Final answer:
The student is asked to evaluate the difference quotient (f(x+h)-f(x))/h for f(x) = x² - 5x + 6. First, f(x + h) is expanded and simplified, then the function f(x) is subtracted from it, and finally the result is divided by h.
Step-by-step explanation:
The student is asked to evaluate the function (f(x+h)-f(x))/(h) given that f(x)=x² -5x+6. This is a common question in mathematics related to the concept of a difference quotient, which is a fundamental component in the study of calculus and represents the average rate of change of the function over the interval from x to x + h.
To evaluate the function, first calculate f(x + h):
- f(x + h) = (x + h)² - 5(x + h) + 6
- Expand the square and distribute the -5: f(x + h) = x² + 2xh + h² - 5x - 5h + 6
Next, find f(x):
Now, calculate the difference quotient:
- (f(x+h)-f(x))/(h) = (x² + 2xh + h² - 5x - 5h + 6 - (x² - 5x + 6)) / h
- Simplify the expression by canceling terms and dividing by h: (2xh + h² - 5h) / h = 2x + h - 5
In the limit as h approaches 0, this expression would give the derivative of f(x), but without taking the limit, the difference quotient simply represents the average rate of change over the interval [x, x+h].