Derivative Introduction as Slope of Tangent Line and Speed. Fun Help Math Game by SliderMath.

Math Help - What is a Derivative?

A Derivative is an Instantaneous Rate of Change.

You can think of a derivative as the Slope of a Curve at point P.

You can also interpret a derivative as a Speed.

In this game, the derivative at point P is the slope of the tangent line at point P.

- Method 1:   Notice the slope of the secant QP changes as point Q moves along the curve towards point P.
    The slope of the secant QP approaches the slope of the tangent line at P.
    Recall that the slope of QP = rise/run = (change in height)/(change in time).

    Average speed is equal to change in distance divided by change in time.
    Average speed is represented by the slope of secant QP.
    Instantaneous vertical speed at point P equals the slope of the line tangent to the curve at P.

- Method 2: In Calculus, there is a derivative rule that calculates speed = instantaneous rate of change.
    The derivative of the curve y = a(x-p)²+q is stated as dy/dx = 2a(x-p) = slope of the tangent line.
    For example, the rate of change of the curve y = -4(x-3)²+6 is stated as dy/dx = 2(-4)(x-3).
    At a point P on this curve, say where x=1, the speed = derivative = dy/dx = 2(-4)(1-3) = 16.

The above 2 Methods work for smooth continuous curve sections such as parabolic arcs.
The derivative = slope of a tangent can be described as the limit of a sequence of secant slopes.

In this game, the tangent line touches the curve at one point P,
and the secant line touches the curve at two points Q and P.

Method 1 is an example of a numerical method using software calculations to approximate an answer.

- The score report automatically appears after you have made 10 choices.
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