Concavity and Rates of Change Lesson 2.5

Changing Rate of Change Note that the rate of change of the curve of the top of the gate is changing Consider how it is changing … Increasing? Decreasing? What effect does that have on the curve?

Concavity and Rates of Change What do the graphs below have in common? How are they different? Specifically, look at the changing rate of change Both are increasing functions The slope (rate of change) is increasing The slope (rate of change) is decreasing

Concavity and Rates of Change Slope is becoming more positive An increasing positive rate of change is concave UP A decreasing positive rate of change is concave DOWN Slope is becoming less positive

Concavity and Rates of Change Increasing negative rate of change is concave DOWN Decreasing negative rate of change is concave UP Slope is becoming more negative Slope is becoming less negative

Try It Out Given the following table Now Plot the points Calculate the rates of change for successive pairs of points – are the slopes increasing or decreasing? Predict concave up or down Now Plot the points Concave up or concave down? x 12 15 18 21 H(x) 21.40 21.53 21.75 22.02

Assignment Lesson 2.5 Page 86 Exercises 1 – 19 odd