Question

Difference quotient: f(x)=X^3 + 2x^2 - 5

Answer 1

`3x^2+(4+3h)x +(2h+h^2)`

Explanation:

Evaluate the difference quotient in the usual way: put the function arguments where the variables are and simplify.

Difference quotient

The formula for the difference quotient is ...

`(f(x+h)-f(x))/(h)`

Application

For f(x) = x^3 +2x^2 -5, the difference quotient is ...

`(((x+h)^3+2(x+h)^2-5)-(x^3+2x^2-5))/(h)\\\\=(x^3+3hx^2+3h^2x+h^3+2(x^2+2hx+h^2)-5-x^3-2x^2+5)/(h)\\\\=((1-1)x^3+(3h+2-2)x^2+(3h^2+4h)x+(h^3+2h^2-5+5))/(h)\\\\=(3hx^2+h(3h+4)x+h(h^2+2h))/(h)\\\\=\boxed{3x^2+(4+3h)x +(2h+h^2)}`

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Additional comment

We have shown h-terms with increasing powers to the right. That is because we're usually concerned with small values of h, so higher-degree terms become insignificant and can be neglected. If the expression were written in "standard form", it might be ...

h^2 +3hx +3x^2 +2h +4x . . . . . lexicographical order of decreasing degree