approximate small change

[little urchin]

y = -4x² -5x -4
express approximate change in y, in termsof k, when x changes from -2 to -2+k, where k is a small value.

i tried doing dy = f(-2+k) - f(-2) (im lazy to write all the equation)
the answer i get is -4k² + 36k - 40

but then there is also the δy/δx = dy/dx
δy= (dy/dx)*(δx)
which is (-8(-2)-5)(k)
and the answer i get is 16k - 5k

both are different answer so which is correct or both are wrong

[kindred jetty]

you calculated f(-2 + k) - f(-2) incorrectly

you should get ∆y = f(-2 + k) - f(-2) = -4k^2 + 11k
∆y represents the exact change of y in terms of k, where k can be any value
for example, if k = 9 (not a small value at all), you get that ∆y = -225, which lines up with that f(7) = f(-2) - 225

you need the approximate change in y instead
approximate = simpler answer

remember that dy/dx = the limit as h -> 0 of ∆y/∆x
so calculating (dy/dx)(∆x) would amount to doing:
= (lim h -> 0 of (f(-2 + h) - f(-2))/h)(k)
= (lim h -> 0 of (-4h^2 + 11h)/h) k
= (lim h -> 0 of (-4h + 11)) k
= (-4 * 0 + 11) k
= 11 k

notice here that this resembles the -4k^2 + 11k, except the -4k^2 is removed
if youre not comfortable with that, you can consider that k^2 goes to 0 twice as fast as k does, so that -4k^2 will be so small that it will barely affect -4k^2 + 11k for small values of k

[little urchin]

I see but what about the (-8(-2)-5)(k) is that also wrong?

[kindred jetty]

calculate -8(-2) - 5 for me

[little urchin]

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