Final answer:
To find the slope of the curve y = x² at point P, we can find the limit of the secant slopes through point P. By calculating the difference in y-coordinates divided by the difference in x-coordinates, we can determine the slope of the secant line passing through point P. Taking the limit as the difference approaches zero, we can find the slope of the curve at point P.
Step-by-step explanation:
The slope of a curve at a point is equal to the slope of a straight line tangent to the curve at that point. To find the slope of the curve y = x² at the point P, we can find the slope of the secant line through point P. Let's choose point P as (a, a²) where a is any real number. Now, we need to find the limit of the secant slopes as the distance between the two points approaches zero. The slope of the secant line passing through point P can be found by calculating the difference in y-coordinates divided by the difference in x-coordinates, i.e., (a² - p²) / (a - p), where p is very close to a. Taking the limit as p approaches a, we can find the slope of the curve at point P.