This week, we really started to get into integration. We learned how to approximate the area under a curve (which is integration) by using four different methods: LRAM, RRAM, MRAM, and Trapezoidal. I feel as though everyone has a hang of it so I am really not going to go into much detail. The biggest thing is probably going to be substitution. We just started learning it and the matter is a little "iffy".
Substitution:
1. Label the parts of your integral. (x^2)(2x) u= (x^2) du=(2x)
2. Integrate (u du). (1/2)(u^2) +c
3. Plug u back in. (1/2)((x^2)^2) +c
Your final answer should be (1/2)(x^4) +c or (x^4)/2 +c
For substitution to work, one must recognize derivative properties that applies the product rule, quotient rule, or chain rule. Say if a problem asks you to find the integral of (x^3)(x). You would have to bring out a 1/3 then multiply your x by 3, making it look like (1/3) integrate (x^3)(3x). Then you just follow yours steps of substitution, but make sure to leave the 1/3 out until the last step then you distribute it in.
Apparently, I'm missing my Second Derivative Test notes. Can anyone explain to me the Second Derivative Test?
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for the second derivative test, you take the derivative of a function two times. Lets say you have the function..
ReplyDelete25x^3+12x+8
the first derivative is..
75x^2+12.. then you would take the derivative again which would be..
150x.. then you would solve for x and get critical values... then plug back in to find changes in concavity and points of inflection...
hope this helps!
1. Take the first derivative
ReplyDelete2. Take the second derivative
3. Solve for x
4. Set up intervals with the x values you get. These will show points of inflection and where the graph concaves up or down.
first you take the deriviative then you take the second derivative. you solve for x. then set up intervals and you find points of inflectiong and concave up or down.
ReplyDeleteall you do is just simply take the first derivative. then after that you take the second derivative. After that you set equal to zero solve for x. You then set up intervals and plug in from those intervals backinto the equation. positive concacave up. negative concave down.
ReplyDelete