Calculus question (derivatives)!?

Question

The Following limit represents f '(a). determine f(x), a and then evaluate f '(a).

lim {[4(2+h)^3 + 5(2+h)] - 42} / h

h->0

thankss!

Answer 1

f'(x) = lim (h -> 0) { [ f(x + h) - f(x) ] / h }.

Comparing this formula with the one in the question:

f(x) = 4x^3 + 5x

a = 2

f'(a) = lim (h -> 0) { [ f(a + h) - f(a) ] / h }

= lim (h -> 0) { [ 4(2 + h)^3 + 5(2 + h) - 42 ] / h }

= lim (h -> 0) { [ 4(8 + 12h + 6h^2 + h^3) + 10 + 5h - 42 ] / h }

= lim( h -> 0) { [ 32 + 48h + 24h^2 + 4h^3 + 10 + 5h - 42 ] / h }

= lim (h -> 0) { [ 53h + 24h^2 + 4h^3 ] / h }

= lim (h -> 0) { 53 + 24h + 4h^2 }

= 53.