**EDIT: So, apparently, I forgot to post this yesterday. Whoops.
What a weird week.
From #Peppergate to winning at Quizbusters, this has been an eventful, if not slightly confusing, week. And what we are learning in math has definitely contributed to the confusion. The topics of this week's learning: "u" substitution and implicit differentiation!
Because of the whole pepper spray incident, we didn't get to do a whole lot on Monday, but we started off Tuesday learning how to anti-derive chain rule functions using "u" substitution, which involves identifying a part of the derivative as "u" and using substitution to come out with the anti-derivative. If, for example, S 6x^3(3x^4-8)^3 dx then, to solve for the anti-derivative:
1. Define "u" in terms of x ("u" should be the "inside")
u=3x^4-8
2. Find the derivative of "u" in terms of x
du=12x^3 dx
3. Solve so that du can be substituted into the derivative
du/12= x^3 dx
4. Substitute "u" and "du" into the derivative
S 6(3x^4-8)^3*x^3 dx
S 6u^3 du/12
5. Anti-derive that thing you got in the last step.
1/12*6u^3 du
1/8u^4+c
6. Substitute "x" back in.
1/8(3x^4-8)^4+c
Tada!! (Hopefully I did that right; I made the example up). That concept took up the first half of our week. It wasn't particularly difficult to learn, but sometime I lost track of what parts I was deriving and what parts I was anti-deriving, which was a bit of a nuisance. But, like I said earlier, not too hard.
The next thing we did was implicit differentiation, or finding the derivative when y isn't explicitly defined, like when y^2=x. I'm not gonna go into too much detail explaining this process (the picture has my notes on this stuff, if you really want to see it), but basically, differentiate y with respect to x by using the chain rule, factor out dy/dx, and then solve for dy/dx. That's the gist of it.
Again, like most of the stuff we learned these past few weeks, this concept wasn't all too difficult to grasp. My only problem is that there are times when I forget when and what I'm supposed to be differentiating. You can get a loooonng stream of numbers in implicit differentiation. I'm talking about longer than the quotient rule long. It can get kinda confusing keeping the numbers straight, so I have to go back and check my work to make sure I didn't make any mistakes. Which, you know, I should probably do anyway.
What a weird week.
From #Peppergate to winning at Quizbusters, this has been an eventful, if not slightly confusing, week. And what we are learning in math has definitely contributed to the confusion. The topics of this week's learning: "u" substitution and implicit differentiation!
Because of the whole pepper spray incident, we didn't get to do a whole lot on Monday, but we started off Tuesday learning how to anti-derive chain rule functions using "u" substitution, which involves identifying a part of the derivative as "u" and using substitution to come out with the anti-derivative. If, for example, S 6x^3(3x^4-8)^3 dx then, to solve for the anti-derivative:
1. Define "u" in terms of x ("u" should be the "inside")
u=3x^4-8
2. Find the derivative of "u" in terms of x
du=12x^3 dx
3. Solve so that du can be substituted into the derivative
du/12= x^3 dx
4. Substitute "u" and "du" into the derivative
S 6(3x^4-8)^3*x^3 dx
S 6u^3 du/12
5. Anti-derive that thing you got in the last step.
1/12*6u^3 du
1/8u^4+c
6. Substitute "x" back in.
1/8(3x^4-8)^4+c
Tada!! (Hopefully I did that right; I made the example up). That concept took up the first half of our week. It wasn't particularly difficult to learn, but sometime I lost track of what parts I was deriving and what parts I was anti-deriving, which was a bit of a nuisance. But, like I said earlier, not too hard.
The next thing we did was implicit differentiation, or finding the derivative when y isn't explicitly defined, like when y^2=x. I'm not gonna go into too much detail explaining this process (the picture has my notes on this stuff, if you really want to see it), but basically, differentiate y with respect to x by using the chain rule, factor out dy/dx, and then solve for dy/dx. That's the gist of it.
Again, like most of the stuff we learned these past few weeks, this concept wasn't all too difficult to grasp. My only problem is that there are times when I forget when and what I'm supposed to be differentiating. You can get a loooonng stream of numbers in implicit differentiation. I'm talking about longer than the quotient rule long. It can get kinda confusing keeping the numbers straight, so I have to go back and check my work to make sure I didn't make any mistakes. Which, you know, I should probably do anyway.