Question

find each of the following for f(x) = 5x² − 8x + 5. (a) f(x + h) (b) f(x + h) − f(x) (c) f(x + h) − f(x) / h (a) f(x + h) = \boxed{}

Explanation:

Step1: Compute f(x+h)

Substitute \(x = x+h\) into \(f(x)\):
\(f(x+h) = 5(x+h)^2 - 8(x+h) + 5 = 5(x^2 + 2xh + h^2) - 8x - 8h + 5 = 5x^2 + 10xh + 5h^2 - 8x - 8h + 5\)

Step2: Compute f(x+h)-f(x)

Subtract \(f(x)\) from f(x+h):
\((5x^2 +10xh +5h^2 -8x -8h +5) - (5x^2 -8x +5) = 10xh +5h^2 -8h\)

Step3: Compute the difference quotient

Divide by \(h\) (h≠0):
\(\frac{10xh +5h^2 -8h}{h} = 10x +5h -8\)

Answer:

(a) \(5x^2 + 10xh + 5h^2 - 8x - 8h + 5\)
(b) \(10xh + 5h^2 - 8h\)
(c) \(10x + 5h - 8\)