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Geometry

Calculus is useful in geometry. Already mentioned is the ability to find areas under curves and the gradient of a curve exactly. Another useful application of differential calculus is to find minimum, maximum and inflexion points on a curve. Such points are known as stationary points. From geometric considerations, we can easily see that at a stationary point, the gradient of the curve is equal to 0. Hence, to find the stationary points on a curve, we need to solve the equation

\begin{displaymath}
\frac{\mathrm{d}y}{\mathrm{d}x} = 0
\end{displaymath}

To identify the nature of the points - minimum, maximum or inflexion point, we need to evaluate the second derivative - that is - the derivative of the derivative at the stationary point. This is written

\begin{displaymath}
\frac{\mathrm{d}}{\mathrm{d}x} \biggl ( \frac{\mathrm{d}y}{\mathrm{d}x}
\biggr ) = \frac{\mathrm{d}^2 y}{\mathrm{d} x^2}
\end{displaymath}

If the point is a maximum, then the gradient of the curve decreases, so at a maximum, \( \frac{\mathrm{d}^2 y}{\mathrm{d} x^2} < 0 \). Similarly, if the point is a minimum, \( \frac{\mathrm{d}^2
y}{\mathrm{d} x^2} > 0 \). In the case when \( \frac{\mathrm{d}^2
y}{\mathrm{d} x^2} = 0 \), the point could be any of the three types of points and the nature has to be determined by a different method. The simplest method is to consider the signs of the function close to the stationary point.

For example, to find the stationary points of the curve \( y = x^3 +
3 x^2 + 2 x - 1 \), we proceed as follows.

\begin{eqnarray*}
y & = & x^3 + 3 x^2 + 2 x - 1 \\
\frac{\mathrm{d}y}{\mathrm{d...
...c{-6 \pm \sqrt{12}}{6} \\
& = & -1 \pm \frac{1}{3} \sqrt{3} \\
\end{eqnarray*}



This shows that the curve has two stationary points. To find their nature, we find and evaluate the second derivative at these points

\begin{eqnarray*}
\frac{\mathrm{d}^2 y}{\mathrm{d} x^2} & = & 6 x + 6 \\
\frac{...
...rm{d} x^2} \Biggm \vert _{x=-1+\sqrt{3}/3} & \approx
&
3.464 > 0
\end{eqnarray*}



Hence, the first is a maximum and the second is a minimum.

Calculus has lots more applications in geometry in the field known as ``differential geometry'' concerning curvature, solids of revolution, arc length, etc.


next up previous
Next: Mean value of a Up: Applications of calculus Previous: Applications of calculus
Alexander Frolkin 2001-03-13