# Approximating Slopes Using Technology

Instructor: Michael Eckert

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

Depending on the function in question, determining the slope of its tangent line (i.e. a derivative) can be difficult. Slopes of functions can often be determined quickly and correctly using a scientific calculator with a graphing feature. This lesson will explain how one such calculator (the Texas Instruments TI-84) allows for approximating slopes with speed and accuracy.

## Approximating Slopes with the Texas Instruments TI-84 Graphing Calculator

Some slopes of functions, more specifically tangent lines to functions (i.e. derivatives), can be difficult to approximate, as an expression might prove difficult to simplify. An efficient method for approximating such slopes exists. This method implements various commands on the TI-84 scientific graphing calculator. Keep in mind that the derivative of a function is merely the slope of the line tangent to that specific function at a specific point. We can therefore find the slope of the tangent line to y = x2 with the TI-84 via the dy/dx command under 2nd CALC.

### MODE and WINDOW

Before we input our function into Y= to be graphed, it is important that we set up our MODE and WINDOW to deal with this type of function.

#### MODE

MODE simply determines the functional settings of the calculator. Are we in Normal, Scientific, or Engineering mode? How many decimals will the calculator round to? Are we set in radians or degrees? Are we dealing with polar coordinates or Cartesian coordinates, imaginary or real numbers, etc.? MODE sets these parameters. MODE should be set as follows:

#### WINDOW

WINDOW sets the bounds, symmetry, and overall look of our Cartesian coordinate system; this sets what our minimum x and y values (xmin, ymin) and our maximum x and y values (xmax, ymax) are. How many tick-marks do we want to have along each axis? With our function y = x2, WINDOW should be set as:

So, Xmin = -10, Xmax = 10, Xscl = 1, Ymin = -10, Ymax = 10, Yscl = 1, and Xres = 1. Xres determines resolution (number of pixels per unit length) and can be set as 1 in most cases.

Now that we have our MODE and WINDOW set, we can input y = x2 into our Y=:

We press GRAPH:

We press 2nd CALC and select the 6: dy/dx option. If we wish to find the slope of the tangent to y = x2 at x = 1, we enter x = 1:

After pressing ENTER, we are given dy/dx = 2, ergo the slope of the tangent to y=x2 via x = 1 is 2.

To check this result against the graph, we merely input y = 2x into Y = with y=x2 and press GRAPH. A picture of y = 2x superimposed onto y=x2 will be displayed:

y = 2x clearly looks tangent to the graph of y = x2 with a slope of 2.

For the slope of the tangent to the graph via x = 2, we follow the same procedure. We use the dy/dx command and input x = 2.

We are given the slope of 4:

Checking this result, we input y = 4x into Y = with y = x2 and press GRAPH.

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