Estimating the Derivative at a Point (College Board AP® Calculus AB): Revision Note

Estimating derivatives at a point using a graph

How can I estimate a derivative at a point using points on the graph?

• The coordinates of the points that lie on the graph of a function can be used to estimate the derivative at a point

• Recall that the derivative of $f\left(x\right)$ at the point where $x=a$, denoted as $f^{\text{'}}\left(a\right)$,

  • is equal to the slope of the tangent to the graph of $f\left(x\right)$ at $x=a$

• To approximate the slope of the tangent to the graph of $f\left(x\right)$ at $x=a$:

  • Find the slope of line segments joining nearby points that lie on the graph

• The function must be continuous and differentiable within the relevant interval for this method to be valid

• Consider the graph of $f\left(x\right)$ below, where points $A,B,C,D,E$ are labeled with their coordinates

• To estimate the derivative at point $C$ you can find the slope of the nearest secant line

  • Finding the slope between $B$ and $D$

    • $\frac{0--6}{4-2}=\frac{6}{2}=3$

How can I estimate a derivative at a point using a tangent line?

• To estimate the derivative of a function at a point

  • Draw a tangent to the curve at the point

  • Pick two points on the line

  • Calculate the slope

Estimating derivatives at a point using a table

How can I estimate a derivative at a point using a table?

• The same method can be used as for a graph, but with a table of values instead

• Recall that the derivative of $f\left(x\right)$ at the point where $x=a$, denoted as $f^{\text{'}}\left(a\right)$,

  • is equal to the slope of the tangent to the graph of $f\left(x\right)$ at $x=a$

• To approximate the slope of the tangent to the graph of $f\left(x\right)$ at $x=a$:

  • Find the average rate of change using points on either side of $x=a$

• Consider the table of values below for the function $g$

  • $g$ is a continuous and differentiable function within this interval

  • This must be true to use this method

• To find an estimate for $g^{\text{'}}\left(7\right)$:

  • Find the average rate of change between $x=5$ and $x=9$

  • $\frac{0-\left(-24\right)}{9-5}=6$

• To find an estimate for $g^{\text{'}}\left(6\right)$:

  • Find the average rate of change between $x=5$ and $x=7$

  • $\frac{-16-\left(-24\right)}{7-5}=4$