Section 4.10 Antiderivatives
References.
- Calculus, Early Transcendentals by Stewart, Section 4.9.
Probably the greatest use of calculus is in problems where one knows something about the derivatives of a function and wishes to learn about the function: going from knowledge about the rate of change of a quantity to knowing the quantity itself. For example, the laws of physics often describe acceleration (second derivative of position), from which position as a function of time is determined. Also in biological, chemical and economic models, rates of changes are often the measured or known information, from which we seek to make predictions of how a quantity will change over time.
We have already seen one simple but important example: when the acceleration of a falling body is constant, its velocity is linear in time, and its position is a suitable quadratic function of time.
Our interest here is the “opposite” of derivatives:
For example, for function \(f(x)=3x^2\text{,}\) the function \(F(x)=x^3\) is an antiderivative, and so is the alternative \(F(x)=x^3+7\text{,}\) or indeed \(F(x)=x^3 + C\) for any constant \(C\text{.}\)
Position and Velocity from (Constant) Acceleration.
When acceleration
\begin{equation*}
a(t) = v'(t) = -9.8m/s^2\text{,}
\end{equation*}
one possible anti-derivative of \(a\) is
\begin{equation*}
v(t) = -9.8t\text{,}
\end{equation*}
and since this \(v(t)\) is \(s'(t)\) for \(s(t)\) the displacement, one possible antiderivative of \(v\) is displacement
\begin{equation*}
s(t) = -4.9t^2\text{.}
\end{equation*}
But there are other antiderivatives: one can have
\begin{equation*}
v(t) = -9.8t+v_0 \text{ for any constant } v_0 \text{, the velocity when } t=0.
\end{equation*}
Then this velocity has as one antiderivative the position function \(s(t)=-4.9t^2+v_0t\text{,}\) and again, adding any constant is allowed, giving the family of of possible position functions
\begin{equation*}
s(t)=-4.9t^2+v_0t+s_0 \text{ for any constant } s_0\text{, the displacement when } t=0.
\end{equation*}
Checkpoint 4.10.2.
Find formulas for all possible antiderivatives of
- \(\displaystyle f(x)=\sin x\)
- \(f(x)=x^n\text{,}\) \(n \neq -1\) a constant
- \(f(x)=x^{-1}=1/x\text{.}\)
Hint.
Be careful with the cases where the domain is not an interval because it excludes \(x=0\text{.}\)
The “quirks” with domains like \(x \neq 0\) illustrate why we usually work with derivatives and antiderivatives on intervals. This is natural; domains are intervals in most applications of antiderivatives (and indeed in most applications of calculus).
Checkpoint 4.10.3. Hunting and Collecting Antiderivatives (the start of an ongoing activity).
Every derivative formula gives the anti-derivative of a function, so this is how we start our collection of antiderivatives:
- Write down all the the formulas you know for derivatives of basic functions, giving pairs \(F(x)\text{,}\) \(f(x)=F'(x)\text{.}\)
- Multiply each by a constant if necessary so that the function \(f(x)\) is as simple as possible.
- Turn each pair around into a function-antiderivative pair \(f(x)\text{,}\) \(F(x)\text{,}\) and gather these in a table.
- Add new antiderivatives to this table as we discover them.
This table will be useful for computing derivatives and anti-derivatives: keep it with your notes.
Theorem 4.10.4. Antiderivatives on an interval differ only by a constant.
If \(F\) and \(G\) are both antiderivatives of the same function \(f\) on the same interval \(I\text{,}\) they differ only by a constant: \(G(x)-F(x)\) is a constant \(C\text{,}\) so
\begin{equation}
G(x)=F(x) + C.\tag{4.10.1}
\end{equation}
Proof.
This follows from the Mean Value Theorem:
Firstly, the difference \(G-F\) has derivative \((G-F)'=G'-F'=f-f=0\) everywhere.
We saw in Section 4.4 that the only function with derivative equal to zero everywhere on an interval is a constant, so \(G(x)-F(x)\) is a constant: call it \(C\text{.}\)
As a result, the graphs of two different antiderivatives of \(f\) on an interval never pass through the same point, so once you know one point on the graph of the desired antiderivative, there is only one choice.
The Geometry of Antiderivatives.
We have seen how derivatives relate to the “geometry” of a function; the shape or of the graph. This ideas is useful in the reverse direction too, using that fact that the value of \(f(x)\) describes the slope of the antiderivative graph at \(x\text{.}\)
The above theorem says that the graphs of any two different antiderivatives of the same function on an interval differ simply by a vertical shift.
Checkpoint 4.10.5. Sketching Antiderivatives.
- Sketch a few simple functions \(f\) like \(f(x)=x^2\) or \(f(x)=\sin x\text{,}\) for which we know an antiderivative, and from the information there about the slope of the antiderivative, try to sketch an antiderivative \(F(x)\text{,}\) passing through the origin. Place the sketch of \(F\) directly below that of \(f\text{.}\)
- Sketch the known antiderivatives and see how well you did.
- Draw a graph of some function \(f\) for which you do not know a formula, and try to sketch several antiderivatives for it.
Rectilinear Motion (motion along a line).
If we know the acceleration of an object moving on a line, its velocity is an antiderivative. Knowing the velocity at any one time allows one to choose the correct antiderivative:
Knowing acceleration at all times plus velocity at any one time determines velocity at all times.
In turn, knowing velocity tells as the position (an antiderivative of velocity) up to a constant and knowing position at one time determines the position function. Putting this all together,
Knowing acceleration at all times plus position and velocity at any one time determines position at all times.
This is the basic form of the single most important mathematical contribution to physics in the last few centuries, and one of the main reasons why calculus is so important in physical science.Exercises Exercises
Study Calculus Volume 1, Section 4.10 2 ; including all the Examples and Checkpoints, and a selection from Exercises 465–469, 470–473, 474–489, 490–498, 499–503 and 504–508.
Here the exercises are grouped in ranges by "question type", so start by trying several from each of the ranges; some suggested selections are Exercises 465, 467, 469, 471, 477, 487, 491, 493, 499, 501 and 505.
Hint: It often helps to simplify the function first, and then use the list of derivatives and indefinite integrals in the online test.
openstax.org/books/calculus-volume-1/pages/4-10-antiderivativesopenstax.org/books/calculus-volume-1/pages/4-10-antiderivatives
