Gradients & Curves On a straight line the gradient remains constant, however with curves the gradient changes continually, and the gradient at any point is in fact the same as the gradient of the tangent at that point. The sides of the half-pipe are very steep(S) but it is not very steep near the base(B). B S

Consider the following A Gradient of tangent = gradient of curve at A B Gradient of tangent = gradient of curve at B

To find the gradient at any point on a curve we need to modify the gradient formula viz m = y 2 - y 1 x 2 - x 1 For the function y = f(x) we do this by taking the point (x, f(x)) and another “very close point” ((x+h), f(x+h)). Then we find the gradient between the two. (x, f(x)) ((x+h), f(x+h)) True gradient Approx gradient

The gradient is not exactly the same but is quite close to the actual value. We can improve the approximation by making the value of h smaller. This means the two points are closer together. (x, f(x)) ((x+h), f(x+h)) True gradient Approx gradient

We can improve upon this approximation by making the value of h even smaller. (x, f(x)) ((x+h), f(x+h)) True gradient Approx gradient So the points are even closer together.