Exploring gradient functions of polynomials

Use the text boxes at the side to choose coefficients to plot particular polynomials up to degree 4. Then point M will trace the gradient of the curve at particular x values. Toggle the trace box to see the gradient function plotted.
Start with the function y = x^3 + x^2. If you turn the trace off, use the graph to find the gradient of the curve at the point (1,2). Now turn the trace on, how would you describe the gradient function that is formed as you vary the x coordinate? Use differentiation to work out the equation of dy/dx. How could you use this to calculate the gradient at a specific point? Now plot your own curve - you choose the coefficients. Predict what the gradient function will look like. See if you're correct. Find out the gradient of the curve at x=0, 1, 2. Use the equation of dy/dx to check your answers.