I need to find the most common anti-derivative of f '' (x) = x^-2 , where x > 0, f(1) = 0 and f(2) = 0
As usual, this is how far I've gotten:
f'(x) = - x^-1 + C
f(x) = -Inx + Cx + D
I think that's right. Now what am I supposed to do?
I tried finding f(1) and f(2) of the function f(x) but it's not coming out as it should.
The answer is supposed to be f(x) = -Inx + (In2)x - In2
You came up with:Originally Posted by zachb
f(x) = -Inx + Cx + D
=> f(1) = -ln(1) + C + D = 0
=> C + D = 0 ...................(1) since, ln(1) = 0
=> f(2) = -ln(2) + 2C + D = 0
=> 2C + D = ln(2).............(2)
So now to find C and D, we must solve the system:
C + D = 0 ...................(1)
2C + D = ln(2) .............(2)
=> C = ln(2) ...........(2) - (1)
but C + D = 0
=> ln(2) + D = 0
=> D = -ln(2)
so f(x) = -lnx + ln(2)x - ln(2)
Hello,
all your calculations are correct!
You only have to plug in the values you know into the equation of f:
f(1) = 0 = -ln(1) + C*1 + D ===> 0 = C + D (note that ln(1) = 0)
f(2) = 0 = -ln(2) + C*2 * D ===> 0 = -ln(2) + 2C + D
Subtract equ1 from equ2 and you'll get:
0 = -ln(2) + C ===> C = ln(2). Plug in this value into the first equation D = -ln(2).
The equation of f becomes:
f(x) = -ln(x) +(ln(2))x - ln(2)
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