Defining the Derivative

Question

Let f(x) be the function 7x² - 5x + 8. Then the quotient f(1+h) - f(1) ÷ h can be simplified to ah+ b for

a =

b =

Answer 1

f(x) = 7x² - 5x + 8 = 7(x)² – 5(x) + 8

f(1) = 7(1)² – 5(1) + 8 = 7 – 5 + 8 = 10

f(1+h) = 7(1+h)² – 5(1+h) + 8 = 7(1+2h+h²)– 5(1+h) + 8 = 7+14h+7h²– 5–5h + 8 = 7h² + 9h + 10

[f(1+h)] – [f(1)] = [7h² + 9h + 10] – [10] = 7h² + 9h + 10 – 10 =  7h² + 9h

f(1+h) - f(1) ÷ h = ([f(1+h)] – [f(1)])/( h) =  (7h² + 9h)/( h) = (h)(7h + 9)/( h) = 7h + 9

Therefore, the quotient f(1+h) - f(1) ÷ h can be simplified to: ah + b = 7h + 9 for a = 7 and b = 9

Answer 2

f(1+h) = 7(1+h)²-5(1+h)+8 = 7+14h+7h²-5-5h+8 = 7h²+9h+10

f(1)=7-5+8 = 10

f(1+h)-f(1) = 7h²+9h+10-10 = 7h²+9h

(f(1+h)-f(1))/h = 7h+9 = ah+b

a=7

b=9