Similarly, a function can be concave down and either increasing or decreasing. Here is the mathematical definition of concavity. To show that the graphs above do in fact have concavity claimed above here is the graph again blown up a little to make things clearer. So, as you can see, in the two upper graphs all of the tangent lines sketched in are all below the graph of the function and these are concave up. In the lower two graphs all the tangent lines are above the graph of the function and these are concave down.
This is important to note because students often mix these two up and use information about one to get information about the other. Now that we have all the concavity definitions out of the way we need to bring the second derivative into the mix.
We did after all start off this section saying we were going to be using the second derivative to get information about the graph. The following fact relates the second derivative of a function to its concavity.
So, what this fact tells us is that the inflection points will be all the points were the second derivative changes sign. We saw in the previous chapter that a function may change signs if it is either zero or does not exist. It is simply a fact that applies to all functions regardless of whether they are derivatives or not. We will only know that it is an inflection point once we determine the concavity on both sides of it. It will only be an inflection point if the concavity is different on both sides of the point.
If you think about it this process is almost identical to the process we use to identify the intervals of increasing and decreasing.
Find the intervals where is concave up. The intervals where a function is concave up or down is found by taking second derivative of the function. Use the power rule which states:. Now, set equal to to find the point s of infleciton. In this case,. To find the concave up region, find where is positive. This will either be to the left of or to the right of. To find out which, plug in a test point in each of those regions. If we plug in we get , which is negative, so that cannot be concave up.
If we plug in , we get , which is positive, so we know that the region will be concave up. Find the intervals that are concave down in between the range of. Now, find which values in the interval specified make. In this case, and. A test value of gives us a of. This value falls in the range , meaning that interval is concave down. The function is concave-down for what values of over the interval?
The derivative of is. Given the equation of a graph is , find the intervals that this graph is concave down on. To find the concavity of a graph, the double derivative of the graph equation has to be taken. To take the derivative of this equation, we must use the power rule,. Setting the equation equal to zero, we find that.
This point is our inflection point, where the graph changes concavity. In order to find what concavity it is changing from and to, you plug in numbers on either side of the inflection point.
Plugging in 2 and 3 into the second derivative equation, we find that the graph is concave up from and concave down from. To find the invervals where a function is concave down, you must find the intervals on which the second derivative of the function is negative.
To find the intervals, first find the points at which the second derivative is equal to zero. The first derivative of the function is equal to. Solving for x,. The intervals, therefore, that we analyze are and. On the first interval, the second derivative is negative, which means the function is concave down. On the second interval, the second derivative is positive, which means the function is concave up. View All Related Lessons.
Alex Federspiel. So let's look at an example to see how this all works. I'm not gonna do these out, but here's what you should get. If we plot this graph we can see that this gave us the right answer.
You've reached the end. How can we improve? Send Feedback. The graph shows that the total cost of a certain activity increases sharply at the beginning and then rises more and more slowly until a point when the total cost begins to rise more sharply again. The blue colour indicates a region where the slope of the tangent decreases. That is, in this region the rate at which the cost function increases, decreases. The red colour indicates a region where the slope of the tangent increases, i.
By our previous definitions, the blue area is concave downward and the red area is concave upwards. The green point is the point at which the rate of change of the slope changes from decreasing to increasing. It is also the point at which the concavity of the function changes from downward to upward. This point is called a point of inflection POI. In general, note that regardless of the sign of the slope positive, negative or zero , the slopes of the tangent are decreasing as we move from left to right when the graph is concave down and increasing from left to right when it is concave up.
Look at the applet below.
0コメント