Graph the vector-valued function and identify the common curve. r(t) = ½t²i + tj - √3/2 t²k

Find the velocity vector for the function $\mathbf { r } ( t ) = \left\langle e ^ { t } , e ^ { - t } , 0 \right\rangle$.

In the following exercise, divide. Then check by multiplying.

$42 \div 7$

If a varies directly as b and b = 18 when a = 27, find a when b = 10.

Solve the initial value problem for $\mathbf{r}$ as a vector function of $t$.
Differential equation: $\frac{d \mathbf{r}}{d t}=(180 t) \mathbf{i}+\left(180 t-16 t^2\right) \mathbf{j}$
Initial condition: $\quad \mathbf{r}(0)=100 \mathbf{j}$

At $1000. K$, $\Delta H_R^{\circ}=-123.77 \mathrm{~kJ} \mathrm{~mol}^{-1}$ for the reaction $\mathrm{N}_2(g)+3 \mathrm{H}_2(g) \longrightarrow 2 \mathrm{NH}_3(g)$, with $C_{P, m}=3.502 R, 3.466 R$, and $4.217 R$ for $\mathrm{N}_2(g), \mathrm{H}_2(g)$, and $\mathrm{NH}_3(g)$, respectively. Calculate $\Delta H_f^{\circ}$ of $\mathrm{NH}_3(g)$ at $450 . \mathrm{K}$ from this information. Assume that the heat capacities are independent of temperature.

Solve each system of equations.

$$\left\{\begin{aligned} x+2 y+3 z &=5 \\ y+11 z &=21 \\ 5 y+9 z &=13 \end{aligned}\right.$$

Evaluate (if possible) the vector-valued function at the indicated value of $t$. \

$$\mathbf{r}(t)=(2 t+1) \mathbf{i}+t^2 \mathbf{j}-\dfrac{1}{3} t^3 \mathbf{k}$$

\

$$\mathbf{r}(1+\Delta t)-\mathbf{r}(1)$$

10 raised to power x means that 10 is to be used as a factor x times, and $10^{-x}$ is equal to

$$\frac{1}{10^x}$$

A very large or very small number, therefore, is frequently written as a decimal multiplied by $10^x$, where x is an integer. Which, if any, are false?

(a) $470,000=4.7 \times 10^5$

(b) 450 billion $=4.5 \times 10^{11}$

(c) $0.00000000075=7.5 \times 10^{-10}$

(d) 86 hundred-thousandths $=8.6 \times 10^2$

Simplify:

(a) $\frac{-45}{9} \quad\qquad$ (b) $\frac{-450}{15}$

(c) $15(-20)\quad$ (d) $-15(-12)$

Evaluate the integrals.

$$\displaystyle\int_0^{\ln 5} e^r(3 e^r+1)^{-3 / 2} d r$$

Describe how you could use the prime factorization of 450 to write √450 in simplest form.

In this exercise, sketch the graph of the polar equation.

$$r=10 \cos \theta$$

Write an equation that shows what your next step would be in solving this matrix equation for x, y, and z.

$$\left[ \begin{array} { r r r } { - 1 } & { 2 } & { - 3 } \\ { 2 } & { - 13 } & { 9 } \\ { - 4 } & { 12 } & { - 6 } \end{array} \right] \cdot \left[ \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right] = \left[ \begin{array} { r } { 2 } \\ { - 7 } \\ { 2 } \end{array} \right]$$