For the function ƒ(x) = x^3 - 8x + 3, calculate where a = 5 and h = 2. Simplify before substitution (the simplifying is where I am having problems...not sure if my answer is right). Thanks.
Using the limits equation.
3 answers
Knowing what a and h are is what is confusing me. They are not defined and do not appear in f(x)
yes....f(a+h)-fa)/h
OK, [f(a+h)-f(a)]/h is the approximation to the derivative at x = 5. It is the average slope of the function from f(5) to f(7).
Perhaps they want it expressed in terms of a and h. In that case,
[f(a+h)-f(a)]/h =
[a+h)^3 -a^3 -8(a+h)+a]/h
= [a^3 + 3a^2h + 3ah^2 + h^3 -a^3 -8h]/h
= 3a^2 + 3 ah + h^2 -8
in the limit as h goes to 0, this becomes the derivative 3a^2 -8, or the value of f'(x) at x = a.
Perhaps they want it expressed in terms of a and h. In that case,
[f(a+h)-f(a)]/h =
[a+h)^3 -a^3 -8(a+h)+a]/h
= [a^3 + 3a^2h + 3ah^2 + h^3 -a^3 -8h]/h
= 3a^2 + 3 ah + h^2 -8
in the limit as h goes to 0, this becomes the derivative 3a^2 -8, or the value of f'(x) at x = a.
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