Consider the following situation. We have a function *f* and we want to
know its value at a certain point *x*. However, substituting *x*
into the expression for *f* is not exactly pleasant, on the other hand
there is a point *a* near *x* that we would not mind putting into
*f*. Is there a way to use this to at least approximate
*f* (*x*)?

In cases when substituting *a* into *f* is simple we usually also
do not have a problem with substituting *a* into *f* ′. This
means that we can find the
tangent line *T* to the graph
of *f* at *a*. If *x* is really close to *a*, then
*f* (*x*)*T*(*x*),

**Solution:**
Although there is an algorithm for calculating square roots with a pencil
and paper, almost nobody remembers it. Thus we try a different trick. We
will denote the square root of *x* as *f* (*x*)*f* (5),*a* that is close to
*x* = 5*a* = 4.*f* at *a*:

We therefore approximate the square root of 5, in other words
*f* (5),*T*(5) = 9/4 = 2.25.

Such approximations by tangent lines are actually quite useful. People who often work with square roots of numbers close to 1 may like the approximation

which works pretty well for *x*| < 0.2.

**Example:**
Consider a solid body of weight *m*, we will be interested in its
potential energy with respect to the force of gravity of Earth. We will
assume that Earth is round (with radius *R*) and homogeneous, which is
not exactly true but close enough. Then the force *F*, acceleration
*a* and potential energy *U*_{g} of such a body
that has distance *r* from the center of the Earth is given by

However, this is not what students are taught in high-school physics. Let's
simplify situation a bit. We do not really intend to bore into the Earth or
move too high, so we take the radius of Earth *R* as the starting point.
Acceleration at this radius is traditionally denoted as *g*, so we can
put this into the acceleration equation above, express *M* and
substitute into *U*_{g}:

Then it also makes sense to measure potential energy with respect to the
surface of Earth. We will use *h* for the height of the solid above
the surface, so *r* = *R*+*h*,

*U*(*h*) = *U*_{g}(*R*+*h*)
- *U*_{g}(*R*).

We get a difference of two ratios where the variable *h* is in the
denominator, which is not too pleasant. If we expect that the distance will
not change substantially, it makes sense to try to approximate
*U*_{g} by its rangent line taken at
*r* = *R*,*h* = 0.

When we substitute this instead of
*U*_{g}(*R*+*h*),

*U*(*h*) = *g**m**h*.

In fact, quite a few "laws" that we were taught in high school physics are just approximations by tangent lines of more general laws that are more complicated. The approximations usually work well within reasonable limits, for instance the potential energy formula above works reasonably well within heights that we humans most often move.

Of course, when approximating by tangent lines, we always make an error and we do not really know how large, for instance we do not know how good an estimate we got for the root of 5. For some answers (and better approximations) see the next section. For some examples see Solved Problems - Applications, an interesting application an be also found in this problem in Series - Solved Problems - Testing convergence.