#Estimating Derivatives at a Point

#Table of Contents

1. Estimating Derivatives Using a Graph
2. Estimating Derivatives Using a Table
3. Glossary
4. Practice Questions
5. Summary and Key Takeaways

#Estimating Derivatives Using a Graph

#How Can I Estimate a Derivative at a Point Using a Graph?

To estimate the derivative of a function at a particular point using its graph, follow these steps:

1. Identify the coordinates of the points that lie on the graph of the function.
2. Recall that the derivative of a function $f(x)$ at $x=a$, denoted as $f'(a)$, is equal to the slope of the tangent to the graph of $f(x)$ at $x=a$.

#Example

Consider the graph of $f(x)$ with points $A, B, C, D, E$ labeled with their coordinates. To approximate the derivative of $f(x)$ at $x=C$, we can find the slope of nearby line segments.

1. Finding the slope between $A$ and $E$: $$\frac{6 - (-6)}{5 - 1} = \frac{12}{4} = 3$$

2. Finding the slope between $B$ and $D$: $$\frac{0 - (-6)}{4 - 2} = \frac{6}{2} = 3$$

3. Finding the slope between $C$ and $D$: $$\frac{0 - (-4)}{4 - 3} = \frac{4}{1} = 4$$

Depending on which line segment is used, an approximation for the derivative of $f(x)$ at $C$ is 3 or 4.

#Estimating Derivatives Using a Table

#How Can I Estimate a Derivative at a Point Using a Table?

A similar method can be used to estimate the derivative at a point from a table of values.

1. Identify the coordinates from the table of values.
2. Recall that the derivative of $f(x)$ at $x=a$, denoted as $f'(a)$, is equal to the slope of the tangent to the graph of $f(x)$ at $x=a$.

#Example

Consider the table of values for the function $g$:

To find an estimate for the derivative of $g(x)$ at $x=7$, i.e., to find $g'(7)$, find the slope of line segments close to the point (7, -16).

1. Between (3, -24) and (7, -16): $$\frac{-16 - (-24)}{7 - 3} = \frac{8}{4} = 2$$

2. Between (9, 0) and (7, -16): $$\frac{-16 - 0}{7 - 9} = \frac{-16}{-2} = 8$$

3. Between (5, -24) and (9, 0): $$\frac{0 - (-24)}{9 - 5} = \frac{24}{4} = 6$$

Depending on which line segment is used, an approximation for the derivative of $g(x)$ at $x=7$ is 2, 6, or 8.

#Glossary

• Continuous: A function is continuous if there are no breaks, holes, or gaps in its graph.
• Differentiable: A function is differentiable at a point if it has a defined derivative at that point.
• Slope: The measure of the steepness or incline of a line, calculated as the ratio of the vertical change to the horizontal change.

#Practice Questions

#Using a Graph

Practice Question

1. Given the graph of $h(x)$ with points at $(1, 2)$, $(3, 6)$, $(5, 8)$, and $(7, 10)$, estimate the derivative at $x=3$.

#Using a Table

Practice Question

2. Given the table of values for $k(x)$:

   x	k(x)
   2	-8
   4	-4
   6	0
   8	8
   10	18

Estimate the derivative at $x=6$.

#Summary and Key Takeaways

#Summary

• Estimating derivatives using a graph involves finding the slope of line segments joining points near the point of interest.
• Estimating derivatives using a table involves calculating the slope between nearby points.
• Both methods require the function to be continuous and differentiable in the interval of interest.

#Key Takeaways

• The derivative at a point is approximated by the slope of the tangent line.
• Ensure the function is continuous and differentiable.
• Use nearby points to calculate slopes for a better approximation.