- 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 vectors $A$ and $B$ and a vector-valued function $F$ such that $F''(t) = tA + B,$ determine $F(t)$ if $F(0) = D$ and $F'(0) = C.$
- Using the definition of an integral for a vector-valued function along with the second fundamental theorem, we have: $$\displaylines{F'(t) = \int_0^tF''(t)\,dt + F'(0)\\ =\frac{t^{2}}{2}A + Bt + C}$$Integrating and applying the second fundamental theorem again: \begin{align*} F(t) &= \int_0^t F'(t)\, dt + F(0) \\ &= \frac{t^3}{6}A + \frac{t^2}{2}B + Ct + D\quad\blacksquare \end{align*}