Question

For f(x)=6 x³, construct and simplify the difference quotient f(x+h)-f(x)/h.

Answer 1

The difference quotient for the function f(x) = 6x³ is calculated by expanding f(x+h), subtracting f(x), and dividing by h, resulting in a simplified expression of 18x² + 18xh + 6h².

Step-by-step explanation:

To find and simplify the difference quotient for the function f(x) = 6x³, we calculate f(x+h) and then subtract f(x), all divided by h. Here are the steps:

• Substitute x + h into the function to get f(x + h) = 6(x + h)³.
• Expand 6(x + h)³ to get 6(x³ + 3x²h + 3xh² + h³).
• Calculate f(x) = 6x³.
• Determine f(x +h) - f(x) by subtracting 6x³ from 6(x³ + 3x²h + 3xh² + h³) to get 6(3x²h + 3xh² + h³).
• Divide by h to simplify and cancel out the h in the numerator, resulting in 6(3x² + 3xh + h²).

The fully simplified difference quotient for f(x) = 6x³ is 18x² + 18xh + 6h².