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In lesson
3.3, we found the velocity formula from the distance (or
position) formula by finding the derivative.
Often in scientific research, we have velocity data and need to find
the distance formula. We need to find the formula whose derivative (or
derivative data) we know. This inverse operation is called finding an antiderivative.
Definition:
| Antiderivative | An antiderivative of f is a function F such that F' = f. |
Whereas we found that functions had only one derivatives, it is also true that many functions can have the same derivative. The diagram illustrates this:
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Antiderivatives of
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A direction field
as shown above indicates the common slopes of all of the curves at the
left.
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All of the functions listed and graphed
above have the same derivative, namely: 
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Because of this, there are infinitely many antiderivatives for a function. If you look closely at all of the antiderivative formulas in the illustration above, you will notice that they all differ only by a constant. Therefore....
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If you have the Journey Through Calculus CD, load and run Resources/Module 6/Antiderivatives/Start of Antiderivatives. |
If one ordered pair from the function is known, then the constant C can be named specifically.
Based on this, we ought to be able to take some of the derivatives that we have learned and find their antiderivatives:
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Function
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Antiderivative
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Sample Problem:
Find f(x)
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Given Problem, #14, Lesson 4.9 |
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For the first part, we use the "reverse power rule" on the 4. We increased the exponent by one and then divided by 1. For the second part we used the product rule and recognized that we had the last antiderivative from the above table. |
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Now we substitute the given point (1,0) from the problem and solve for C. |
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Now that we know the constant C, we can write the complete antiderivative. |
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