The past couple of week in AP Calc we haven't been required to post anything, so there is an abundance of things I could write about in here. Basically, we have been talking about applications of the derivatives that we spent most of the first trimester finding.
First we talked about using derivatives to find critical points, which are where a function's derivative either equals zero or does not exist, and how to identify whether or not it is an extrema (maximum or minimum). The first derivative determines where the function is increasing and decreasing. Where the original function is increasing, the derivative function is above the x-axis or positive and the same goes for when it is decreasing.
Then, we used the second derivative to find important points. These points enabled us to deduce where the graph of the derivative had points of concavity and, subsequently, the points of inflection. The points in which the original function is
We advanced from identifying the extrema, intervals of increasing and decreasing, intervals of concavity, and the zeroes to being able to actually draw a function's graph with only it's equation for information.
So, we took our exam including these topics that we learned, and that went well. I got an A! Woohoo! Anyway, we are now continuing our study of Chapter 4 as well as the application of derivatives with optimization.
At first, optimization was significantly harder for me to grasp than anything else we have done so far. I think this is because optimization is a little more mentally taxing and conceptually trying than any other calculus we have done this year. With guidance, I have begun to better understand how to complete an optimization problem and with confidence.
I've come to the conclusion that the first and one of the most critical steps to solving the problems is to truly understand what the problem is asking you to do. Make sure you are aware of whether you are to find the volume of an object, the area of the object, or the dimensions of the object. Since you are working with a lot of numbers it is crucial that you know what the problem is specifically asking for, and sometimes that means you have to find something you aren't necessarily required to find first. Another very important step to solving optimization problems is to understand whether you are looking to minimize something or maximize something.
To complete most of the problems I begin by...
1. Writing down all the information that I am given and determine what it is I need to find. (Sometimes it is beneficial to draw a sketch of the information you are given.)
2. Then, I solve for the variables that I must find before finding what I am specifically asked to find. Using basic math skills, usually previously learned algebra skills, helps me to write equations and simplify down in order to solve for a single variable.
3. Then, I use the equations I have formed and substitute them into the function for the variable that I am asked to find.
4. Now, I can either take the derivative (depending on how messy the function is) and graph it or just graph the original function, and interpret the critical points. (If I graph the original function, I want to look at the minimums and maximums. If I graph the derivative function, I want to look at the zeroes.)
5. Now, I can solve the optimization problem and decide what the meaning is behind my solution.
Shown in the gallery below:
First we talked about using derivatives to find critical points, which are where a function's derivative either equals zero or does not exist, and how to identify whether or not it is an extrema (maximum or minimum). The first derivative determines where the function is increasing and decreasing. Where the original function is increasing, the derivative function is above the x-axis or positive and the same goes for when it is decreasing.
Then, we used the second derivative to find important points. These points enabled us to deduce where the graph of the derivative had points of concavity and, subsequently, the points of inflection. The points in which the original function is
We advanced from identifying the extrema, intervals of increasing and decreasing, intervals of concavity, and the zeroes to being able to actually draw a function's graph with only it's equation for information.
So, we took our exam including these topics that we learned, and that went well. I got an A! Woohoo! Anyway, we are now continuing our study of Chapter 4 as well as the application of derivatives with optimization.
At first, optimization was significantly harder for me to grasp than anything else we have done so far. I think this is because optimization is a little more mentally taxing and conceptually trying than any other calculus we have done this year. With guidance, I have begun to better understand how to complete an optimization problem and with confidence.
I've come to the conclusion that the first and one of the most critical steps to solving the problems is to truly understand what the problem is asking you to do. Make sure you are aware of whether you are to find the volume of an object, the area of the object, or the dimensions of the object. Since you are working with a lot of numbers it is crucial that you know what the problem is specifically asking for, and sometimes that means you have to find something you aren't necessarily required to find first. Another very important step to solving optimization problems is to understand whether you are looking to minimize something or maximize something.
To complete most of the problems I begin by...
1. Writing down all the information that I am given and determine what it is I need to find. (Sometimes it is beneficial to draw a sketch of the information you are given.)
2. Then, I solve for the variables that I must find before finding what I am specifically asked to find. Using basic math skills, usually previously learned algebra skills, helps me to write equations and simplify down in order to solve for a single variable.
3. Then, I use the equations I have formed and substitute them into the function for the variable that I am asked to find.
4. Now, I can either take the derivative (depending on how messy the function is) and graph it or just graph the original function, and interpret the critical points. (If I graph the original function, I want to look at the minimums and maximums. If I graph the derivative function, I want to look at the zeroes.)
5. Now, I can solve the optimization problem and decide what the meaning is behind my solution.
Shown in the gallery below: