Test for symmetry with respect to each axis and to the origin. $y=\left|x^{3}+x\right|$

Evaluate the integrals. $\int_{1 / 2}^{1} \frac{\sin ^{-1} x}{x^{2}} d x$

Show that $x_{1}(t) \text { and } x_{2}(t)$ are solutions to the initial-value problem $d \mathbf{x} / d t=A \mathbf{x} \text { with } \mathbf{x}(0)=\mathbf{x}_{0}.$

$$A=\left[\begin{array}{rr} {1} & {-1} \\ {0} & {1} \end{array}\right] \quad \mathbf{x}_{0}=\left[\begin{array}{l} {2} \\ {1} \end{array}\right]$$

$x_{1}(t)=2 e^{t}-t e^{t}, x_{2}(t)=e^{t}$

When a man's face is in front of a concave mirror of radius 100 cm, the lateral magnification of the image is +1.5. What is the image distance?

Evaluate the integrals. Assume a, b, A, B, $P_{0}$, h, and k are constants. $\int \frac{e^{x}}{2+e^{x}} d x$

Indicate whether the statement is true or false. If S(t) is an income stream, then S(t) has units of dollars.

Consumer resource models often have the following general form $R^{\prime}=f(R)-g(R, C) \quad C^{\prime}=\varepsilon g(R, C)-h(C)$ where R is the number of individuals of the resource and C is the number of consumers. The function $f(R)$ gives the rate of replenishment of the resource, g(R, C) describes the rate of consumption of the resource, and h(C) is the rate of loss of the consumer. The constant $\varepsilon,$ where $0<\varepsilon<1,$ is the conversion efficiency of resources into consumers. Find all equilibria of the following examples and determine their stability properties. A chemostat is an experimental consumer-resource system. If the resource is not self-reproducing, then it can be modeled by choosing $f(R)=\theta, g(R, C)=b R C,$ and $h(C)=\mu C.$ Suppose $\theta=2, b=1, \varepsilon=1, \text { and } \mu=1.$

Find the derivative of each function. $f(x)=\frac{x^{2}}{e^{x}}$

Solve the equation. Approximate the result to three decimal places. $2 \log _{3} 4 x=15$

Verify Stokes’s Theorem by evaluating $\int_{C} \mathbf{F} \cdot d \mathbf{r}$ as a line integral and as a double integral. $\mathbf{F}(x, y, z)=x y z \mathbf{i}+y \mathbf{j}+z \mathbf{k}$, $S: 6 x+6 y+z=12$, first octant

Evaluate the integrals by using the properties of the definite integral and interpreting integrals as areas. $\int_{-3}^{3}(2+t) \sqrt{9-t^{2}} d t$

Evaluate the limits. $\lim _{x \rightarrow 1} \frac{x^{1 / 3}-1}{x^{2 / 3}-1}$

What is a doubling time? Suppose a population has a doubling time of 25 years. By what factor will it grow in 25 years? in 50 years? in 100 years?

Algebraically determine the general solution to the equation $4 \cos ^{2} x-1=0$. Give your answer in radians.