# Analyzing graphs and finding derivatives

20. The graph below displays growth of a town's population y = P(t) over the next 3 years, where t is time in months.

a. Estimate how fast the population is increasing 5 months, and 20 months from now.

b. Graph y = P'(t).

In graphs of questions 2, 4 and 6, determine which is the f(x) function and which is the derivative?

Please see the attached document for graphs. Please explain your answer well i.e. how do you arrive at the answer/decision.

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#### Solution Preview

20. The rate of change in the value of the function with respect to a change in the value of x is the derivative of the function. The derivative at a point is the slope of the tangent line to the graph of y=f(x) at that point. So, to estimate the growth rate 5 and 20 months from now, we need to estimate the slopes of the tangent lines to the given graph at x=5 and x=20. The first one ...

#### Solution Summary

The process of analyzing the graph of a function to find its values and estimate the values of its derivative is illustrated with several examples. Examples of matching the graph f a function to the graph of its derivative are also given.