If f(x)= 2x^2 +7x +18. Find f(a+h) - f(a) / h
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Instead of being "super hard", it's actually fairly straight forward.
f(x) = 2x² + 7x + 18
f(a+h) = 2(a+h)² + 7(a+h) + 18 = 2(a²+2ah+h²) + 7a + 7h + 18
= 2a² +7a + 18 + h(4a +7 + 2h)
= f(a) + h(4a+7+2h)
[f(a+h) - f(a)]/h = [f(a) + h(4a+7+h) - f(a)]/h = 4a+7+2h
This is the "difference equation" and is one step in the direction of finding the derivative but it is not the derivative. It becomes the derivative at x = a when you take the limit of the difference equation as h goes to zero.
f(x) = 2x² + 7x + 18
f(a+h) = 2(a+h)² + 7(a+h) + 18 = 2(a²+2ah+h²) + 7a + 7h + 18
= 2a² +7a + 18 + h(4a +7 + 2h)
= f(a) + h(4a+7+2h)
[f(a+h) - f(a)]/h = [f(a) + h(4a+7+h) - f(a)]/h = 4a+7+2h
This is the "difference equation" and is one step in the direction of finding the derivative but it is not the derivative. It becomes the derivative at x = a when you take the limit of the difference equation as h goes to zero.
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f(a + h) is 2(a+h)^2 + 7(a+h) + 18. Is simplifying that the thing you are finding "super hard"?
(a + h)^2 is a^2 + 2ah + h^2. Just plug that in, multiply by the constants and combine like terms.
f(a) is 2a^2 + 7a + 18. Nothing to do there, so that can't be the "super hard" part.
Subtracting them is no big deal. Just subtract the second polynomial from the first. Combine like terms.
And then dividing by h is not "super hard" since you'll find a factor of h in each term remaining.
So which is the step that has you stuck?
(a + h)^2 is a^2 + 2ah + h^2. Just plug that in, multiply by the constants and combine like terms.
f(a) is 2a^2 + 7a + 18. Nothing to do there, so that can't be the "super hard" part.
Subtracting them is no big deal. Just subtract the second polynomial from the first. Combine like terms.
And then dividing by h is not "super hard" since you'll find a factor of h in each term remaining.
So which is the step that has you stuck?
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Instead of that long process, it's much easier to simply find the derivative which can be explained on pretty much any website that comes out of googling, "how to find the derivative of a polynomial"... so once you learn this that answer is f'(x)=4x+7
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f(x) = 2 x^2 + 7x + 18
f(a) = 2 a^2 + 7a + 18
f(a+h) = 2(a+h)^2 + 7(a+h) + 18 = 2( a^2 + 2ah + h^2) + 7a + 7h + 18
= 2 a^2 + 7 a +18 + 4 ah + 2h^2 + 7 h
f(a+h) - f(a) = (4 a + 7) h + 2 h^2
(f(a+h) - f(a)) / h = 4 a +7 + 2h
f(a) = 2 a^2 + 7a + 18
f(a+h) = 2(a+h)^2 + 7(a+h) + 18 = 2( a^2 + 2ah + h^2) + 7a + 7h + 18
= 2 a^2 + 7 a +18 + 4 ah + 2h^2 + 7 h
f(a+h) - f(a) = (4 a + 7) h + 2 h^2
(f(a+h) - f(a)) / h = 4 a +7 + 2h
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f ' = 4x+7