Final answer:
The slope of the curve y = 5x² - 4x at the point P(2,12) can be found using the definition of average rate of change. Taking the limit as a approaches 2, we find the slope to be 16.
Step-by-step explanation:
To find the slope of the curve y = 5x² - 4x at the point P(2,12), we can use the definition of average rate of change. The average rate of change is the slope of the secant line passing through two points on the curve. In this case, we'll use the points P(2,12) and Q(a,5a² - 4a), where a is some other point on the curve. We can calculate the slope of the secant line using the formula: (y2 - y1) / (x2 - x1).
Substituting the coordinates of the points P and Q into the formula, we get: (5a² - 4a - 12) / (a - 2).
To find the slope of the curve at the point P(2,12), we need to find the limit of the average rate of change as a approaches 2. Taking the limit as a approaches 2, we find the slope of the curve at P(2,12) to be 16. Therefore, the correct answer is option a) 16.