1D Gradient Fields

Watch the YouTube video where I explain this applet: http://youtu.be/TMmM0-KbtrY We first hear of gradient fields when studying function of more than one variable. Here we look at the 1D equivalent. In a gradient field (regardless of dimension): 1. The vector always points towards growth (maximum). 2. The vector magnitude is rate of change. ---- Look at the 1D gradient field below. * Where are the maximums, i.e. where do the vectors point towards each other? * Where are the minimums, i.e. where do the vectors point away from each other? Now, select the checkbox Function Points. *How would you draw the slope field of this function at each of these points? The slope field is line segments of equal length tangent to the function. So the slope of each segment is the corresponding value of the derivative, i.e. the vector magnitude (divided by Vector scale - see below). Select the other checkboxes to check your answers.
Vector Scale: Look at the table of derivative values at the right. Notice that the magnitude of each vector drawn on the x-axis has been multiplied by vector scale. This is done so that the vectors will not overlap! This is ALWAYS done! So when looking at ANY vector field you must always calculate the magnitude of the derivative at one point and then check the corresponding vector to find this vector scale! For example: http://en.wikipedia.org/wiki/Gradient. Look at the colored picture for the function . The actual magnitude of the gradient at (0,0) is e, but the vector drawn at (0,0) has magnitude ≈0.2. So the vector scale v≈0.1. ---- Input a different function and follow the steps above. Change vector factor to avoid overlapping of the vectors. Here are some good sample functions: