Sketching Functions From The Graphs Of Derivatives
Sketching the graphs of functions is an important skill in calculus. One useful tool in graphing functions is to use the graphs of their derivatives. In this tutorial, we will explore how to sketch functions from the graphs of their derivatives.
What is a Derivative?
A derivative represents the rate at which a function changes at a given point. It is defined as the slope of the tangent line to the function at that point. In other words, the derivative is the instantaneous rate of change of the function.
Why Use the Graphs of Derivatives?
The graphs of derivatives can be used to help sketch the graphs of functions. This is because the derivative gives information about the behavior of the function at a given point. For example, if the derivative is positive at a point, then the function is increasing at that point. If the derivative is negative at a point, then the function is decreasing at that point.
How to Sketch Functions from the Graphs of Derivatives
To sketch a function from its derivative, we need to follow these steps:
Step 1: Identify Critical Points
Critical points are points where the derivative is zero or undefined. These points are important because they can tell us about the behavior of the function. For example, if the derivative changes sign at a critical point, then the function has a local maximum or minimum at that point.
Step 2: Determine the Sign of the Derivative
The sign of the derivative tells us whether the function is increasing or decreasing at a given point. If the derivative is positive, then the function is increasing. If the derivative is negative, then the function is decreasing.
Step 3: Sketch the Function
Using the information from steps 1 and 2, we can sketch the function. We start by plotting the critical points on the x-axis. Then, we use the sign of the derivative to determine the behavior of the function between the critical points. We can also use the second derivative to determine the concavity of the function.
Example
Let's consider the function f(x) = x^3 - 3x^2 + 2x. We can find its derivative by taking the derivative of each term:
To sketch the graph of f(x), we need to follow the steps outlined above.
Step 1: Identify Critical Points
To find the critical points, we need to solve for when the derivative is zero or undefined.
Therefore, the critical points are x = 1/3 and x = 2.
Step 2: Determine the Sign of the Derivative
To determine the sign of the derivative, we need to evaluate it at points between the critical points.
Therefore, the sign of the derivative is:
Step 3: Sketch the Function
Using the information from steps 1 and 2, we can sketch the graph of f(x):
We can see that the function has a local maximum at x = 1/3 and a local minimum at x = 2.
Conclusion
Sketching functions from the graphs of their derivatives is a useful technique in calculus. By following the steps outlined in this tutorial, we can determine the behavior of a function at critical points and sketch its graph.
Remember to practice and check your work with a graphing calculator or software to ensure accuracy.
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