Techlab 15.3 - Graphmatica

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This Exploration demonstrates how to find the value of a in the function f(x) = ax, such that the gradient of the tangent at x = 0 is 1, as referenced on page 15-23 of your text, using Graphmatica a shareware programme downloadable from Keith Hertzer. (Disclaimer: While I have had no difficulties using this program, please undestand that you use it at your own risk.)

Instructions
If you haven't already done so, download and install Graphmatica. Run Graphmatica and minimise the window. In this Activity, you will switch between this page and Graphmatica, by minimising and maximising the Graphmatica window. Follow the instructions below.

  • Finding the value of a in the function f(x) = ax, such that the gradient of the tangent at x = 0 is 1:
    1. Firstly, we set up the axes. Click the "Options" button on the toolbar. Then select "Settings" from the menu. Next click the "Change Range" button and enter the following values in the dialog box: Left "-3", Right "3", Bottom "-0.5", Top "8 ". Click "OK" and then "OK" again to return to the Home screen. Next click the "Labels" button on the toolbar. Then select "Legends" from the menu and enter the following values in the dialog box: X axis "1", Y axis "1". Click "OK".
    2. Next, we sketch the function to ensure that it is continuous over the required bounds. Where the cursor is flashing in the input box type y=2.5^x and press the "Enter" key on the keyboard.
    3. To find the value for the gradient of f(x) at x=0. Click the "Calculus" button on the toolbar. Then select "Draw Tangent" from the menu. Move the crosshair to the y-intercept of y=2.5^x and left-click the mouse button. The value of the slope will appear at bottom of screen.
    4. If the gradient does not equal 1 then change the value for a. Continue finding the gradient at x = 0 until the gradient at x = 0 is 1.
    5. What value for a has a gradient of 1 at x = 0?