Approximating Slopes of Curves on a Graphing Calculator

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.

The scientific calculator -specifically the Texas Instrument TI-84 graphing calculator- is a tool, which can increase the efficiency at which one can perform a wide variety of mathematical operations. One such operation that is made easy with this tool is the approximation of slope (or gradient).

Approximating Slopes of Curves Using a Graphing Calculator

The scientific calculator, specifically the Texas Instruments (TI-84) graphing calculator is a tool, which can increase the speed or efficiency that one can perform on a wide variety of mathematical operations. Where slope might be easy to derive by hand for linear regression curves, some exponential function can be quite a chore for many other functions. We can use the TI-84 to

  1. graph a function in question and
  2. approximate the slope of this specific curve / slope of a tangent line to this curve

For this lesson, we will use a simple non-linear equation to illustrate this procedure: y = x2

Before we begin using the TI-84 to begin approximating slope, it is best that we set both the WINDOW and MODE. MODE determines whether we can deal with decimals or just whole numbers, radians or degrees, general functions or parametric equations, and real numbers or imaginary. In this case, the MODE should be set with the following parameters:


We must also make sure that the WINDOW is set correctly. Before we put our function/s into the TI-84 to be graphed, we must make sure that our Cartesian coordinate system is suitable for its projection. Therefore, we set up the WINDOW as follows:


Xmin = -10, Xmax = 10, Xscl = 1, Ymin = -10, Ymax = 10, Yscl = 1 and Xres = 1.

Xscl and Yscl determine how many tic-marks that we will have along the x and y-axis. Along each axis, we will have 10 in the negative direction from the origin (0,0) and 10 in the positive direction from the origin along both the x and y-axis. Xres is a variable we can just take to be 1.

Solving for Slope of Curve (Slope of Tangent Line dy/dx) at Any Point x.

What if we are given y = x2 and asked to solve for dy/dx (the slope of the tangent line) at any point x?

We start by putting our function into Y = :


After pressing the GRAPH command:


We press 2nd CALC and scroll down to the derivative (dy / dx) command 6:


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