The graph of y = f ′(x), the derivative of f(x), is shown below. Given f(2) = 8, evaluate f(–2).

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The graph of y = f ′(x), the derivative of f(x), is shown below. Given f(2) = 8, evaluate f(–2).

Mathematics
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let me try to do it on my own,tell me if i'm doing something wrong
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Remember, use the information that f(2)=8, to find f(-2)
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that feels wrong
Hmm, you don't need that graph at all
okay i have no idea where to start :/
Look at the middle line segment, going from (-2,-2,) to (2,2) is the line segment you need to look at
okay is the answer 8?
im saying 8 only because the graph seems mirrored and if it
Idk, I haven't calculated the answer yet use the formula \[\int\limits_{-2}^{2}f'(x)dx=f(2)-f(-2)\] \[f(-2)=f(2)-\int\limits_{-2}^{2}f'(x)dx=8-\int\limits_{-2}^{2}y.dx\] Have you studied about the definite integral?
okay that seems way better, one sec
and yes i have
Good, you need to find y(x) for the interval -2 to 2, it will be the equation of the middle line segment, again since it's passing through origin it's c will be 0 \[y=mx+c=mx+0=mx=(\frac{y_{2}-y_{1}}{x_{2}-x_{1}})x\] m=slope (x1,y1) (x2,y2) are any points on the line segment
x
so y = x?
yes!! y=x \[f(-2)=8-\int\limits_{-2}^{2}x.dx\] Now it's a simple matter of integration
8!!
I LOVE YOU!!
Hey! allternatively we could also use the symmetry to conclude that the integral is 0
good job jeb!!
-_- how did i not see that
Oh yeah, the same amount of area under the line is negative as it is positive so it cancels out
oh well it doesn't matter i get it! :D
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but it is also important that you know the method
thanks so much guys, i should probably head to bed its 5:20 am here
anyway thanks again!
Another alternative, Since \(f'(x)\) is an odd function, it follows that \(f(x)\) is an even function. Therefore \(f(-2)=f(2)=8\)
amazing

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