examples examples examplesssssssss.
EXAMPLE 1:Given the equation y = (x-3)/(2-5x). Find dy/dx.
dy/dx = ((2-5x)(1) -(-5)(x-3))/(2-5x)^2
= (2-5x + 5x -15)/(2-5x)^2
= - 13/(2-5x)^2
take the derivative of the function using the quotient rule
EXAMPLE 2: What is the maximum value for the following: f(x) = xe^-x
because i finally understand e's!
Take derivative using product rule (multiplying two things).
x(-e^-x) + e^-x(1)remember that the der. e is e^w/e times the der. of the exponent
Simplify:
-xe^-x + e^-x
You can factor out an e^-x
(-x+1)e^-x
To find the critical values (possible maxima) set (-x+1) equal to zero and solve for x.
This yields x = 1. Now plug in that one to e^-x.
This then gives you 1/e====>your maximum value.
Example 3: The average value of f(x)= -1/x^2 on [1/2, 1].
average value means take the integral!
1/b-a 1S1/2 (-1/x^2)
1/(1-1/2)S x^-2
2[(-x^-1)/-1] = 2/x still have to integrate from 1/2 to 1
(2/1)-(2/(1/2) = -2
i could use a refresher on anything and every thing that has to do with ln.
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taking the derivative of ln is pretty easy let's say you have... ln(x^3+2x) the derivative of this would be 1/(x^3 + 2x) times (3x^2+3) so the derivative of ln u is 1/u times u'.
ReplyDeleteHope this helps.
since john went over derivative of ln, i'll help with integration. ln integration is super easy, all you have to do is recopy the bottom of the fraction and put an ln in front of it.
ReplyDeletesoooo
(int) .../6x+2
= ln(6x+2)
but, the bad thing is i don't remember how you know when you need to use ln integration,. b rob went over all this one day before she left, so it's in the notes. i just dont have it with me.
also, if you need to substitute in a 3 or something, just put 1/3 in front of the ln