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Next: Impulse and Momentum Up: Some Math Tricks Previous: Differentials

Antiderivatives

Suppose we have a function g(x) which we know is the derivative [with respect to x] of some other function f(x), but we don't know which - i.e. we know g(x) explicitly but we don't know [yet] what f(x) it is the derivative of. We may then ask the question, "What is the function f(x) whose derivative [with respect to x] is g(x)?" Another way of putting this would be to ask, "What is the antiderivative of g(x)?"11.1 Another name for the antiderivative is the integral, which is in fact the "official" version, but I like the former better because the name suggests how we go about "solving" one.11.2

For a handy example consider   $g(x) = k \, x$.  Then the antiderivative [integral] of g(x) with respect to x is   $f(x) = {1\over2} \, k \, x^2 \; + \; f_0$  [where  f0  is some constant] because the derivative [with respect to x] of x2 is 2x and the derivative of any constant is zero. Since any combination of constants is also a constant, it is equally valid to make the arbitrary constant term of the same form as the part which actually varies with x, viz.   $f(x) = {1\over2} \, k \, x^2 \; + \; {1\over2} \, k \, x_0^2$. Thus f0 is the same thing as ${1\over2} \, k \, x_0^2$ and it is a matter of taste which you want to use.

Naturally we have a shorthand way of writing this. The differential equation

\begin{displaymath}df = g(x) \; dx \end{displaymath}

can be turned into the integral equation

 \begin{displaymath}f(x) = \int_{x_0}^x g(x) \; dx
\end{displaymath} (11.2)

which reads, "f(x) is the integral of g(x) with respect to x from x0 to x." We have used the rule that the integral of the differential of f [or any other quantity] is just the quantity itself,11.3 in this case f:

 \begin{displaymath}\int df = f
\end{displaymath} (11.3)

Our example then reads

\begin{displaymath}\int_{x_0}^x k \, x \, dx = k \int_{x_0}^x x \, dx
= {1\over2} \, k \, x^2 \; - \; {1\over2} \, k \, x_0^2 \end{displaymath}

where we have used the feature that any constant (like k) can be brought "outside the integral" - i.e. to the left of the integral sign   ${\displaystyle \int}$.

Now let's use these new tools to transform Newton's SECOND LAW into something more comfortable.


next up previous
Next: Impulse and Momentum Up: Some Math Tricks Previous: Differentials
Jess H. Brewer - Last modified: Sat Nov 14 12:38:58 PST 2015