Indefinite Integrals (Antiderivatives)

What is an "antiderivative?" So far in this course we have studied many aspects about the rates of change of functions and models: average rates as slopes of secant lines, instantaneous rates as slopes of tangent lines i.e., derivatives, yada yada yada. ...We can now investigate models of real world situations and determine when important amounts will increase or decrease; change slowly, rapidly or not at all, and what trends we might expect for the future. But what accumulates as a result of the changes that occur? How might we calculate that? 

Suppose we begin with a rate function, that is, a derivative. Is it possible to "work backwards" to reverse the derivative process and reconstruct the original function? And if so, couldn't we call the resulting function an "antiderivative" ?

We investigate the idea of accumulations a few lessons hence. . We first begin with with the second question: Is it possible to "work backwards" to reverse the derivative process and reconstruct the original function? 

Example 1: See if you can guess a function that would give the derivative shown. Roll your mouse over the corresponding antiderivative to check your answer.
 
Derivative
Antiderivative (original function)

Example 2 Sketch 3 possible curves for the graph of the original function  given the derivative.
 
Slope Graph or Derivative Graph 
Possible Original Graphs


 

What changes from function to function above is the value of the constant, referred to as "any number", in example 1 above. We will call this constant C.The graph will shift up or down depending on the value of C.  In general the antiderivative of a line y = mx + b is a quadratic function, but the exact placement of the parabola on the plane is unknown without a little more information. 
Example 3Rules for Finding Antiderivatives for Polynomials and Power Functions 
Here are a few new symbols and a little new vocab that you need to learn before we proceed: 

(1) Antiderivatives are also called integrals.

(2) The mathematical notation for antiderivatives is as follows:



To generalize the process in example 1 and put it into a formula,

Using the above rule (or your common sense), find the antiderivative of the given function. Roll your mouse over the images to check your answer.



A Special Case:

Example:

                           

Note: since there is a possibility that x could be negative, we must restrict x to positive values, since you can not take the log of a negative number. Thus, we should write the solution with the absolute value of x, as shown.
 
 

OK, let's try a couple of the "special case" problems: find the antiderivative of the given function. Roll your mouse over the images to check your answer.

                     

Example 4:Rules for Finding Antiderivatives for Exponential Functions 

Since the derivative of exis ex , then the antiderivative of ex is ex .

Using the above rule about ex  (or your common sense), find the antiderivative of the given function. Roll your mouse over the images to check your answer. Remember to completely reverse the derivative! (The opposite of multiplication is division).
 

                                      

Example 5: Rules for Finding Antiderivatives for Exponential Functions with a general base "b"

                          

Let's try a couple of problems like these: work the problems and then roll your mouse over the images to check your answers.

                 

Example 6 - Our first application :The Marginal Revenue Function for a popular software company (see sample of their work below :-) is where x is sales in hundreds and marginal revenue is in thousands of dollars. 
 
a) Find the original revenue function R(x). 

As mentioned above,.unless more information is provided, we have no way to solve for the constant "C". However,

we know that if there are no sales, then there is no revenue. Thus, R(0) = 0. We substitute this information into R(x) in order to solve for C.

final solution: 

b) What is the revenue if x = 150?  Include the correct units in your answer!

solution:    R(150) = 1.75(150) 2 + 15 (150) = 41,625

If sales are 15,000 software products, then the revenue is $41,625,000

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Last Update: January 9, 2010, Leslie Arce copyright 2010 (c) Sharon Walker and theDepartment of Mathematics and Statistics at ASU - all rights reserved