$x(t)=t^{1 / 3}, y(t)=t^{2}$ from t=1 to t=1.1.

Simplify:

$$t-\{t-[3t-(2t-t)-t]-4t\}-t.$$

Complete the truth table.

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

$$\mathbf{r}(t)=\left\langle\sin t-t \cos t, \cos t+t \sin t, t^2\right\rangle$$

Read about the disjunctive normal form of a statement in a logic text.

a. What is the disjunctive normal form of a statement that has the following truth table?

Explain why the disjunctive normal form is a valuable concept.

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

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

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 T(t), N(t),

$$a_T$$

, and

$$a_N$$

at the given time t for the plane curve r(t).

$$(t) = (t - t³) \hat{i} + 2t² \hat{j}$$

, t = 1

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

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

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

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).

is φ{f(t)+g(t)}=φ{f(t)}+φ{g(t)}? φ{f(t)g(t)}φ{f(t)φ{g(t)}? Explain.

Prove: If x=x(t) and y=y(t) are differentiable at t, and if z=f(x, y) is differentiable at the point (x(t), y(t)), then

$$\frac { d z } { d t } = \nabla z \cdot \mathbf { r } ^ { \prime } ( t )$$

where r(t)=x(t)i+y(t)j.

Determine the Laplace transforms of these functions: (a) $f(t)=(t-4) u(t-2)$ (b) $g(t)=2 e^{-4 t} u(t-1)$ (c) $h(t)=5 \cos (2 t-1) u(t)$ (d) $p(t)=6[u(t-2)-u(t-4)]$