Determine the system of differential equations corresponding to each compartment model and analyze the stability of the equilibrium (0, 0). a = 1.7, b = 0.6, c = 0.1, d = 0.3

Convert each fraction to a decimal. $\frac{1}{10}$

Maximize the function $f(x, y)=x y(15-5 y-3 x)$ on the triangle bounded by the line 5y + 3x = 15, the x-axis, and the y-axis.

Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. $f(x, y)=x y^{2} ; x^{2}-y=0$

Let

$$A=\left[\begin{array}{rrr} 1 & 3 & -1 \\ 2 & 4 & 1 \\ 0 & -2 & 2 \end{array}\right], B=\left[\begin{array}{rrr} 5 & -1 & 4 \\ 2 & 0 & 1 \\ 1 & -3 & -3 \end{array}\right], C=\left[\begin{array}{rrr} -2 & 0 & 4 \\ 1 & -3 & 1 \\ 0 & 0 & 2 \end{array}\right].$$

Determine D so that A+B+C+D=0.

Let $\mathbf{x}=[1,-2,4]^{\prime}.$ Find $\mathbf{y}$ so that $\mathbf{x}$ and $\mathbf{y}$ are perpendicular.

Find the transpose of

$$A=\left[\begin{array}{rrr} -1 & 0 & 3 \\ 2 & 1 & -4 \end{array}\right].$$

Differentiate the functions with respect to the independent variable. $f(x)=2^{x}$

Find the linear approximation of f(x) at x = 0. $f(x)=e^{2 x}$

Use l’Hospital’s rule to find the limits. $\lim _{x \rightarrow \infty} x^{5} e^{-x}$

Differentiate $y=\frac{e^{x-1} \sin ^{2} x}{\left(x^{2}+5\right)^{2 x}}.$

Use a graphing calculator to determine all local and global extrema of the functions on their respective domains. $f(x)=(x-2)^{2}, x \in[0,3]$

Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. $y=x^{2}+5 x, x \in \mathbf{R}$

The half-life of $C^{14}$ is 5730 years. If a sample of $C^{14}$ has a mass of 20 micrograms at time t = 0, how much is left after 2000 years?