How do you find the slope of a tangent line using secant lines?

Question

Isabella Ackerman · Accepted Answer

The slope of a tangent line can be approximated by the slope of a secant line with one of the end point coincides with the point of tangency.  So,  if the slope of the secant line from a to a+h is #{f(a+h)-f(a)}/{h}#, then we can better approximate the slope of the tangent line by the slope of secant line by making #h# smaller and smaller.  Hence, we can find the slope of the tangent line #m# at #x=a# by #m=lim_{h to 0}{f(a+h)-f(a)}/{h}#

Charles Anctil · Answer

undefined To find the slope of a tangent line using secant lines, you need to follow these steps:

1. Select two points on the curve that are very close to each other.
2. Calculate the slope of the secant line passing through these two points using the formula: slope = (change in y-coordinates) / (change in x-coordinates).
3. Gradually move the two points closer together, making the interval between them smaller and smaller.
4. As the interval approaches zero, the secant line will become closer to the tangent line at the point of interest.
5. The slope of the tangent line can be approximated by the limit of the slopes of the secant lines as the interval approaches zero.

This process is known as the method of secant lines or the method of "taking the limit." It allows us to estimate the slope of the tangent line at a specific point on a curve.