- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.4 Exercises
- Compute the derivatives $F'(t)$ and $F''(t)$ for: $F(t) = \cosh t\mathbf{i} + \sinh 2t\mathbf{j} + e^{-3t}\,\mathbf{k}$
- Recall from section 6.18 that $$\text{sinh}(t) = \frac{e^t - e^{-t}}{2};\quad \text{cosh}(t) = \frac{e^t + e^{-t}}{2}$$This means that $$F(t) = \left(\frac{e^t + e^{-t}}{2}, \frac{e^{2t} - e^{-2t}}{2}, e^{-3t}\right)$$
- $$F(t) = \left(\frac{e^t + e^{-t}}{2}, \frac{e^{2t} - e^{-2t}}{2}, e^{-3t}\right)$$$$\displaylines{F'(t) &= \left(\frac{e^t - e^{-t}}{2}, e^{2t} + e^{-2t}, -3e^{-3t}\right)\\ &=\left(\text{sinh}(t), 2\text{cosh}(t), -3e^{-3t}\right)}$$$$\displaylines{F''(t) &= \left(\frac{e^t + e^{-t}}{2}, 2e^{2t} - 2e^{-2t}, 9e^{-3t}\right)\\ &= \left(\text{cosh}(t), 4\text{sinh}(2t), 9e^{-3t}\right)}$$