The Difference Quotient is the average rate of change between two points from a function where one point is the point of interest. Graphically, this is the slope of the secant line between the point of interest (a,f(a)) and a nearby point (a+h, f(a+h)). Notice that here h = delta x and delta y = f(a+h)-f(a). h can be positive or negative.[br]Difference Quotient = [br][math]m=\frac{\bigtriangleup y}{\bigtriangleup x}=\frac{f\left(a+h\right)-f\left(a\right)}{h}[/math][br][u]In the activity[/u].[br] Enter a desired formula for the function in the input box for f(x).[br] Choose an x-value for the point of interest by moving the slider for a or typing into its input box.[br] h is adjusted by its slider or input box.[br] Check the Difference Quotient check box to see the difference quotient illustrated.[br] The slope of the purple secant line is the value of the difference quotient.[br] Adjust h to make it closer to 0. Experiment with several values of h, a, and f(x). [br]
The Symmetric Difference Quotient is the average rate of change between two points[br] from a function where the points are equally spaced to the left and right of the point of interest. Graphically,[br] this is the slope of the secant line between the points[br](a-h,f(a-h)) and (a+h, f(a+h)). Notice that here delta x = 2h[br] and delta y = f(a+h)-f(a-h). h can be positive or negative.[br]Symmetric Difference Quotient = [br][math]m=\frac{\bigtriangleup y}{\bigtriangleup x}=\frac{f\left(a+h\right)-f\left(a-h\right)}{2h}[/math][br][u]In the activity[/u].[br] Enter a desired formula for the function in the input box for f(x).[br] Choose an x-value for the point of interest by moving the slider for a or typing into its input box.[br] h is adjusted by its slider or input box.[br] Check the Symmetric Difference Quotient check box to see the difference quotient illustrated.[br] The slope of the orange secant line is the value of the difference quotient.[br] Adjust h to make it closer to 0. Experiment with several values of h, a, and f(x).
What happens to the graphs of the secant lines and the values of the difference quotient and symmetric difference quotient as we let h approach 0?
If this limit exits then both secant lines are approaching a tangent line to the curve at the point (a, f(a)). The Difference Quotient and Symmetric Difference Quotient are both approaching a number which is the slope of this tangent line.
The Derivative is the instantaneous rate of change at a single point on a function. Graphically, it is the slope of the tangent line to the function at that point. We obtain the derivative by applying a limit as h goes to 0 to either the Difference Quotient or Symmetric Difference Quotient. Formally, we define the derivative as the limit as h approaches 0 of the Difference Quotient:[br][math]f'\left(a\right)=\frac{lim}{h\longrightarrow0}\frac{f\left(a+h\right)-f\left(a\right)}{h}[/math]
Notice that we can use either a difference quotient or symmetric [br]difference quotient to approximate the value of the derivative.
How do we make an approximation of the derivative better?
Regardless of whether we use a difference quotient or symmetric difference quotient, the closer the value of h is to 0, the better the approximation gets. Experiment with the activity to see this.
For the same value of h, which is typically a better approximation of the derivative, the difference quotient or symmetric difference quotient.
For the same value of h, the symmetric difference quotient usually has a value that is closer to the value of the derivative. It is better to take information on both sides of the point of interest and apply the slope formula in order to approximate the value of the derivative.
We can visualize each of these as directed (signed) vertical lengths.[br]Recall that the slope of a line is the amount of vertical change ([math]\bigtriangleup y[/math]) when [math]\bigtriangleup x=1[/math]. So the derivative is seen as the amount of vertical change in the tangent line when we go to the right 1. Similarly, the difference quotient is the amount of vertical change in its secant line when we go to the right 1, and the symmetric difference quotient is the amount of vertical change in its secant line when we go to the right 1. Notice that these are all approximations of the amount of change in the function ([math]\bigtriangleup f[/math]) when we go to the right 1 from (a, f(a)). In particular, notice that this change in the function when we increase from a to a+1 is close to, but not the same as, the derivative. We often use one of these quantities to approximate the other.[br][br]In the activity Click on the check boxes in the right column to see these illustrated. These check boxes also give a way to get to the secant line for the difference quotient and the tangent line with less clutter.[br]
Dr. Jackson's YouTube Channel: [url=https://www.youtube.com/c/DrJackLJacksonII]https://www.youtube.com/c/DrJackLJacksonII[/url][br]Dr. Jackson's Video using this activity: [br][url=https://www.youtube.com/watch?v=-O-McaKpRrE&list=PLEjUUlEpP-BMcyQGoQIGyskXTlPO_i-_8&index=5]Calculus[br]Video 2.5 using GeoGebra to Investigate the Difference Quotient, Symmetric[br]Difference Quotient, and the Derivative[/url] [br][br][br]