Final answer:
The difference quotient of the function f(x) = 2x² - 5x at x = 3 is calculated by evaluating the expression [f(3 + h) - f(3)] / h as h approaches 0, resulting in the value 7.
Step-by-step explanation:
The difference quotient is a method for determining the slope of the secant line between two points on a graph of a function. It is calculated using the formula:
f(x + h) - f(x) / h, where h is the difference in the x-values of the two points (usually h approaches 0).
For the function f(x) = 2x² - 5x, the difference quotient at x = 3 involves considering an increment h:
f(3 + h) - f(3) / h = [2(3 + h)² - 5(3 + h)] - (2(3)² - 5(3)) / h
After expanding the terms and simplifying, we find:
f(3 + h) - f(3) / h = [2(9 + 6h + h²) - 5(3 + h)] - (18 - 15) / h
f(3 + h) - f(3) / h = [18 + 12h + 2h² - 15 - 5h] - 3 / h
f(3 + h) - f(3) / h = [3 + 7h + 2h²] / h
Now, divide every term by h:
f(3 + h) - f(3) / h = 7 + 2h
As h approaches 0, the difference quotient specifically at x = 3 is simply 7. Therefore, the correct answer is: a) 7