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

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 $\theta=2, b=1, \varepsilon=1, \text { and } \mu=1.$

Differentiate the given function and simplify the answer whenever possible. $F(x)=\left(1+x^{2}\right) \tan ^{-1} x$

Why are chromatic and spherical aberrations important factors in refracting telescopes, but not in reflecting telescopes?

A slit 0.025 mm wide is illuminated with red light $(\lambda=680\ \mathrm{nm}).$ How wide are (a) the central maximum and (b) the side maxima of the diffraction pattern formed on a screen 1.0 m from the slit?

Determine the convergence or divergence of the series and identify the test used. $\sum_{n=1}^{\infty} \frac{8}{\sqrt[3]{n}}$

Find $\lim _{x \rightarrow 2}\left(\frac{x^{2}}{x-2}-\frac{2 x}{x-2}\right)$. Then comment on the statement that this limit is given by $\lim _{x \rightarrow 2} \frac{x^{2}}{x-2}-\lim _{x \rightarrow 2} \frac{2 x}{x-2}$.

Explain why the function is differentiable at the given point. Then find the linearization of the function at that point. $f(x, y)=x \sqrt{y}, \quad(1,4)$

When a thin film of kerosene spreads out on water, the thinnest part looks bright. The index of refraction of kerosene is (a) greater than, (b) less than, (c) the same as that of water.

a) Prove the change of base formula, $\log _{c} x=\frac{\log _{d} x}{\log _{d} c},$ where c and d are positive real numbers other than 1. b) Apply the change of base formula for base d = 10 to find the approximate value of $\log _{2} 9.5$ using common logarithms. Answer to four decimal places. c) The Krumbein phi ($\varphi$) scale is used in geology to classify the particle size of natural sediments such as sand and gravel. The formula for the $\varphi$-value may be expressed as $\varphi=-\log _{2} D$, where D is the diameter of the particle, in millimetres. The $\varphi$-value can also be defined using a common logarithm. Express the formula for the $\varphi$-value as a common logarithm. d) How many times the diameter of medium sand with a $\varphi$-value of 2 is the diameter of a pebble with a $\varphi$-value of -5.7? Determine the answer using both versions of the $\varphi$-value formula from part c).

Identify the appropriate base rational function, $y=\frac{1}{x}$ or $y=\frac{1}{x^{2}},$ and then use transformations of its graph to sketch the graph of each of the following functions. Identify the asymptotes.

$$\begin{array}{ll}{\text { a) } y=\frac{1}{x+2}} & {\text { b) } y=\frac{1}{x-3}} \\ {\text { c) } y=\frac{1}{(x+1)^{2}}} & {\text { d) } y=\frac{1}{(x-4)^{2}}}\end{array}$$

Find the directional derivative of $f(x, y)=\sqrt{x y} \text { at } P(2,8)$ in the direction of Q(5, 4).

Use an iterated integral to find the area of the region bounded by the graphs of the equations. $y=9-x^{2}$, y=0

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