- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.15 Exercises
- Refer to the curves described in Exercises 1 through 6 of Section 14.9 and in each case determine the curvature $\kappa(t)$ for the value of $t$ indicated.
- Recall from Section 14.14, (equation 14.20) that the length of the curvature vector, or the curvature $\kappa$ of a curve is defined as: $$ \begin{align} \\ \kappa(t) = \frac{\|T'(t)\|}{v(t)} \tag{14.20} \end{align} $$
- 1. $$ \begin{align*} \\ \|T'(t)\| &= \frac{\sqrt{2}}{1 + t^2} \\ \\ v(t) &= 3\sqrt{2}\left(1 + t^2\right) \\ \\ \kappa(t) &= \frac{\|T'(t)\|}{v(t)} \\ &= \frac{1}{3\left(1 + t^2\right)^2} \\ \kappa(2) &= \frac{1}{75} \quad \blacksquare \end{align*} $$ 2. $$ \begin{align*} \\ \|T'(\pi)\| &= \frac{\left[\left(1 + e^{2\pi}\right)\left(1 + 2e^{2\pi}\right)\right]^{1/2}}{\left(1 + e^{2\pi}\right)^{3/2}} \\ \\ v(\pi) &= \sqrt{1 + e^{2\pi}} \\ \\ \kappa(\pi) &= \frac{\left(1 + 2e^{2\pi}\right)^{1/2}}{\left(1 + e^{2\pi}\right)^{3/2}} \quad \blacksquare \end{align*} $$ 3. $$ \begin{align*} \\ \|T'(0)\| &= \frac{6}{5} \\ \\ v(0) &= 5 \\ \\ \kappa(0) &= \frac{6}{25} \quad \blacksquare \end{align*} $$ 4. $$ \begin{align*} \\ \|T'(\pi)\| &= \frac{\sqrt{2}}{2} \\ \\ v(\pi) &= 2 \\ \\ \kappa(\pi) &= \frac{1}{4} \sqrt{2} \quad \blacksquare \end{align*} $$ 5. $$ \begin{align*} \\ \|T'(1)\| &= \frac{2}{3} \\ \\ v(1) &= 9 \\ \\ \kappa(1) &= \frac{2}{27} \quad \blacksquare \end{align*} $$ 6. $$ \begin{align*} \\ \left\|T'\left(\frac{\pi}{4}\right)\right\| &= \frac{\sqrt{2}}{2} \\ \\ v\left(\frac{\pi}{4}\right) &= \sqrt{2} \\ \\ \kappa\left(\frac{\pi}{4}\right) &= \frac{1}{2} \quad \blacksquare \end{align*} $$