Graphing an antiderivative

In summary, to graph the original function when given the derivative, you can use the anti-derivative (the original function) which represents the area under the curve or the slope of the curve. Keep in mind that the derivative only tells you the original function up to an additive constant, so you will need to choose a starting height for your function. If the derivative function is positive, the original function is increasing and the larger the value of the derivative, the greater the slope of the original function. If the derivative graph has an "M" shape, the original function will be curved upward and getting steeper until it reaches a maximum, then becoming less steep until the derivative is 0 again. The second half of the "M" follows
  • #1
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hey all
if I have the function for a derivative (which it is impossible to find an integral for, by the way), how do I graph the original?
This graph is shaped kind of like an M, meaning it has some sharp points...on the original graph, would these be asymptotes or also sharp points??
thanx in advance
 
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  • #2
Remember that the anti-derivative (the original function) is the area under the curve or, conversely, that the derivative is the slope of the curve.

And, of course, knowing the derivative only tells you the original function up to an additive constant. Choose some arbitrary starting height for your function- 0 will do.

As long as your derivative function is positive, you know that the original function is increasing and the larger the value of the derivative the greater slope of the orginal function. If you derivative graph looks like an "M" (starting at y'= 0?) then your original function will be a graph curved upward (concave upward) getting steadily steeper until the derivtive reaches a maximum. As your derivative graph comes back down from the maximum, the graph of the original curve will still go up but now becoming less steep until, when the derivative function is 0 again, the graph of the original curve is horizontal. The second half of the M repeats that- increasing from the new height of course.
 
  • #3


To graph an antiderivative, you can use the fundamental theorem of calculus. This states that the antiderivative of a function is the original function. So, if you have a function for the derivative, you can simply integrate it to find the antiderivative and then plot it on a graph. The sharp points on the derivative graph will correspond to abrupt changes or discontinuities on the antiderivative graph. These points will not necessarily be asymptotes, as they could also be points of discontinuity or cusps. It is important to carefully consider the behavior of the derivative function in order to accurately graph the antiderivative. I hope this helps!
 

1. What is an antiderivative?

An antiderivative is a function that, when differentiated, gives the original function. It is also known as the indefinite integral of a function.

2. How do you graph an antiderivative?

To graph an antiderivative, first find the antiderivative of the given function. Then, plot points on a graph by choosing different values for x and calculating the corresponding values for the antiderivative. Connect the points to create a smooth curve.

3. What are the important features of an antiderivative graph?

The important features of an antiderivative graph include the x-intercepts, where the antiderivative equals zero, and the points where the slope of the graph changes, known as the points of inflection. The area under the curve of the antiderivative graph also represents the original function.

4. How does graphing an antiderivative help in understanding a function?

Graphing an antiderivative can help in understanding a function by showing the overall shape and behavior of the function. It can also help in identifying key features, such as the maximum and minimum points, and understanding the relationship between the original function and its antiderivative.

5. Can an antiderivative graph have multiple solutions?

Yes, an antiderivative graph can have multiple solutions as there can be infinite antiderivatives for a given function. However, all of these antiderivatives will only differ by a constant value.

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