You can enter any function that you want in the box at the bottom and it will be plotted in green. [br][br]The dashed green line shows a [b]tangent[/b] to the curve at a particular point. [br][br]You can move the green cross to see the tangent at different points or press [b]Animate[/b] and watch the tangent change as the green cross moves from left to right. Press [b]Stop[/b] to stop the animation. [br][br]The blue cross shows the value of the gradient at a particular point. If you select show gradient you can record how the gradient varies. This blue dotted line can be described as the [b]gradient function [/b]associated with the green function[b]. [br][br][/b]Try some different functions and see what you notice.
Note that I have not formally defined what a [b]tangent[/b] is, but you may be familiar with the word. Try to give your own definition of a [b]tangent[/b] below.
Try entering the function [math]f\left(x\right)=x^2[/math] and describe what you notice about the gradient.
Try entering the function [math]f\left(x\right)=x^2+3[/math]and describe what you notice about the gradient.
Try entering the function [math]f\left(x\right)=x^2-1[/math]and describe what you notice about the gradient.
Try entering the function [math]f\left(x\right)=2x+3[/math]and describe what you notice about the gradient.
Try entering the function [math]f\left(x\right)=x^3[/math]and describe what you notice about the gradient.
Now choose a function of your own. Try to predict what the gradient will look like and then see if your prediction fits the data. Describe anything that you notice below.