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

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.

If possible. compute the following:

(a) $(2A)^T$
(b) $(A - B )^T$
(c) $(3 B^T - 2A)^T$
(d)$(3A^T - 5B^T)^T$
(e) $(- A )^T \text{and} - (A^T)$
(f) $(C + E + F^T)^T$.

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

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

Prove the property. In each case, assume r, u, and v are differentiable vector-valued functions of t in space, w is a differentiable real-valued function of t, and c is a scalar. d/dt{r(t) · [u(t) × v(t)} = r′(t) · [u(t) × v(t)] + r(t) · [u′(t) × v(t)] + r(t) · [u(t) × v′(t)]

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 the convolution $f(t)*g(t)$ for the given problems:

$$a)\; f(t)=t,\; g(t)=1,\; b)\; f(t)=g(t)=\sin t,\; c)\; f(t)=g(t)=e^{at}.$$

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

n each of these cases, find the percentage rate of change of the function $f(t)$ with respect to $t$ at the given value of $t$.

a. $f(t)=t^2-3 t+\sqrt{t}$ at $t=4$

b. $f(t)=\frac{t}{t-3}$ at $t=4$

The beam in Fig. supports a concentrated downward load $P=20 \mathrm{kN}$ at its left end and a uniformly distributed downward load $w=40 \mathrm{kN} / \mathrm{m}$ over the span $B C$. If the beam is a $\mathbf{W} 310 \times 52$ wide-flange beam, (a) determine the maximum flexural stress, $\sigma_{\max }$, and (b) determine the maximum transverse shear stress, $\tau_{\max }-$

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

Find the roots of $f(t)$ assuming $f(t)$ has an integer root: (a) $f(t)=t^3-2 t^2-6 t-3$; (b) $f(t)=t^3-t^2-11 t-10$; (c) $f(t)=t^3+2 t^2-13 t-6$.

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) $f(t)=8 e^{-3 t} \cosh t u(t-2)$

If $f(t)=t^{10}$, find f'(t), f''(t), and f'''(t).