Question

Compute and simplify the difference quotient (f(x+h)-f(x))/(h) when h is not equal to 0. f(x)=11-6x^(3)

Answer 1

The difference quotient for f(x) = 11 - 6x^3 is calculated by finding f(x+h), expanding and subtracting f(x), and then dividing by h to simplify the expression. The final simplified form of the difference quotient is -6(3x^2 + 3xh + h^2).

Step-by-step explanation:

The difference quotient for the function f(x) = 11 - 6x3 when h is not equal to 0 is computed by the expression (f(x+h) - f(x)) / h. To find this, we first evaluate f(x+h) which gives us 11 - 6(x+h)3. Expanding this expression and then subtracting f(x) from it gives us a new expression that still includes h. Upon simplifying this new expression, we can then divide by h to find the simplified difference quotient.

Here is a step-by-step process:

1. Calculate f(x+h): f(x+h) = 11 - 6(x+h)3
2. Expand the cube: f(x+h) = 11 - 6(x3 + 3x2h + 3xh2 + h3)
3. Subtract f(x): f(x+h) - f(x) = -(6x3 + 18x2h + 18xh2 + 6h3)
4. Divide by h: ((f(x+h) - f(x)) / h) = -6(3x2 + 3xh + h2)