Find $\mathbf{r}'(t)$ and $\mathbf{r}''(t)$.

$$\mathbf{r}(t)=\left<t\cos t-\sin t, t \sin t+\cos t\right>$$

Use the formula B(t) = T(t) x N(t) to find B(t), and then check your answer by using the formula below to find B(t) directly from r(t).

Formula: B(t)=[(r'(t) x r''(t))/(||r'(t) x r''(t)||)]

r(t) = e raised to ti + e raised to t cos tj + 3k

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

$M(t)=t+\sqrt t$, $N(t)=3t+7$, $C(t)=M(N(t))$ and $D(t)=N(M(t))$

Compute $C(3)$ and $D(4)$.

Find the convolution $f ( t ) * g ( t )$.

$$f ( t ) = t ^ { 2 } , g ( t ) = \cos t$$

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

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

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

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

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

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

Let the moment-generating function M(t) of X exist for −h < t < h. Consider the function R(t) = ln M(t). The first two derivatives of R(t) are, respectively, R'(t) = M'(t)/M(t) and R"(t) = M(t) M"(t) - [M'(t)]²/[M(t)]². Setting t = 0, show that a. μ = R'(0). b. σ² = R"(0).

In the following exercise, find T(t), N(t), a$_{\mathrm{T}}$, and a$_{\mathrm{N}}$ at the given time t for the plane curve r(t).

$$\mathbf{r}(t)=\langle\cos \omega t+\omega t \sin \omega t, \sin \omega t-\omega t \cos \omega t\rangle, \quad t=t_0$$

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. $\lim _{t \rightarrow 2} \frac{t^{3}-5 t^{2}+6 t}{t^{4}-4 t^{2}}$.

If $F(t)=t(\ln t)-t$, find $F^{\prime}(t)$.

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

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

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