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?

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

An increasing positiverate of change is concave UP

A decreasing positiverate of change isconcave DOWN

Slope is becoming less positive

Slope is becoming more positive

Concavity and Rates of Change

Increasing negativerate of change is concave DOWN

Decreasing negativerate of change isconcave UP

Slope is becoming less negative

Slope is becoming more negative

Try It Out

Given the following table 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