AP Calculus

The Getting Off on a Tangent lab showed a number of things. It first pointed out that the average rate of change between two points on a function is the slope of the secant line between the two points. Then as one point approaches the other, the average rate approaches the instantaneous rate. The tangent’s slope = instantaneous rate at P = f’(x_p). For increasing functions the derivative is greater than zero. For decreasing functions the derivative is less than zero. At the high and low points in the graph the derivative equals zero. The derivative is the instantaneous rate of change of f(x) with respect to x at x = c. Also, when the tangent line is above, it is concave down and the slope is decreasing.

The Rate of Change Lab demonstrated visually what the differences are between average rate and instantaneous rate, in that you could see when one of the two methods would be most effective. The average rate of a long period of time does not effectively summarize the variable rates during the course of the time. The instantaneous rate is effective when trying to find what the rate was at a given time. As the intervals got smaller and smaller, the average rate became much more accurate and was also closer to the instantaneous rate. The last parts of the lab also showed how you can find the derivative by taking two points, and essentially finding the slope of the segment. The change in y over the change in x equals the average rate, and the slope of the secant line equals f(x) – f(c) over x – c.

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