09Eratesofchange

= Rates of Change =


 * recall that the ** rate of change ** of a function is the same as the __**gradient**__ of that function


 * a linear function has a constant gradient so it has a constant rate of change


 * a non-linear function has a varying gradient, so its rate of change is varying


 * Average Rate of Change **


 * Recall that the ** average rate of change ** between two points is the __gradient of a straight line segment__ joining those two points

math . \qquad \qquad \text{Ave Rate of Change } = \dfrac{y_2-y_1}{x_2-x_1} math


 * OR, in function notation:

math . \qquad \qquad \text{Ave Rate of Change } = \dfrac{f(x_2)-f(x_1)}{x_2-x_1} math


 * Instantaneous Rate of Change **
 * Recall that the ** instantaneous rate of change ** at a point is the __gradient of the tangent__ to the curve at that point
 * Often the instantaneous rate of change is simply called the rate of change at a point.


 * The __**derivative**__ of a function gives us the gradient of a tangent at any point on the curve


 * So we can find the rate of change at a point by using the derivative.


 * Example 1 **

... ... An oil spill on a calm lake is spreading out in an ever-increasing circle ... ... ** (a) ** .. Express the area of the oil spill in terms of its radius ... ... ** (b) ** .. Hence find the rate of change of the area with respect to its radius ... ... ** (c) ** .. Hence find the rate at which the area is increasing when the radius is 7 metres


 * Solution:**

... ... ** (a) ** .. Express the area of the oil spill in terms of its radius

math . \qquad \qquad A = \pi r^2 math

... ... ** (b) ** .. Hence find the rate of change of the area with respect to its radius

math . \qquad \qquad \dfrac{dA}{dr} = A'(r)= 2 \pi r \qquad \big\{ \text{derivative of area} \big\} math

... ... ** (c) ** .. Hence find the rate at which the area is increasing when the radius is 7 metres

math . \qquad \qquad A'(7) = 14 \pi \;\; m^2 / m math