Today we focused on Anti-Deriving composite functions with U-Substitution or Simple Substitution. Class started with Mr. Hansen asking some questions about the previous night's homework, we got to a couple and saved a few for the end of class. We mainly focused on how to Anti-Derive composite functions. Composite Functions were the functions we had to use the chain rule to derive, essentially functions within functions like sin(2x) or sqrt(6x+1).
The first step of the process is to assign the variable "u" to the inner function.
sqrt(6x+1) -> sqrt(u) or (u)^(1/2)
The second step is to derive "u" and solve dx in terms of "u" so dx can be subbed out of the integral
u=6x+1 du/dx (Derivative of "u") = 6 du=6*dx du/6=dx
After that, we have to rewrite the integral, substituting dx in terms of "u". We'll imagine we're looking at the indefinite integral of sqrt(6x+1) because I have absolutely no idea how to write a script s on a computer
integral(sqrt(6x+1)dx) -> integral(sqrt(u)(du/6))
Then we anti-derive with respect to "u"
sqrt(u)(du/6) -> (1/6)(2/3(u^(3/2))
Finally, we replace the original inner function for "u", and we're finished!
(1/6)(2/3(u^(3/2)) -> (1/6)(2/3((6x+1)^(3/2))
For the rest of class we did a few practice problems and reviewed one or two more homework questions from last night
Today in class we went over test review and homework questions, and then we started something new. We looked at using the FTC to find values and graphs of derivative functions that we were unable to solve algebraically. We reviewed that f=Aprime, and that A is the anti-derivative of f. For equations where we could not find the anti-derivative algebraically, we graphed the integral on our calculators and then used a table to find values at certain points. When we wanted a certain function at a certain value, say, f(3), to equal a number such as 8, we would adjust the integral to start at 3 and then, since the integral would equal 0 at 3, we would add 8 to the equation so f(3)=8 would be true. Then we used the Table button to find other values.
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February 18th, 2009
Today we focused on Anti-Deriving composite functions with U-Substitution or Simple Substitution. Class started with Mr. Hansen asking some questions about the previous night's homework, we got to a couple and saved a few for the end of class. We mainly focused on how to Anti-Derive composite functions. Composite Functions were the functions we had to use the chain rule to derive, essentially functions within functions like sin(2x) or sqrt(6x+1).
The first step of the process is to assign the variable "u" to the inner function.
sqrt(6x+1) -> sqrt(u) or (u)^(1/2)
The second step is to derive "u" and solve dx in terms of "u" so dx can be subbed out of the integral
u=6x+1
du/dx (Derivative of "u") = 6
du=6*dx
du/6=dx
After that, we have to rewrite the integral, substituting dx in terms of "u". We'll imagine we're looking at the indefinite integral of sqrt(6x+1) because I have absolutely no idea how to write a script s on a computer
integral(sqrt(6x+1)dx) -> integral(sqrt(u)(du/6))
Then we anti-derive with respect to "u"
sqrt(u)(du/6) -> (1/6)(2/3(u^(3/2))
Finally, we replace the original inner function for "u", and we're finished!
(1/6)(2/3(u^(3/2)) -> (1/6)(2/3((6x+1)^(3/2))
For the rest of class we did a few practice problems and reviewed one or two more homework questions from last night
Today in class we went over test review and homework questions, and then we started something new. We looked at using the FTC to find values and graphs of derivative functions that we were unable to solve algebraically. We reviewed that f=Aprime, and that A is the anti-derivative of f. For equations where we could not find the anti-derivative algebraically, we graphed the integral on our calculators and then used a table to find values at certain points. When we wanted a certain function at a certain value, say, f(3), to equal a number such as 8, we would adjust the integral to start at 3 and then, since the integral would equal 0 at 3, we would add 8 to the equation so f(3)=8 would be true. Then we used the Table button to find other values.
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