Answer: To find the value of g(2+h)-g(2) divided by h, we can use the definition of the derivative. The derivative of a function at a point is a measure of the slope of the function at that point, and it can be calculated by taking the limit of the difference quotient as h approaches 0.
The difference quotient is defined as:
[g(2+h) - g(2)] / h
So, to find the derivative of g at x=2, we can substitute the value of x in the function g(x) and take the limit as h approaches 0:
lim h→0 [g(2+h) - g(2)] / h
Substituting the value of x in the function g(x), we get:
lim h→0 [(2+h)³ - 2(2+h) - (2² - 2*2)] / h
This simplifies to:
lim h→0 [8+6h+h²-2h - 4] / h
Which simplifies to:
lim h→0 [h²+6h+4] / h
And, finally:
lim h→0 [h(h+6)] / h
The limit of a quotient is equal to the quotient of the limits, as long as the limit of the denominator is not 0. In this case, the limit of the denominator (h) is 0, but the limit of the numerator (h(h+6)) is not. Therefore, we can safely take the limit:
h+6
So, the derivative of g at x=2 is h+6. When h=0, the derivative is equal to the function's value at that point, so the value of g(2) is 6.
Therefore, the value of g(2+h)-g(2) divided by h is:
(h+6) - 6 / h
Which simplifies to:
h / h
Which is equal to:
1
So, the final answer is 1.
Explanation: