This week we started working on U substitution again. A quick outline of how to do this is as follows:
1. Find an inside function (u) of the integral
2. Derive the u
3. Make it so it matches something in the function (du for dx)
4. Plug it back in
5. Anti-derive that integral with the u
6. Plug the expression with x back in for the u
The step that I have the most trouble with is finding the u. Even though this may seem like a quite simple step, it is actually a bit harder. Not every function has an easily spotted inside function, and sometimes I have to try three different u's before I find one that fits write. And to make it more complicated, sometimes you have rewrite the function in different way so it is easier to find a u that works. These are the types of things that you might run into when doing u substitution that could make the problem take a lot longer than expected.
We also started taking derivatives of functions and anti deriving those. The part of these problems that can be kind of tricky is finding the C of the function. You must find the C because you are finding a universal function that has that derivative, but there are many functions that can have that derivative, depending on what point they go through and what interval, etc. In order to find the C, you can plug in a point once you find the anti-derivative (the x and y values). Then you can solve for the C and plug it back into the function. Another way you can do this by plugging in the x value for both the a and b of the integral, adding C at the end, and then setting that equal to the y value. You know that the integral will be 0, and that allows you to solve for C. You would then write the answer as an integral. Both ways are just as effective, but the second way may be faster because you wouldn't have you plug in the x a bunch of times. And pretty much every time the C will equal the y value which is nice.
And finally, we learned about slope fields. Slope fields basically are where you take the derivative of a function, plug in x and y values, and then plot whatever that gives you as a tiny slop line on the dot that represents that point. This seems kind of pointless at first, but it allows you to see the shape of a graph of something if the only thing you have is the derivative of the function.
1. Find an inside function (u) of the integral
2. Derive the u
3. Make it so it matches something in the function (du for dx)
4. Plug it back in
5. Anti-derive that integral with the u
6. Plug the expression with x back in for the u
The step that I have the most trouble with is finding the u. Even though this may seem like a quite simple step, it is actually a bit harder. Not every function has an easily spotted inside function, and sometimes I have to try three different u's before I find one that fits write. And to make it more complicated, sometimes you have rewrite the function in different way so it is easier to find a u that works. These are the types of things that you might run into when doing u substitution that could make the problem take a lot longer than expected.
We also started taking derivatives of functions and anti deriving those. The part of these problems that can be kind of tricky is finding the C of the function. You must find the C because you are finding a universal function that has that derivative, but there are many functions that can have that derivative, depending on what point they go through and what interval, etc. In order to find the C, you can plug in a point once you find the anti-derivative (the x and y values). Then you can solve for the C and plug it back into the function. Another way you can do this by plugging in the x value for both the a and b of the integral, adding C at the end, and then setting that equal to the y value. You know that the integral will be 0, and that allows you to solve for C. You would then write the answer as an integral. Both ways are just as effective, but the second way may be faster because you wouldn't have you plug in the x a bunch of times. And pretty much every time the C will equal the y value which is nice.
And finally, we learned about slope fields. Slope fields basically are where you take the derivative of a function, plug in x and y values, and then plot whatever that gives you as a tiny slop line on the dot that represents that point. This seems kind of pointless at first, but it allows you to see the shape of a graph of something if the only thing you have is the derivative of the function.