Mathematical Immaturity

14.4 Exercises

1. Compute the derivatives $F'(t)$ and $F''(t)$ for: $F(t) = (t, t^2, t^3, t^4)$
2. Compute the derivatives $F'(t)$ and $F''(t)$ for: $F(t) = (\cos t, \sin^2 t, \sin 2t, \tan t)$
3. Compute the derivatives $F'(t)$ and $F''(t)$ for: $F(t) = (\arcsin t, \arccos t)$
4. Compute the derivatives $F'(t)$ and $F''(t)$ for: $F(t) = 2e^t \mathbf{i} + 3e^t \mathbf{j}$
5. Compute the derivatives $F'(t)$ and $F''(t)$ for: $F(t) = \cosh t\mathbf{i} + \sinh 2t\mathbf{j} + e^{-3t}\,\mathbf{k}$
6. Compute the derivatives $F'(t)$ and $F''(t)$ for: $F(t) = \log(1 + t^2)\mathbf{i} + \arctan t\mathbf{j} + \frac{1}{1+t^2}\,\mathbf{k}$
7. Let $F$ be the vector-valued function given by: $$ \begin{align*} \\ F(t) = \frac{2t}{1+t^2}\mathbf{i} + \frac{1-t^2}{1+t^2}\mathbf{j} + \mathbf{k} \end{align*} $$ Prove that the angle between $F(t)$ and $F'(t)$ is constant, that is, independent of $t.$
8. Compute the vector-valued integral:$$\int_0^1 (t, \sqrt{t}, e^t)\,dt$$
9. Compute the vector-valued integral:$$\int_0^{\pi/4} (\sin t, \cos t, \tan t)\,dt$$
10. Compute the vector-valued integral:$$\int_0^1 \left(\frac{e^t}{1+e^t}\mathbf{i} + \frac{1}{1+e^t}\mathbf{j}\right)\,dt$$
11. Compute the vector-valued integral:$$\int_0^1 (te^t\mathbf{i} + t^2e^t\mathbf{j} + te^{-t}\,\mathbf{k})\,dt$$
12. Compute $A \cdot B,$ where $A = 2\,\mathbf{i} - 4\,\mathbf{j} + \mathbf{k}$ and $B = \int_0^2 (te^{2t}\,\mathbf{i} + t\cosh 2t\,\mathbf{j} + 2te^{-t}\,\mathbf{k})\,dt$
13. Given a nonzero vector $B$ and a vector-valued function $F$ such that $F(t) \cdot B = t$ for all $t,$ and such that the angle between $F'(t)$ and $B$ is constant (independent of $t$). Prove that $F''(t)$ is orthogonal to $F'(t).$
14. 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 $F(t).$
15. If $G = F \times F',$ compute $G'$ in terms of $F$ and derivatives of $F.$
16. If $G = F \cdot F' \times F'',$ prove that $G' = F \cdot F' \times F'''.$
17. Prove that $\lim_{t\to p} F(t) = A$ if and only if $\lim_{t\to p} \|F(t) - A\| = 0.$
18. Prove that a vector-valued function $F$ is differentiable on an open interval $I$ if and only if for each $t$ in $I$ we have:$$F'(t) = \lim_{h\to 0} \frac{1}{h}[F(t + h) - F(t)]$$
19. Prove the zero-derivative theorem for vector-valued functions. If $F'(t) = O$ for each $t$ in an open interval $I,$ then there is a vector $C$ such that $F(t) = C$ for all $t$ in $I.$
20. 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 $F'(0) = C.$
21. A differential equation of the form $Y'(x) + p(x)Y(x) = Q(x),$ where $p$ is a given real-valued function, $Q$ a given vector-valued function, and $Y$ an unknown vector-valued function, is called a first-order linear vector differential equation. Prove that if $p$ and $Q$ are continuous on an interval $I,$ then for each $a$ in $I$ and each vector $B$ there is one and only one solution $Y$ which satisfies the initial condition $Y(a) = B,$ and that this solution is given by the formula: $$ \begin{align*} \\ Y(t) = Be^{-q(t)} + e^{-q(t)}\int_a^t Q(x)e^{q(x)}\,dx \end{align*} $$ where $q(x) = \int_a^x p(t)\,dt.$
22. A vector-valued function $F$ satisfies the equation $tF'(t) = F(t) + tA$ for each $t \geq 0,$ where $A$ is a fixed vector. Compute $F''(1)$ and $F(3)$ in terms of $A,$ if $F(1) = 2A.$
23. Find a vector-valued function $F,$ continuous on the interval $(0, +\infty),$ such that: $$ \begin{align*} \\ F(x) = xe^xA + \frac{1}{x}\int_1^x F(t)\,dt \end{align*} $$ for all $x > 0,$ where $A$ is a fixed nonzero vector.
24. A vector-valued function $F,$ which is never zero and has a continuous derivative $F'(t)$ for all $t,$ is always parallel to its derivative. Prove that there is a constant vector $A$ and a positive real-valued function $u$ such that $F(t) = u(t)A$ for all $t.$