Antiderivatives

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Definition: A function F is called an antiderivative of f on an interval I if F ' ( x ) = f ( x ) for all x in I .

It is clear that if is an antiderivative of f , and if for some constant C , then is also an antiderivative of f .

 

Is the converse true ?

Recall that we showed, by the Mean Value Theorem, if two functions have identical derivative on an interval I , then they must differ by a constant. 

If both and are antiderivatives of f , then ' ( x ) = ' ( x ) = f ( x ).

so there exists a constant C such that .

 

Theorem: If F is an antiderivative of f on the interval I , then all the antiderivatives of f on the interval I are of the form where C is an arbitrary constant.

 

Suppose that and are antiderivatives of f and g , respectively, since

( cF + G ) ' = c f + g ,

we get that is an antiderivative of for all real number c .
Example:  It is easy to check that for all x , is an antiderivative of and is an antiderivative of .  Thus, is an antiderivative of .  By the theorem, the antiderivatives of are , where C is an arbitrary constant.

The notation is traditionally used for an antiderivative of f  (we shall see the reason later !!)

With this notation.

Given a function f on an interval I, does the antiderivative of f always exist ?

 

Theorem (Darboux)

Let and I be an interval containing a , b . 

Suppose that F ' ( x ) = f( x ) on I , and . 

If d is a number between f( a ) and f( b ), then there exists a number c in () such that .

 

Proof:  Without loss of generality, we may assume .  Let , then H is differentiable so is continuous on I. By the Extreme Value Theorem, H has both maximum value and minimum value on .

Since H ' ( a ) < 0 and H ' ( b ) > 0, we get that neither H( a ) nor H( b ) is the minimum value of H. This implies that there is a number c in () such that H( c ) is the the minimum value of H. Since H is differentiable on I , by Fermat's Theorem, we get

H ' ( c ) = = 0.

That is, .

 

Darboux's Theorem gives a necessary condition on f so that f has an antiderivative.

 

Example: Let .

We see that , and there exists no number c in () such that . By Darboux's Theorem, f does not have an antiderivative.

 

We shall see later that if f is continuous on I , then f has antiderivatives. An equation that involves the derivatives of a function is called a differential equation . Given a function f on the interval I, the existence of antiderivatives of f is equivalent to the existence of solutions to the differential equation

 

Hence, if f is continuous on I , then the differential equation above has solutions.

 

If we want to sketch the graph of the antiderivative F of with . We surely are guided by the fact that the slope of is f( x ). We start at () and draw F as an increasing function since for all (why ??). To see whether the graph has a point of inflection, we need to compute the derivative of f.

> diff( sqrt(1+x^3)-x , x );

   solve( %=0 , x );

 

It would be a pain in the ass to solve f ' ( x ) = 0. And we could try all day to come up with a formula for the antiderivative of f and still in vain. Another possibility is to draw a graph of the tangent lines of then use it to draw F as shown in the animation below :

 

 

This works, we could get a more accurate graph using what is called a direction field .

 

 

Example:  A particle moves in a straight line and has acceleration given by .  Its initial velocity is cm/s and its initial displacement is cm. Find its position function .

Since v ' ( t ) = , we have

The fact that gives and so

Since s ' ( t ) = , we have

 

But we are given that , so and

.

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