In examining graphs, we will also look at the concept of CONCAVITY and its relat
ID: 2866251 • Letter: I
Question
In examining graphs, we will also look at the concept of CONCAVITY and its relationship to derivatives (namely, the second derivative, or "rate of the rate"). Graphs that are CONCAVE UP are shaped like a cup. Graphs that are CONCAVE DOWN are shaped like a frown. The goal of this activity/class prep is to look at a couple of increasing graphs and examine the rate of the rate (the second derivative) and how it connects to concavity (the curvature of the graph).
2.
What is happening to the slopes of the tangent lines as you read from left to right?
What can you conclude about the rate of change of the slopes themselves?
What is happening to the slopes of the tangent lines as you read left to right?
What can you conclude about the rate of change of the slopes themselves?
2.
Consider the graph below:
What is happening to the slopes of the tangent lines as you read from left to right?
What can you conclude about the rate of change of the slopes themselves?
Examine the graph below (with mini tangent lines sketched along the curve):
What is happening to the slopes of the tangent lines as you read left to right?
What can you conclude about the rate of change of the slopes themselves?
Explanation / Answer
Considering the case 1: Concave UP curve
As we move from left to right.
We encounter an increasing slope. Initially the slope is zero and then it keeps on increasing.
This can als be proved by first derivative test if we write the equation on curve and differentiate it w.r.t x. We will notice an increasing slope.
Since the starting point of curve can be called as local minima. Hence the second derivative gives us a positive number. Which then decreases to zero and so on.
Hence in this curve rate of change of the slopes changes from some positive value to zero and so on.
Considering the case 2: Concave DOWN curve
As we move from left to right.
We encounter an decreasing slope. Initially the slope is maximum ( or infinite) and then it keeps on decreasing to zero.
This can als be proved by first derivative test if we write the equation on curve and differentiate it w.r.t x. We will notice an decreasing slope.
Since the curve attains a maxima. Hence the second derivative gives us a negative number at the end.
Hence in this curve rate of change of the slopes changes from zero and to a negative value at the maxima.