Find f(a), f(a + h),
and the difference quotient
f(a + h) − f(a)
------------------
h
,
where h ≠ 0.
f(x) = 4 − 6x + 7x2
f(a)=
f(a + h) =
f(a + h) − f(a)
----------------- =
h
f(x) = 4 − 6x + 7x2
7(x + h)^2 -6(x+h) +4-(7x^2 -6x +4))/h
7( x^2 xh+ xh + h^2) -6x -6h + 4 -7x^2 +6x -4))/h
(7x^2 +14xh + 7h^2 -6x -6h +4 -7x^2 + 6x -4)/h
(14xh + 7h^2 -6h)/h
h(14x +7h -6)/h
14x + 7h -6
lim(h->0) 14x + 7h -6 = 14x -6
To find f(a), substitute a into the function f(x):
f(a) = 4 - 6a + 7a^2
To find f(a + h), substitute (a + h) into the function f(x):
f(a + h) = 4 - 6(a + h) + 7(a + h)^2
Now, let's simplify f(a + h):
f(a + h) = 4 - 6a - 6h + 7(a^2 + 2ah + h^2)
= 4 - 6a - 6h + 7a^2 + 14ah + 7h^2
The difference quotient is calculated by subtracting f(a) from f(a + h) and dividing the result by h:
[f(a + h) - f(a)] / h = [4 - 6a - 6h + 7a^2 + 14ah + 7h^2 - (4 - 6a + 7a^2)] / h
= [4 - 6a - 6h + 7a^2 + 14ah + 7h^2 - 4 + 6a - 7a^2] / h
= (-6h + 14ah + 7h^2) / h
= -6 + 14a + 7h
Therefore, f(a) = 4 - 6a + 7a^2, f(a + h) = 4 - 6a - 6h + 7a^2 + 14ah + 7h^2, and the difference quotient is -6 + 14a + 7h.