Determine the following limits. a. $\lim _{t \rightarrow-2^{+}} \frac{t^{3}-5 t^{2}+6 t}{t^{4}-4 t^{2}}$. b. $\lim _{t \rightarrow-2^{-}} \frac{t^{3}-5 t^{2}+6 t}{t^{4}-4 t^{2}}$. c. $\lim _{t \rightarrow-2} \frac{t^{3}-5 t^{2}+6 t}{t^{4}-4 t^{2}}$. d. $\lim _{t \rightarrow 2} \frac{t^{3}-5 t^{2}+6 t}{t^{4}-4 t^{2}}$.

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

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

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

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) $p(t)=1, q(t)=2 t+3$

Suppose $f(t)$ is a periodic function with period $T$; that is,

$$f(t+T)=f(t) \text { for all } t .$$

Show that

$$\mathscr{L}[f]=\frac{1}{1-e^{-T s}} \int_0^T f(t) e^{-s t} d t$$

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 $b_{n}=\frac{2}{T} \int_{0}^{T} F(t) \sin n \omega_{T} t d t \quad n=1,2, \ldots.$

Find T(t) and N(t) at the given point. x=

$$e^t$$

cost, y=

$$e^t$$

sin t, z=

$$e^t$$

; t=0

Evaluate the following integrals $(r \neq 0)$ :

(a) $\int_0^T b t e^{-r t} d t$

(b) $\int_0^T(a+b t) e^{-r t} d t$

(c) $\int_0^T\left(a-b t+c t^2\right) e^{-r t} d t$

Find F(s) if: (a) $f(t)=6 e^{-t} \cosh 2 t$ (b) $f(t)=3 t e^{-2 t} \sinh 4 t$ (c) $f(t)=8 e^{-3 t} \cosh t u(t-2)$

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

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

Superposition Principle. Let $y_1$ be a solution to

$$y^{\prime \prime}(t)+p(t) y^{\prime}(t)+q(t) y(t)=g_1(t)$$

on the interval $I$ and let $y_2$ be a solution to

$$y^{\prime \prime}(t)+p(t) y^{\prime}(t)+q(t) y(t)=g_2(t)$$

on the same interval. Show that for any constants $k_1$ and $k_2$, the function $k_1 y_1+k_2 y_2$ is a solution on $I$ to

$$y^{\prime \prime}(t)+p(t) y^{\prime}(t)+q(t) y(t)=k_1 g_1(t)+k_2 g_2(t) \text {. }$$

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 $\frac { A ( t + h ) - A ( t ) } { A ( t ) } = e ^ { k h } - 1$

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) $x(t)=\cos t, y(t)=1-\sin ^{2} t ; 0 \leq t \leq \pi$