Nonetheless, not all functions are this simple, and if they were, there would be no need for calculus.Ĭalculus is the mathematics of change, so now is a good time to move on to parabolas, curves with changing slopes.

So far, we have considered the simplest example of having a straight line, and hence a linear function, with an unchanging slope. If, otherwise, the value of x remains constant as y increases, we would consider the range of change to be infinite, because the slope of a vertical line is considered undefined. Furthermore, when the value of y remains constant as x increases, we would say that we have zero rate of change. However, the rate of change can also be negative if the direction of the line decreases, which means that the value of y would be decreasing as the value of x increases. Slope = 𝛿 y / 𝛿 x = ( y 2 – y 1) / ( x 2 – x 1) = (16 – 4) / (8 – 2) = 2įor this particular example, the rate of change, represented by the slope, is positive since the direction of the line is increasing rightwards. If we had to consider two particular points, P 1 = (2, 4) and P 2 = (8, 16), on this straight line, we may confirm the slope to be equal to: Rate of change = 𝛿 y / 𝛿 x = rise / run = slope The previous rate examples can be graphed on an x-y coordinate system, where each rate appears as a slope. Rates and slopes have a simple connection. Since the line can be seen to rise 2 units for each single unit that it runs to the right, then its rate of change, or its slope, is equal to 2. In this graphical representation of the object’s movement, the rate of change is represented by the slope of the line, or its gradient.

Calculus of infinitesimals