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
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
MEAN VALUE THEOREM:
If f is continous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that F'(c) = f(b) - f(a) / b-a\
EXTREME VALUE THEOREM:
the EVT states that a continuous function on a close interval [a,b], must have both a minimum and a maximum on the interval. However, the max and min can occur at the endpoints
steps for related rates:
1. Identify all variables
2. Identify what you are looking for
3. Sketch & label that graph
4. write an equation using all of the variables
5. Take the derivative of this equation
6. Substitute everything back in
7. Solve
Limits:
If the degree on top is bigger than the degree on the bottom, the limit is infinity
If the degree on top is smaller than the degree on the bottom, the limit is zero.
If the degree on the top is the same as the degree on the bottom, you divide the coefficients to get the limit.
If they give you a limit problem where there is any letter going to 0 and they have a huge problem with parenthesis in it, you take the derivative of what is behind the parenthesis and plug in for x if needed.
Saturday, April 10, 2010
Sunday, April 4, 2010
Ash's 33rd Post
Soo...almost the end of the year...the AP is getting closer and closer
(btw: the key between the a and d is going out...forgive my errors)
So this week I figured out some things:
How to find bounds via calculator:
Step 1: graph the equation(s)
Step 2: 2nd->calc
Step 3: hit "intersect" (4)
Step 4: go left
Step 5: enter
Step 6: go right
Step 7: enter
Step 8: go in between the two
Step 9: enter
TADA!!
How to find bounds with no calculator:
Step 1: Set them equal
Step 2: Solve for x...
(What do you do after this)
I also figured out that the derivative of (cos(2x+4))^3 is actually 3(cos(2x+4))^2X(sin(2x+4))(2)
OH!
I'm so confused on the binder assignment...can I borrow someone's binder when we get back?
Also, don't forget to study for the super beast AP!
(btw: the key between the a and d is going out...forgive my errors)
So this week I figured out some things:
How to find bounds via calculator:
Step 1: graph the equation(s)
Step 2: 2nd->calc
Step 3: hit "intersect" (4)
Step 4: go left
Step 5: enter
Step 6: go right
Step 7: enter
Step 8: go in between the two
Step 9: enter
TADA!!
How to find bounds with no calculator:
Step 1: Set them equal
Step 2: Solve for x...
(What do you do after this)
I also figured out that the derivative of (cos(2x+4))^3 is actually 3(cos(2x+4))^2X(sin(2x+4))(2)
OH!
I'm so confused on the binder assignment...can I borrow someone's binder when we get back?
Also, don't forget to study for the super beast AP!
Posting...#33
Substitution:
1. State your u and du
2. Integrate
3. Plug u back in
also a lot of the time, you need to substitue in a number. when doing this, simply put it in front of the problem when you are done.
LRAM-delta x[f(a)+f(a+delta x)+...f(b-delta x)]
RRAM-delta x[f(a+delta x)+...f(b)]
MRAM-delta x[f(mid)+f(mid)+...]
TRAM-delta x/2[f(a)+2f(a+delta x)+2f(a+2 delta x)+...f(b)]
*for TRAM, delta x is [b-a/# of subintervals]
volume by disks:
pi S [function]^2 dx.
solve it by taking the integral of it and then pluging in the numbers they give you.
REMEMBER TO GRAPH
volume by washers:
pi S [top function]^2 - [bottom function]^2 dx.
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
1. State your u and du
2. Integrate
3. Plug u back in
also a lot of the time, you need to substitue in a number. when doing this, simply put it in front of the problem when you are done.
LRAM-delta x[f(a)+f(a+delta x)+...f(b-delta x)]
RRAM-delta x[f(a+delta x)+...f(b)]
MRAM-delta x[f(mid)+f(mid)+...]
TRAM-delta x/2[f(a)+2f(a+delta x)+2f(a+2 delta x)+...f(b)]
*for TRAM, delta x is [b-a/# of subintervals]
volume by disks:
pi S [function]^2 dx.
solve it by taking the integral of it and then pluging in the numbers they give you.
REMEMBER TO GRAPH
volume by washers:
pi S [top function]^2 - [bottom function]^2 dx.
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
REMEMBER TO GRAPH
i still need help with intergration.
Post #33
HAPPEE EASTER!
Tangent lines:
To find the equation of a tangent line. you are usually given an equation and a x-value. To find the equation, you need a slope, and a point. To find the point, plug in the x-value into the original equation to find the y. To find the slope, take the derivative of the equation and plug in the x- value into the derivative. The final step is to plug the point and the slope into point slope form which is y-y1 = m(x-x1).
Examples:
Find the equation of the line tangent to y=4x^3 - 7x^2 at x=3.
1. Find y by plugging x into original: 4(3)^3 - 7(3)^2 = 45
The point is (3, 45)
2. Find slope by taking derivative and plugging in x: 12x^2 - 14x
12(3)^3 - 14(3) = 66
3. Point slope form; y-45= 66 (x-3)
The answer may be in other forms, so you might have to solve the equation in order to get an answer given.
Other possible answers are: y-45 = 66x - 198 = y= 66x - 153
OR 66x - y = 153
Normal lines:
To find the equation of a normal line, follow the same steps as you would finding the equation of a tangent line EXCEPT when you find slope, you have to take the negative reciprocal of it.
Example:
An equation of the line normal to the graph of y = (3x^2 +2x)^1/2 at (2,4) is
1. Since you already have a point, you do not need to plug in the x to find the y.
2. Derivative: 1/2 (3x^2 + 2x) ^ -1/2 (6x +2)
Plug in x to find slope: 6(2) +2 / 2 (3(2)^2 + 2(2)) ^1/2
14/8 = 7/4
Since this is normal, we need the negative reciprocal of the slope, which is -4/7
3. Now we can plug in: y-4 = -4/7 (x-2)
Lastly, manipulate the equation to get an answer choice.
In this case, it helps to get rid of the fraction.
7y - 28 = -4 (x-2)
7y - 28 = -4x + 8
7y = -4x + 36
4x + 7y = 36
Something to remember: Position, velocity, acceleration
It helps when given a problem such as
A particle's position is given by s=t^3 - 6t^2 + 9t. What is its acceleration at time t=4?
Since we are given position and are looking for acceleration, we know we have to take the second derivative and then plug in 4.
3t^2 - 12t + 9
6t - 12
6(4) - 12 = 12
Another example: A particle moves along the x-axis so that its position at time t, in seconds, is given by x(t) = t^2 -7t +6. For what value(s) of t is the velocity of the particle zero?
Given position and looking for velocity means take derivative.
2t - 7
We are looking for where the velocity equals zero, so set the derivative equal to zero and solve for t.
2t-7 = 0
t = 7/2 OR 3.5
I can use a review on problems such as number 2 on the calculator portion of the last AP (the one with invertible and give the derivative of f-1). I know we went over this in class, but I didn't quite catch on to what we have to do.
I can also use a review on optimization and related rates.
Tangent lines:
To find the equation of a tangent line. you are usually given an equation and a x-value. To find the equation, you need a slope, and a point. To find the point, plug in the x-value into the original equation to find the y. To find the slope, take the derivative of the equation and plug in the x- value into the derivative. The final step is to plug the point and the slope into point slope form which is y-y1 = m(x-x1).
Examples:
Find the equation of the line tangent to y=4x^3 - 7x^2 at x=3.
1. Find y by plugging x into original: 4(3)^3 - 7(3)^2 = 45
The point is (3, 45)
2. Find slope by taking derivative and plugging in x: 12x^2 - 14x
12(3)^3 - 14(3) = 66
3. Point slope form; y-45= 66 (x-3)
The answer may be in other forms, so you might have to solve the equation in order to get an answer given.
Other possible answers are: y-45 = 66x - 198 = y= 66x - 153
OR 66x - y = 153
Normal lines:
To find the equation of a normal line, follow the same steps as you would finding the equation of a tangent line EXCEPT when you find slope, you have to take the negative reciprocal of it.
Example:
An equation of the line normal to the graph of y = (3x^2 +2x)^1/2 at (2,4) is
1. Since you already have a point, you do not need to plug in the x to find the y.
2. Derivative: 1/2 (3x^2 + 2x) ^ -1/2 (6x +2)
Plug in x to find slope: 6(2) +2 / 2 (3(2)^2 + 2(2)) ^1/2
14/8 = 7/4
Since this is normal, we need the negative reciprocal of the slope, which is -4/7
3. Now we can plug in: y-4 = -4/7 (x-2)
Lastly, manipulate the equation to get an answer choice.
In this case, it helps to get rid of the fraction.
7y - 28 = -4 (x-2)
7y - 28 = -4x + 8
7y = -4x + 36
4x + 7y = 36
Something to remember: Position, velocity, acceleration
It helps when given a problem such as
A particle's position is given by s=t^3 - 6t^2 + 9t. What is its acceleration at time t=4?
Since we are given position and are looking for acceleration, we know we have to take the second derivative and then plug in 4.
3t^2 - 12t + 9
6t - 12
6(4) - 12 = 12
Another example: A particle moves along the x-axis so that its position at time t, in seconds, is given by x(t) = t^2 -7t +6. For what value(s) of t is the velocity of the particle zero?
Given position and looking for velocity means take derivative.
2t - 7
We are looking for where the velocity equals zero, so set the derivative equal to zero and solve for t.
2t-7 = 0
t = 7/2 OR 3.5
I can use a review on problems such as number 2 on the calculator portion of the last AP (the one with invertible and give the derivative of f-1). I know we went over this in class, but I didn't quite catch on to what we have to do.
I can also use a review on optimization and related rates.
POST 33
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
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
Easter Post
First off, Happy Easter to everyone! I hope you are enjoying your time off...I know I am trying to get the most out of it. :-).
Second off, I have no recollection whatsoever of what we did this week in Calculus...I think we did like a practice test or something... I don't know. Therefore, I guess I will just explain something random as I have nothing to work off of for what you guys did not know...
Wait, I just had a glimpse :-).
So, on the practice test we took, we worked a problem that involved acceleration, velocity, position, and distance traveled. So, in an earlier post on this blog I kind of messed something up that I want to clear up..
I had previously said that the integral of the position function is displacement; this is wrong of course.
What I had meant to say, is that the integral of the absolute value of velocity is the distance traveled, and the integral of velocity is simply the displacement. Now, the absolute value is important to note because 1) you will get it wrong if you don't use it (for distance traveled) and 2) you have to use your calculator to integrate it (at least as far as I know). The reason it is the absolute value is because if you Physics students will recall, displacement and distance are two different things. If I move 15 meters right, 13 meters left, then 4 meters right, that would have a displacement of 15-13+4=6 meters to the right; however, the distance traveled will be 15+13+4=32 meters. To handle this in Calculus, we will simply use the absolute value so that when taking the integral, nothing will cancel out. Hopefully that explains the concept a little better than my previous post on the subject.
Enjoy the holidays :-).
Second off, I have no recollection whatsoever of what we did this week in Calculus...I think we did like a practice test or something... I don't know. Therefore, I guess I will just explain something random as I have nothing to work off of for what you guys did not know...
Wait, I just had a glimpse :-).
So, on the practice test we took, we worked a problem that involved acceleration, velocity, position, and distance traveled. So, in an earlier post on this blog I kind of messed something up that I want to clear up..
I had previously said that the integral of the position function is displacement; this is wrong of course.
What I had meant to say, is that the integral of the absolute value of velocity is the distance traveled, and the integral of velocity is simply the displacement. Now, the absolute value is important to note because 1) you will get it wrong if you don't use it (for distance traveled) and 2) you have to use your calculator to integrate it (at least as far as I know). The reason it is the absolute value is because if you Physics students will recall, displacement and distance are two different things. If I move 15 meters right, 13 meters left, then 4 meters right, that would have a displacement of 15-13+4=6 meters to the right; however, the distance traveled will be 15+13+4=32 meters. To handle this in Calculus, we will simply use the absolute value so that when taking the integral, nothing will cancel out. Hopefully that explains the concept a little better than my previous post on the subject.
Enjoy the holidays :-).
something else.
I have recently learned/rediscovered a few things:
1. how to guess my way through some AP questions.
2. I'm ready to leave.
3. I need help on the rate problems (the ones with A knot...)
4. My mother thinks my father is trying to poison her...?? (I'm just as confuzzled as you...)
5. I might not fail the AP....
6. The minute I turn 18, I am out of that house...
7. I have no more math problems really...
UMMMM>>>>OOOKKAAAYYYY THEN!
STEPS:
1. take the deriv.
2. set the deriv. equal to zero.
3. solve for the xmax, mins, horizontal tangents and the critical points, (if you don't know how to do this i'll explain in a second.)
4. set up intervals using the step above.
5. plug in to the first deriv. (which is why it's called the 1st deriv. test)
6. plug in values from above to the original function to find the absolute max and min, but only do this if it asks for it.
EXAMPLE:
x^2-6x+8
2x-6
2x-6=0
x=3 is the critical point.
the intervals are (-infinity, 3)u(3,infinity)
.. after you set up the intervals, plug in numbers within the intervals to see whether or not the intervals are increasing or decreasing and if they are a max or min. so in this case you could plug in 2 and 4.
The second derivative test is also really easy.
STEPS:
1. take the first deriv.
2. take the second deriv.
3. set the 2nd deriv equal to zero.
4. set up intervals
5. pic a number and plug in to the second deriv.
(the second deriv. is used to find whether or not an interval is concave up or concave down and where are the points of inflection.)
1. how to guess my way through some AP questions.
2. I'm ready to leave.
3. I need help on the rate problems (the ones with A knot...)
4. My mother thinks my father is trying to poison her...?? (I'm just as confuzzled as you...)
5. I might not fail the AP....
6. The minute I turn 18, I am out of that house...
7. I have no more math problems really...
UMMMM>>>>OOOKKAAAYYYY THEN!
STEPS:
1. take the deriv.
2. set the deriv. equal to zero.
3. solve for the xmax, mins, horizontal tangents and the critical points, (if you don't know how to do this i'll explain in a second.)
4. set up intervals using the step above.
5. plug in to the first deriv. (which is why it's called the 1st deriv. test)
6. plug in values from above to the original function to find the absolute max and min, but only do this if it asks for it.
EXAMPLE:
x^2-6x+8
2x-6
2x-6=0
x=3 is the critical point.
the intervals are (-infinity, 3)u(3,infinity)
.. after you set up the intervals, plug in numbers within the intervals to see whether or not the intervals are increasing or decreasing and if they are a max or min. so in this case you could plug in 2 and 4.
The second derivative test is also really easy.
STEPS:
1. take the first deriv.
2. take the second deriv.
3. set the 2nd deriv equal to zero.
4. set up intervals
5. pic a number and plug in to the second deriv.
(the second deriv. is used to find whether or not an interval is concave up or concave down and where are the points of inflection.)
Post #33
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
Graph Interpretation
Every AP exam will include at least one graph on it's short answer section that requires the test-taker to interpret certain requirements from it in order to receive credit for that question.
Interpreting a graph is very easy but in order to do it properly one must understand certain properties.
1. If the original graph is increasing, the slope is positive.
2. If the original graph is decreasing, the slope is negative.
3. An interval with a positive slope on the first derivative means that there is a downward concavity on that interval in the second derivative.
4. An interval with a negative slope on the first derivative means that there is an upward concavity on that interval in the second derivative.
5. Upward concavity (bowl-shaped) is positive.
6. Downward concavity (umbrella-shaped) is negative.
7. There is a horizontal tangent where the slope=0 on the original graph.
8. Wherever there is a horizontal slope there is either a maximum or a minimum value.
9. A x intercept on the first derivative is either a maximum of a minimum on the original graph.
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
Graph Interpretation
Every AP exam will include at least one graph on it's short answer section that requires the test-taker to interpret certain requirements from it in order to receive credit for that question.
Interpreting a graph is very easy but in order to do it properly one must understand certain properties.
1. If the original graph is increasing, the slope is positive.
2. If the original graph is decreasing, the slope is negative.
3. An interval with a positive slope on the first derivative means that there is a downward concavity on that interval in the second derivative.
4. An interval with a negative slope on the first derivative means that there is an upward concavity on that interval in the second derivative.
5. Upward concavity (bowl-shaped) is positive.
6. Downward concavity (umbrella-shaped) is negative.
7. There is a horizontal tangent where the slope=0 on the original graph.
8. Wherever there is a horizontal slope there is either a maximum or a minimum value.
9. A x intercept on the first derivative is either a maximum of a minimum on the original graph.
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