For the function f(x) = 4 + 3x - x^2, determine (f(3+h) - f(3))/h and simplify the result.
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calendarEducator since 2015
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Given f(x)=4+3x-x^2, then
f(x+h)=4+3(x+h)-(x+h)^2.
Consider the difference f(x+h)-f(x):
-(x+h)^2+x^2 + 3x+3h-3x + 4-4 =
-x^2-2xh-h^2+x^2 + 3h =
(3-2x)*h - h^2.
This divided by h is equal to
3-2x-h.
For x=3 as asked it is -3-h.
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calendarEducator since 2015
write489 answers
starTop subject is Math
Given the function `f(x)=4+3x-x^2`
and the difference quotient `(f(3+h)-f(3))/(h)`
`(4+3(3+h)-(3+h)^2-(4+3(3)-(3)^2))/(h)`
`(4+9+3h-(9+6h+h^2)-(4+9-9))/(h)`
`(4+9+3h-9-6h-h^2-4)/(h)`
`(-3h-h^2)/(h)`
`(h(-3-h))/(h)`
The solution is: `-3-h = -3`
It is given that the function f(x) = 4 + 3x - x^2.
f(3+h) = 4 + 3*(3+h) - (3+h)^2
= 4 + 9 + 3h - 9 - h^2 - 6h
= 4 - h^2 - 3h
f(3) = 4 + 3*3 - 3^2
= 4
(f(3+h) - f(3))/h
= (4 - h^2 - 3h - 4)/h
= -h - 3
= - 3
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