Find the difference quotient add simplify your answer. F(x)=x^2/3 +1, F(x)-f(27)/x-27, X≠27

Answer 1

So, let's write things a bit more carefully:

The DQ is (f(x) - f(27))/(x-27) with x ≠ 27 and f(x) = x²/3 + 1

Substitute

DQ = ((x²/3 +1)-(27³/3 +1))/(x-27)

Simplifying the top and factoring

DQ = (1/3)(x²-27²)/(x-27) = (1/3)(x-27)(x+27)/(x-27) = (1/3)(x+27)

If we take the limit as x goes to 27, we get the derivative at 27: 18

Please consider a tutor. Take care.

Answer 2

F(x) = x^(2/3) + 1

F(27) = 27^(2/3) +1 = 9+1=10

F(x) -F(27) = x^(2/3) +1 -10 = x^(2/3)-9

= (x^1/3 -3)(x^1/3 +3) using difference of squares formula

x-27 = (x^1/3- 3)(x^2/3 + 3x^1/3 + 9) using difference of cubes formula

then cancel the factor x^1/3 -3 from numerator and denominator

[F(x)-F(27)]/(x-27)

= [x^(2/3) -9]/(x-27)

or

= (x^1/3+3)/(x^2/3 +3x^1/3 +9)