- 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
- Given fixed nonzero vectors $A$ and $B,$ let $F(t) = e^{2t}A + e^{-2t}B.$ Prove that $F''(t)$ has the same direction as $F(t).$
- For two vectors $A$ and $B$ to have the same direction, then for some positive scalar $c,$ $A$ = $cB$
- It will suffice to show that for some positive scalar $c,$ that $F''(t) = cF(t).$ Taking the second derivative of of $F(t):$ $$\displaylines{F'(t) &= 2e^{2t}A - 2e^{-2t}B}$$$$\displaylines{F''(t) &= 4e^{2t}A + 4e^{-2t}B\\ &= 4F(t) \quad \blacksquare}$$