Write the given system in the matrix form x' = Ax + f.

$$x'(t)=3x(t)-y(t)+t^2$$

$$y'(t)=-x(t)+2y(t)+e^t$$

Let $V$ be the inner product space in previous example. (a) If $p(t)=\sqrt{t}$, find $q(t)=a+b t \neq 0$ such that $p(t)$ and $q(t)$ are orthogonal. (b) If $p(t)=\sin t$, find $q(t)=a+b e^{\prime} \neq 0$ such that $p(t)$ and $q(t)$ are orthogonal.

Complete the truth table.

In the following exercise, find T(t), N(t), a$_{\mathrm{T}}$, and a$_{\mathrm{N}}$ at the given time t for the space curve r(t). [Hint: Find a(t), T(t), and a$_{\mathrm{N}}$. Solve for N in the equation a(t)=a$_{\mathrm{T}}$T + a$_{\mathrm{N}}$N.]

Function: $\mathbf{r}(t)=e^t \sin t \mathbf{i}+e^t \cos t \mathbf{j}+e^t \mathbf{k}$, Time: $t=0$

Derive equations

$$\int_{0}^{T} \sin n \omega_{T} t \sin m \omega_{T} t d t=\left\{\begin{array}{ll}{0} & {m \neq n} \\ {T / 2} & {m=n}\end{array}\right.$$

,

$$\int_{0}^{T} \cos n \omega_{T} t \cos m \omega_{T} t d t=\left\{\begin{array}{ll}{0} & {m \neq n} \\ {T / 2} & {m=n}\end{array}\right.,$$

and $\int_{0}^{T} \cos n \omega_{T} t \sin m \omega_{T} t d t=0$ and hence verify the equations for the Fourier coefficient given by equations $a_{0}=\frac{2}{T} \int_{0}^{T} F(t) d t,$ $a_{n}=\frac{2}{T} \int_{0}^{T} F(t) \cos n \omega_{T} t d t \quad n=1,2, \ldots,$ and $b_{n}=\frac{2}{T} \int_{0}^{T} F(t) \sin n \omega_{T} t d t \quad n=1,2, \ldots.$

Consider $V$ to be the inner product space of Example 6. Find the cosine of the angle between each pair of vectors in $V$.

(a) $p(t)=t, q(t)=t-1$ (b) $p(t)=t, q(t)=t$ (c) $p(t)=1, q(t)=2 t+3$

Which of the following are odd functions? (a) sin t + cos t (b) t sin t (c) t ln t (d) t³ cos t (e) sinh t

A function f is ________ if f(-t) = -f(t) and ________ if f(-t) = f(t).

Find $\mathbf{r}^{\prime \prime}(t)$.

$$\mathbf{r}(t)=\langle\cos t+t \sin t, \sin t-t \cos t, t\rangle$$

The parametric equations of four plane curves are given. Graph each curve, indicate its orientation, and compare the graphs. (a) $x(t)=t, y(t)=t^{2} ; -4 \leq t \leq 4$ (b) $x(t)=\sqrt{t}, \quad y(t)=t ; 0 \leq t \leq 16$ (c) $x(t)=e^{t}, y(t)=e^{2 t} ; 0 \leq t \leq \ln 4$ (d) $x(t)=\cos t, y(t)=1-\sin ^{2} t ; 0 \leq t \leq \pi$

The relative rate of growth of $A ( t ) = P e ^ { k t }$ in the interval [ t , t + h ] is defined by $\frac { A ( t + h ) - A ( t ) } { A ( t ) }$ a. Show that $\frac { A ( t + 1 ) - A ( t ) } { A ( t ) } = e ^ { k } - 1$ b. Show that $\frac { A ( t + h ) - A ( t ) } { A ( t ) } = e ^ { k h } - 1$

In each of these cases, find the rate of change of f(t) with respect to t at the given value of t. a. $f(t)=t^{3}-4 t^{2}+5 t \sqrt{t}-5$ at t = 4 b. $f(t)=\frac{2 t^{2}-5}{1-3 t}$ at t = -1

Find T(t) and N(t) at the given point. x=cosh t, y=sinh t, z=t, t=ln 2

Find $F(f(t), g(t))$ if $F(x, y)=x^2 y$ and $f(t)=t \cos t$, $g(t)=\sec ^2 t$.