Given a differentiable function , we can have and possibly . Each of these functions has their own characteristics, and each describes the function(s) before it.
is the function in question. It may be a function of any degree and may increase and decrease at some points. The question arises when we want to know when it decreases or increases. This is the characteristic of
tells us the gradient of the tangent at any point of the graph, and it also tells us if the graph is increasing or decreasing at any point, including where the graph changes direction and increases after decreasing or vice versa. We call that point a critical point.
To find the tangent...
Find the equation of the tangent on graph where .
First, we find the gradient of the tangent:
Next, we find the value of when
Lastly, we find which line goes through and has a gradient of Let the equation of the gradient be
Therefore, the tangent has equation
To find if the graph is increasing or decreasing at any point, we just substitute the value of into . If is increasing at . If is decreasing at . If is a critical point.
Find the critical points of
Firstly, we find
Next, we find the points where
Lastly, substitute the values of into We get two points, and
determines the concavity of the graph. A picture on it will be uploaded soon...
When , concavity may change. This point(s) is(are) called inflection point(s).
With the knowledge of and , graph sketching will be easier.
While the tangent line is a very useful tool, when it comes to investigating the graph of a function, the tangent line fails to say anything about how the graph "bends" at a point. This is where the second derivative comes into play.
A second-order derivative is a measure of how the rate of change of a quantity is itself changing, which is obtained by differentiating again the first derivative of a quantity.
- The second derivative of a function is written as .
- In Leibniz's notation, the second derivative of (dependent variable) with respect to (independent variable) is written as
Find the second derivative of the function , where is any constant.
Therefore, the answer is 0.
Find the second derivative of the function .
Therefore, answer is .
Relation with Graphs:
First, we will list all theorems and points related to graphs and second derivatives and then move to examples (from basic to advanced).
If the function is twice differentiable at , then the graph of is concave up at if and concave down if .
An inflection point is a point on a curve where the concavity (sign) of curvature changes. Inflection points may be stationary, but they are not local maxima or minima.
To find a point of inflection, let us see an example:
Let be a function defined as . Find the concavity of different parts of the graph of for different values of . Also calculate the points of inflection.
We have .
For , . Hence the curve is concave up for .
For , . Hence the graph is concave down for .
Since the concavity is opposite for and , the concavity changes at the point where .
Note: We can deduce that point of inflection can be obtained from , which is a necessary condition for point of inflection.
Draw the graph of for .
- Inflection point:
Since the inflection point is obtained by setting , we have because Hence the inflection point is .
Since , we have the following:
- For , . Hence it is concave up.
- For , . Hence it is concave down.
We have the following graph:
Given , plot the graph .
we can differentiate the given equation as follows:
Again differentiating it, we get
- For , . Hence concave down.
- For , . Hence concave up.
Inflection point: Setting implies that the point where is the inflection point.
We get the following graph:
Second Derivative Test:
A second derivative test is a test used to determine whether a stationary point where is a local maximum or minimum.
Performing the test:
In order to calculate higer-order derivatives, we can use Leibniz's formula: