Calculus involves pairs of functions:

1. the integral gives the total change, and
2. the derivative gives the rate of change.

When the rate of change is constant, the derivative is a horizontal line, and the integral is a straight line angled up and to the right. For example, a car driving at a constant speed of 30 kph covers distance uniformly over time - 30 km in 1 hr, 60 km in 2 hrs, etc.

When the rate of change is increasing at a constant rate, the derivative is a straight line angled up, and the integral is a parabola. For example, an object falling at at 32 feet-per-second squared, drops 32 feet in the first second, another 64 feet in the second second, and so on.

The derivative of the sine is the cosine, and the derivative of the cosine is the negative sine.

The derivative of the exponential function is itself. An example of exponential growth is compound interest of a savings account.

The derivative shows us the local optima, and the second derivative gives us the bend.

For a cubic equation, the derivative is quadratic, the second derivative is linear, the third derivative is constant, and all higher-order derivatives are zero.

The polynomial has two local optima - a local minimum and a local maximum.

The derivative shows us where to find the optima - at the points where the derivative line crosses the x-axis (y=0).

The second derivative tells us the bend of the polynomial line - and whether an optimum point is a minimum or maximum. Where the second derivative is negative (y<0), the polynomial line is convex, and an optimum point in that section must be a maximum. Where the second derivative is positive (y>0), the polynomial line is concave, and an optimum point in that section must be a minimum. The point where the second derivative crosses the x axis (y=0), the polynomial line is changing direction from _concave_ to _convex_ and is called the _inflection point_.