91影视

We want to

• Describe in your own words how the slope of a tangent line can be approximated by the slope of a nearby secant line.
• Then describe how the derivative of a function at a point is defined as a limit of slopes of secant lines.
• What is the approximation/limit situation described in this section.

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 $\frac{f(a+h)鈭�f\left(a\right)}{h}$

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 $m=\underset{h鈫�0}{\lim }\frac{f(a+h)鈭�f\left(a\right)}{h}$

Since the derivative is defined as the limit which finds the slope of the tangent line to a function, the derivative of a function f at x is the instantaneous rate of change of the function at x.

The limit of f(x) as x approaches p from above is L if, for every 蔚 > 0, there exists a 未 > 0 such that |f(x) 鈭� L| < 蔚 whenever 0 < x 鈭� p < 未.

The limit of f(x) as x approaches p from below is L if, for every 蔚 > 0, there exists a 未 > 0 such that |f(x) 鈭� L| < 蔚 whenever 0 < p 鈭� x < 未.