Calculus Applets

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Constant Multiple

What happens to the derivative of a function if we multiple the function by some constant?

Try the following:

1. The example shows the line f (x) = kx. We know that the slope for the line f (x) = x is just 1. What does the k do to the slope? Move the k slider and you notice that the new slope is just k.

2. Select the second example, showing a parabola multipled by k. When k > 1, what happens to the shape of the parabola? Move the k slider to see. What happens to the derivative? Set x = 1 and k = 1 and notice the value of f '(1). Now, change k to 2; what happens to the derivative at x = 1? Change k to other values, and see if you can detect a pattern in what happens to the derivative.

3. Select the third example, showing a sine curve multiplied by a constant. What does changing k do to the derivative?

4. Hopefully you have noticed that multiplying a function by a constant just multiplies the derivative by the same constant, or .