Question

Approximations and limits: 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?

Short answer

• The slope of the tangent line m at x=a by$m=\underset{h\to 0}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$
  
• 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 ε > 0, there exists a δ > 0 such that |f(x) − L| < ε whenever 0 < x − p < δ.
  The limit of f(x) as x approaches p from below is L if, for every ε > 0, there exists a δ > 0 such that |f(x) − L| < ε whenever 0 < p − x < δ.

Step-by-step solution

Step 1. Given Information:

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.

Step 2. Solution

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 $\frac{f(a+h)-f\left(a\right)}{h}$

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 $m=\underset{h\to 0}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$

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 ε > 0, there exists a δ > 0 such that |f(x) − L| < ε whenever 0 < x − p < δ.

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