The video above explains in great detail where the formula for the difference quotient comes from. Basically, the difference quotient represents a formula that tells us the slope of a line that is secant to the graph, meaning it touches the graph twice. However, when we EVALUATE THE LIMIT of the difference quotient we are finding the slope of a line tangent to the graph. This is also known as the derivative. Additionally, this is called using the limit definition of the derivative. So, we are basically using the slope formula to derive the difference quotient. As you can see in the picture below, the secant line has two points that touch the graph. If we want to find the slope of these two points we easily use the slope formula derived from delta y over delta x (also known as rise over run) y sub 2 -y sub 1/x sub 2 - x sub 1. From there, we get our difference quotient and if we want to take it to the next level................
....WE FIND THE DERIVATIVE! In the picture below, you can see how as the h approaches 0, the secant line approaches the tangent line and that is why we evaluate the limit as h->0. As you can see, the tangent line only touches the graph ONCE. This new derivative is called f'(x) and is read as "f prime of x".
Works Cited:
No comments:
Post a Comment