone else is as satisfied as me :-)
For my blog I guess I will explain the things I know really well..
Maxs and mins have to be one of the easiest things to do in Calculus. You are given a function, such as 2x^2 + 4x + 5. To find the max or min of this function, you take the derivative. The derivative is 4x + 4. Set this equal to 0, so 4x+4=0 and solve for x. When you solve, you get x=-1. So the x value of the vertex is -1. To find the actual point, you plug this in to the original to get the y-value of the vertex.
Point of inflection, which is the point where concavity changes, is also really easy and is like the same thing. The only difference is you take the second derivative. So for x^3 + 2x^2 + x + 4 ... it would be 3x^2 + 4x + 1 for first, 6x + 4 for second derivative. Set 6x+4=0 and solve for x to get that x=-2/3.
For maxs and mins, to find out if it is a max or a min, you have to use the derivative test. You have to set up intervals and then test them by plugging in points. If you get a positive number, the function is increasing. If you get a negative number, the function is decreasing. If it goes increasing, point, decreasing then it is a max. If it goes decreasing, point, increasing, then it is a min.
Anyway, I don't know what else to talk about for now. Enjoy the rest of your holidays!
AND BTW, according to http://www.wordcounttool.com/, my post is 267 words :-)
Well so my Christmas was pretty good, hope everyone else is as satisfied as me :-)
For my blog I guess I will explain the things I know really well..
Maxs and mins have to be one of the easiest things to do in Calculus. You are given a function, such as 2x^2 + 4x + 5. To find the max or min of this function, you take the derivative. The derivative is 4x + 4. Set this equal to 0, so 4x+4=0 and solve for x. When you solve, you get x=-1. So the x value of the vertex is -1. To find the actual point, you plug this in to the original to get the y-value of the vertex.
Point of inflection, which is the point where concavity changes, is also really easy and is like the same thing. The only difference is you take the second derivative. So for x^3 + 2x^2 + x + 4 ... it would be 3x^2 + 4x + 1 for first, 6x + 4 for second derivative. Set 6x+4=0 and solve for x to get that x=-2/3.
For maxs and mins, to find out if it is a max or a min, you have to use the derivative test. You have to set up intervals and then test them by plugging in points. If you get a positive number, the function is increasing. If you get a negative number, the function is decreasing. If it goes increasing, point, decreasing then it is a max. If it goes decreasing, point, increasing, then it is a min.
Anyway, I don't know what else to talk about for now. Enjoy the rest of your holidays!
AND BTW, according to http://www.wordcounttool.com/, my post is 267 words :-)
Post #19
Alright ladies and gents, here goes nothing. Today i'll explain how to do integrals. Because everyone is going back and expaining derivatives..so, of course, let me do the exact opposite.
Now, the first thing you have to do is locate the problem.
Then, read the problem.
You may ask, is this an integral or a derivative. Well, before you ask that terrible question, you know it's an integral if it has a fancy looking S infront of it.
Next, predict a method of solving; since it's a integral...
Perhaps, you should remember what you learned in class.
Then, perform the steps which include: adding one to the exponent and dividing it by the coefficient.
Finally, simplify fully.
So, lets do it with a problem.
Locate, Read, Predict, Think about the Problem
S 2x^3 + 7x + 6
Solvinggg....
S 2/4x^4 + 7/2x^2 +6x
Simplifying fully...
S 1/2x^4 + 7/2x^2 +6x
And tada. There you have it. You have successfully completed an integral problem. Now, you might thing this is all fun and games, and really easy. But don't fool yourself, it isnt always going to be that easy..such as when substitution comes along. [which i STILL do not understand]. So, if anyone wants to take a wackin at trying to explain it to me, that'd be greattt...or probably not, because we don't have to do comments..so nevermind. hahha, i made a funny on accident!
I hope everyone had a good christmas and got everything they wanteddddd!!!
Now, the first thing you have to do is locate the problem.
Then, read the problem.
You may ask, is this an integral or a derivative. Well, before you ask that terrible question, you know it's an integral if it has a fancy looking S infront of it.
Next, predict a method of solving; since it's a integral...
Perhaps, you should remember what you learned in class.
Then, perform the steps which include: adding one to the exponent and dividing it by the coefficient.
Finally, simplify fully.
So, lets do it with a problem.
Locate, Read, Predict, Think about the Problem
S 2x^3 + 7x + 6
Solvinggg....
S 2/4x^4 + 7/2x^2 +6x
Simplifying fully...
S 1/2x^4 + 7/2x^2 +6x
And tada. There you have it. You have successfully completed an integral problem. Now, you might thing this is all fun and games, and really easy. But don't fool yourself, it isnt always going to be that easy..such as when substitution comes along. [which i STILL do not understand]. So, if anyone wants to take a wackin at trying to explain it to me, that'd be greattt...or probably not, because we don't have to do comments..so nevermind. hahha, i made a funny on accident!
I hope everyone had a good christmas and got everything they wanteddddd!!!
Thursday, December 24, 2009
Post #19
Heyy...hope everyone's christmas holidays are going good!
let's go over Rolle's Theorem:
In order to be able to do Rolle's Theorem you have to have a continuous equation and it has to be differentual. When you plug in the points that are given to you, it has to equal to the same number that way f(a)=f(b)=0 [or any other number]
If all of this is passed, then you can start using the theorem:
find the first derivative, set it equal to zero and solve. The number[s] you get for x are the possible answers. If any of the numbers are not in range of the points given then those numbers cannot be your answer. The numbers that are in the range are your answer so you say c = ____ <--those numbers.
FOR EXAMPLE:
f(x)=x^2-2x [0,2]
it is continuous and differentiable.
plug in: f(0)=0
f(2)=0
take the first derivative: 2x-2=0
now solve: x= -1
since -1 isn't between 0 and 2 it is not an answer. Therefore there is no answers!
&&.. Mean Value Theorem:
In order to be able to do Mean Value Theorem you have to have a continuous equation and it has to be differentual. When you have the points in the problem [a,b] you plug those to the equation. When you solve and find numbers you take [the "b" answer] - [the "a" answer] at the top of the equation, and from the original points given you take [b - a] for the bottom of the equation. When you solve and get a number that's the number that you set the first derivative to. [Take the derivative of the equation given and then set equal to that number] The number[s] you get for the x are the possible answers. If any of the numbers are not in range of the points given then those numbers cannot be you anwer. The numbers that are in the range are your answer so you say c = ____ <--those numbers.
FOR EXAMPLE:
f(x)=x^2 [-2,1]
it is continuous and differentiable.
plug in: f(-2)=4
f(1) =1
so you plug into the equation...1-4 = -3
----- ----
1+2 = 1
since that equals to -3 when you take the first derivative [2x] you will set it equal to -3.
giving you x= -3/2 [which is between [-2,1] so c = -3/2
SANTA CLAUSE COMES TONIGHT!!...
MERRY CHRISTMAS,
FROM MY FAMILY TO YOURS
let's go over Rolle's Theorem:
In order to be able to do Rolle's Theorem you have to have a continuous equation and it has to be differentual. When you plug in the points that are given to you, it has to equal to the same number that way f(a)=f(b)=0 [or any other number]
If all of this is passed, then you can start using the theorem:
find the first derivative, set it equal to zero and solve. The number[s] you get for x are the possible answers. If any of the numbers are not in range of the points given then those numbers cannot be your answer. The numbers that are in the range are your answer so you say c = ____ <--those numbers.
FOR EXAMPLE:
f(x)=x^2-2x [0,2]
it is continuous and differentiable.
plug in: f(0)=0
f(2)=0
take the first derivative: 2x-2=0
now solve: x= -1
since -1 isn't between 0 and 2 it is not an answer. Therefore there is no answers!
&&.. Mean Value Theorem:
In order to be able to do Mean Value Theorem you have to have a continuous equation and it has to be differentual. When you have the points in the problem [a,b] you plug those to the equation. When you solve and find numbers you take [the "b" answer] - [the "a" answer] at the top of the equation, and from the original points given you take [b - a] for the bottom of the equation. When you solve and get a number that's the number that you set the first derivative to. [Take the derivative of the equation given and then set equal to that number] The number[s] you get for the x are the possible answers. If any of the numbers are not in range of the points given then those numbers cannot be you anwer. The numbers that are in the range are your answer so you say c = ____ <--those numbers.
FOR EXAMPLE:
f(x)=x^2 [-2,1]
it is continuous and differentiable.
plug in: f(-2)=4
f(1) =1
so you plug into the equation...1-4 = -3
----- ----
1+2 = 1
since that equals to -3 when you take the first derivative [2x] you will set it equal to -3.
giving you x= -3/2 [which is between [-2,1] so c = -3/2
SANTA CLAUSE COMES TONIGHT!!...
MERRY CHRISTMAS,
FROM MY FAMILY TO YOURS
Monday, December 21, 2009
Post 18
So we're off for two weeks. It's really nice considering we don't have any homework over the holidays except for the blog, which I am extremly greatful for. With the blog I will make an a for the nine weeks. Last week was exam week and it was a lot of reviewing. For review, we took two practice ap tests. We also had a take home portion of the exam. Although this was a part of the exam, it also helped with review. To review, I got together with Chelsea, John, and Mabile. We asked questions, worked problems, and taught each other what we did not know.
In my last blog, I had problems deciphoring between average value, average rate of change, and the mean value theorem. After last week of exams and review, I think I have finally gotten them figured out.
First there is average value. It seems like it would be difficult, but it's really easy to do. It's just taking a definite integral and putting a fraction in front of the integral, just as it would be done if something were missing from the integral. In the front, the fraction is 1/a-b. After the fraction is figured out, it is placed in front of the integral and the integral is solved accordingly.
For average rate of change, you're only finding slope. For average rate of change, slope is found by the formula f(b)-f(a)/b-a
For the mean value theorem, the same formula as average rate of change is to be used except this number is to be set equal to the derivative of the function.
Something I still really don't understand is angle of elevation. I used to really understand linerization, but sometimes I get confused on which number in the problem is what and what formulas to plug it into. Back to angle of elevation, I'm always really lost on how to do the problems. I really don't understand how to work the ones where there is a person and a docking station and a plane flying over it or something? I know I would have to make a right triangle to figure it out and use most of the same rules for linerization, but I get lost after that.
In my last blog, I had problems deciphoring between average value, average rate of change, and the mean value theorem. After last week of exams and review, I think I have finally gotten them figured out.
First there is average value. It seems like it would be difficult, but it's really easy to do. It's just taking a definite integral and putting a fraction in front of the integral, just as it would be done if something were missing from the integral. In the front, the fraction is 1/a-b. After the fraction is figured out, it is placed in front of the integral and the integral is solved accordingly.
For average rate of change, you're only finding slope. For average rate of change, slope is found by the formula f(b)-f(a)/b-a
For the mean value theorem, the same formula as average rate of change is to be used except this number is to be set equal to the derivative of the function.
Something I still really don't understand is angle of elevation. I used to really understand linerization, but sometimes I get confused on which number in the problem is what and what formulas to plug it into. Back to angle of elevation, I'm always really lost on how to do the problems. I really don't understand how to work the ones where there is a person and a docking station and a plane flying over it or something? I know I would have to make a right triangle to figure it out and use most of the same rules for linerization, but I get lost after that.
jessie's 18th post
i was in mississippi this weekend so i hope that mrs robinson sees mine...but i thought i should put the stuff im shaky with on here just to review a little..sooo
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.
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.
post 18
Ok so things ive learned in calculus involve taking derivatives.
take a derivative
1) mulitply exponent by the coefficient
2) subtract one from the exponent.
its fairly simple
Quotient rule
U/v = (v(u)' - u(v)')/ v^2
1) copy the bottom
2) take derivative of the top
3) - the copy the top
4) derivative of bottom
5) put all over the bottom squared
Product rule
UV = u(v)' + v(u)'
1) copy the first
2) derivative the second
3) copy the second
4) derivative of the first
Some things that i still dont fully get are tangent lines. i need help with these
take a derivative
1) mulitply exponent by the coefficient
2) subtract one from the exponent.
its fairly simple
Quotient rule
U/v = (v(u)' - u(v)')/ v^2
1) copy the bottom
2) take derivative of the top
3) - the copy the top
4) derivative of bottom
5) put all over the bottom squared
Product rule
UV = u(v)' + v(u)'
1) copy the first
2) derivative the second
3) copy the second
4) derivative of the first
Some things that i still dont fully get are tangent lines. i need help with these
Post #18
In Calculus 18 weeks ago, I learned about derivatives, average speed, and instantaneous speed. For each of these math terms, there are different steps to follow. Many formulas are used and it is necessary to remember all of them. We learned around ten derivative formulas which include quotient rule, product rule, sum and difference rules, trig function rules, and rules for numbers with and without variables, and taking derivates of quadratics.I am very comfortable with taking derivatives by using the product rule.
The product rule is used when you are multiplying two terms such as this problem x² (x+4). First, you must know the formula, which is uv^1 + vu^1. In simpler terms, this formula means (copy the first term)(derivative of the second term) + (copy the second term)(derivative of the first term). Following along with the problem so far, you would have (x²)(1) + (x+4)(2x). You get x² because as the formula states, the first part of the problem is copying the first term. Next, you have to take the derivative of (x+4).
Knowing the derivative rule for variables without an exponent is equal to 1 and the derivative of any number is equal to 0, you obtain (1+0) for the derivative of (x+4). So far your problem should look like (x²)(1). The next step is to place the addition sign next in the equation. Then you should copy the second term (x+4), and take the derivative of the first. The derivative of x² is obtained by multiplying the exponent by the coefficient and then subtracting one from the exponent. (x²) = (2)(1) x²-1 = 2x.
Therefore your problem will then become (x²)(1) +(x+4)(2x). Simple algebra is used from then on, you should distribute the x² to the one and distribute the 2x to (x+4). Then add like terms to obtain your final answer of 3x² +8x.
One thing I am not comfortable with is finding instantaneous speed. Problems such as finding the instantaneous speed at t=2 where y=16t² confuses me. At first I though you just plug it in, but then I realized there was a special formula that confuses me. I know the formula, but I don’t understand how you decipher what is your “h” and what is the f(x). After the formula is plugged in and you do the algebra, what number are you supposed to plug in?
word..
The product rule is used when you are multiplying two terms such as this problem x² (x+4). First, you must know the formula, which is uv^1 + vu^1. In simpler terms, this formula means (copy the first term)(derivative of the second term) + (copy the second term)(derivative of the first term). Following along with the problem so far, you would have (x²)(1) + (x+4)(2x). You get x² because as the formula states, the first part of the problem is copying the first term. Next, you have to take the derivative of (x+4).
Knowing the derivative rule for variables without an exponent is equal to 1 and the derivative of any number is equal to 0, you obtain (1+0) for the derivative of (x+4). So far your problem should look like (x²)(1). The next step is to place the addition sign next in the equation. Then you should copy the second term (x+4), and take the derivative of the first. The derivative of x² is obtained by multiplying the exponent by the coefficient and then subtracting one from the exponent. (x²) = (2)(1) x²-1 = 2x.
Therefore your problem will then become (x²)(1) +(x+4)(2x). Simple algebra is used from then on, you should distribute the x² to the one and distribute the 2x to (x+4). Then add like terms to obtain your final answer of 3x² +8x.
One thing I am not comfortable with is finding instantaneous speed. Problems such as finding the instantaneous speed at t=2 where y=16t² confuses me. At first I though you just plug it in, but then I realized there was a special formula that confuses me. I know the formula, but I don’t understand how you decipher what is your “h” and what is the f(x). After the formula is plugged in and you do the algebra, what number are you supposed to plug in?
word..
Post #18
Sooooo, Calculus we meet again. Like Stephanie did last week, I'm also kicking it old school.
I'm fully confident in finding the derivatives of -4x^3+2x^2-3x+1. For each term, you take the exponent and multiply it by the constant and subtract one from the exponent. Also, the derivative of any number is 0. for -4x^3: 3(-4)= -12 3-1=2 for 2x^2: 2(2)=4 2-1=1 for -3x: -3(1) = -1 1-1=0 So, then you have -12x^2+4x-3.
Also, I think I'm getting average speed, but can anyone tell me if I'm doing this right: Given the position equation s(t)=3t^2-3t-4, find the anverage velocity from t=2 to t=4. So, I ended up getting (2,4). I then plugged in 2 and then 4 to the equation 3t^2-3t-4. When I plugged in I got 2 and 32. After that I plugged (2,4) and 2, 32 in to the slope formula. For my final answer I got 15.
I think I'm having trouble more with the algebra kind of stuff. for example: [(-3x^2+5x-5)(sinx)] I understand copy the first, derivative of second, minus, copy the second, derivative of first (-3x^2+5x-5)(cosx)-(sinx)(-6x+5) but then what's the next step to solving?
I'm fully confident in finding the derivatives of -4x^3+2x^2-3x+1. For each term, you take the exponent and multiply it by the constant and subtract one from the exponent. Also, the derivative of any number is 0. for -4x^3: 3(-4)= -12 3-1=2 for 2x^2: 2(2)=4 2-1=1 for -3x: -3(1) = -1 1-1=0 So, then you have -12x^2+4x-3.
Also, I think I'm getting average speed, but can anyone tell me if I'm doing this right: Given the position equation s(t)=3t^2-3t-4, find the anverage velocity from t=2 to t=4. So, I ended up getting (2,4). I then plugged in 2 and then 4 to the equation 3t^2-3t-4. When I plugged in I got 2 and 32. After that I plugged (2,4) and 2, 32 in to the slope formula. For my final answer I got 15.
I think I'm having trouble more with the algebra kind of stuff. for example: [(-3x^2+5x-5)(sinx)] I understand copy the first, derivative of second, minus, copy the second, derivative of first (-3x^2+5x-5)(cosx)-(sinx)(-6x+5) but then what's the next step to solving?
Post #18 or something like it
I completely forgot how to do this so..
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:
y= x^1/2 find dy/dt when x=-4 and dx/dt = 3
1. x=4; dx/dt=3
2. dy/dt = ?
3. equation: y=x^1/2
4. dy/dt = 1/2 x ^-1/2 dx/dt
5. 1/2 (-4)^-1/2 (3)
dy/dt = -3/4
The variables x and y are differentiable functions of t and are related by the equation y=2x^3-x+4 when x=2. dx/dt=-1 Find dy/dt when x=2.
1. x=2; dx/dt = -1
2. dy/dt = ?
3. you are already given the equation: y = 2x^3 - x +4
4. Derivative with respect to time: dy/dt= 6x^2 dx/dt - dx/dt
5. Plug in: 6(2)^2 (-1) - (-1)
6. Solve: 6(4)(-1)+1 = -23
A little harder 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. dr/dt = ?
3. Volume of sphere: v=4/3 pi r^3
4. Take 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 know we don't comment this week but for what I am having trouble with:
1. Angle of elevation ( I think I'm just out of practice)
2. Linearization
3. I can always use more help with graphs even though I know we reviewed them plenty of times.
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