Estimate derivatives using the symmetric difference quotient...

estimate derivatives using the symmetric difference quotient $(S D Q),$ defined as the average of the difference quotients at $h$ and $-h :$ $$\begin{array}{r}{\frac{1}{2}\left(\frac{f(a+h)-f(a)}{h}+\frac{f(a-h)-f(a)}{-h}\right)} \\ {=\frac{f(a+h)-f(a-h)}{2 h}}\end{array}$$ The SDQ usually gives a better approximation to the derivative than the difference quotient. In Exercises $69-70,$ traffic speed $S$ along a certain road (in kilometers per hour) varies as a function of traffic density $q$ (number of cars per kilometer of road). Use the following data to answer the questions: \begin{equation} \begin{array}{l}{\text { Estimate } S^{\prime}(80)} \end{array} \end{equation}

estimate derivatives using the symmetric difference quotient $(S D Q),$ defined as the average of the difference quotients at $h$ and $-h :$
$$\begin{array}{r}{\frac{1}{2}\left(\frac{f(a+h)-f(a)}{h}+\frac{f(a-h)-f(a)}{-h}\right)} \\ {=\frac{f(a+h)-f(a-h)}{2 h}}\end{array}$$
The SDQ usually gives a better approximation to the derivative than the difference quotient.
In Exercises $69-70,$ traffic speed $S$ along a certain road (in kilometers per hour) varies as a function of traffic density $q$ (number of cars per kilometer of road). Use the following data to answer the questions:
\begin{equation}
\begin{array}{l}{\text { Estimate } S^{\prime}(80)} \end{array}
\end{equation}

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Key Concepts

Symmetric Difference Quotient

The symmetric difference quotient is a numerical method used to estimate the derivative of a function by taking the average of the forward and backward difference quotients. This method typically provides a more accurate approximation because it uses information from both sides of the point of interest, reducing the approximation error, and is an example of the central difference method.

Numerical Differentiation

Numerical differentiation involves techniques for approximating the derivative of a function using discrete data points. It is useful when an analytical form of the derivative is difficult or impossible to obtain and relies on finite difference approaches to estimate the rate of change of the function.

Finite Difference Methods

Finite difference methods are techniques for approximating derivatives by using the differences between function values at specific points. The symmetric difference quotient is one such method, and these approaches are fundamental in numerical analysis, particularly for solving differential equations and modeling phenomena when data is available only at discrete intervals.

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