Evaluate $\int_C f(\mathbf{r}) d s$ with $f$ and $C$ as follows. (Show the details.) $f=1+y^2+z^2, \quad$ C: $\mathbf{r}=[t, \cos t, \sin t], 0 \leqq t \leqq \pi$.

A particle moves along a horizontal line with position at time t given by $s(t)=\frac{1}{t}+\ln t$ for $t>0$. For which value of t is the acceleration zero?

(A) 2

(B) $e^2$

(C) 1

(D) c

Find $T^{−1}$ for the given isomorphism T. $T: \mathbf{p}^{2} \rightarrow \mathbf{p}^{2}$ with $T\left(a x^{2}+b x+c\right)=c x^{2}-b x+a$

Use the method of variation of parameters (and perhaps a computer algebra system) to solve the initial value problem

$$\mathbf { x } ^ { \prime } = \mathbf { A } \mathbf { x } + \mathbf { f } ( t ) , \quad \mathbf { x } ( a ) = \mathbf { x } _ { a }.$$

In the problem we provide the matrix exponential $e ^ { \mathbf { A } t }$ as provided by a computer algebra system.

$$e ^ { \mathbf { A } t } = \left[ \begin{array} { c c c c c } { 1 } & { 4 t } & { 8 t + 6 t ^ { 2 } } & { 32 t ^ { 2 } + 8 t ^ { 3 } } \\ { 0 } & { 1 } & { 3 t } & { 8 t + 6 t ^ { 2 } } \\ { 0 } & { 0 } & { 1 } & { 4 t } \\ { 0 } & { 0 } & { 0 } & { 1 } \end{array} \right], e ^ { \mathbf { A } t } = \left[ \begin{array} { c c c c } { 1 } & { 4 t } & { 4 \left( - 1 + e ^ { 2 t } \right) } & { 16 t \left( - 1 + e ^ { 2 t } \right) } \\ { 0 } & { 1 } & { 0 } & { 4 \left( - 1 + e ^ { 2 t } \right) } \\ { 0 } & { 0 } & { e ^ { 2 t } } & { 4 t e ^ { 2 t } } \\ { 0 } & { 0 } & { 0 } & { e ^ { 2 t } } \end{array} \right]$$

Given $0<t<\frac{\pi}{2}, \sin t=p,$ and $\cos t=q,$ complete the table. Let x be an angle between 0 and $2 \pi .$

$$\begin{array}{|c|c|c|c|} \hline x & \pi-t & \pi+t & \\ \hline \sin x & & & -p \\ \hline \cos x & & & q \\ \hline \tan x & & & \\ \hline \end{array}$$

$\int_{C} z d s$ where C has the vector equation $a(t)=t \cos t i+\mathrm{tsintj}+t k, \ 0 \leq t \leq t_{0}$

Evaluate the given expression. $\frac{d}{d t}\left(a t^{2}+b t+c\right)$; (a, b, c constant)

Let $f(x, y)=e^{x y}$ and $\mathbf{c}(t)=\left(t^3, 1+t\right)$.

Calculate $\nabla f$ and $\mathbf{c}^{\prime}(t)$.

Assume the variables a = 2, b = 4, and c = 6. Indicate by circling the T or F if each of the following conditions is true or false:

a == 4 || b > 2            T F
6 <= c && a > 3            T F
1 != b && c != 3           T F
a >= -1 || a <= b          T F
!(a > 2)                   T F

The following problem develop properties of matrix transposes. Suppose that A and B are matrices of the same size. Show that: (a) $\left(\mathbf{A}^{T}\right)^{T}=\mathbf{A} ;(\mathbf{b})(c \mathbf{A})^{T}=c \mathbf{A}^{T}$; and (c) $(\mathbf{A}+\mathbf{B})^{T}=\mathbf{A}^{T}+\mathbf{B}^{T}$.

Differentiate the function.

$$F(t)=\frac{A t}{B t^2+C t^3}$$

Let a, b, and c be real numbers with $a \neq 0,$ and define $p(t)=a t^{2}+2 b t+c$. a. Show that p(t) > 0 for every number t if and only if a > 0 and $a c-b^{2}>0$. b. Show that p(t) < 0 for every number t if and only if a < 0 and $a c-b^{2}>0$.

Carry out the step for the given curves C. On the graph of part ( c ), plot $B(t)$ at the points of $C$ corresponding to $t=0$ and $t=\pi / 2$. $C: r(t)=\langle 3 \sin t, 3 \cos t, 4 t\rangle$, for $0 \leq t \leq 2 \pi$

Given

$$T ( x ) = a x ^ { 2 } + b x + c$$

, find a, b and c if T(0) = -4, T(1) = -2 and T(2) = 6.