Find the derivative of y with respect to the given independent variable:$y=\log _{3}\left(\left(\frac{x+1}{x-1}\right)^{\ln 3}\right)$

The numerator of a fraction is five less than the denominator. The sum of the fraction and six times its reciprocal is $\frac { 25 } { 2 } .$ Find the numerator and denominator assuming that each is an integer.

For matrix

$$\begin{bmatrix}2&4\\-1&-2\end{bmatrix}$$

compute its eigenvalues and eigenvector(s), and sketch the eigenspace if the eigenvectors are real.

What is the difference between adaptation and evolution?

Only one of these calculations is correct. Which one? Why are the others wrong? Give reasons for your answers. a. $\lim _{x \rightarrow 0^{+}} x \ln x=0 \cdot(-\infty)=0$ b. $\lim _{x \rightarrow 0^{+}} x \ln x=0 \cdot(-\infty)=-\infty$ c. $\lim _{x \rightarrow 0^{+}} x \ln x=\lim _{x \rightarrow 0^{+}} \frac{\ln x}{(1 / x)}=\frac{-\infty}{\infty}=-1$ d. $\lim _{x \rightarrow 0^{+}} x \ln x=\lim _{x \rightarrow 0^{+}} \frac{\ln x}{(1 / x)}$

$$=\lim _{x \rightarrow 0^{+}} \frac{(1 / x)}{\left(-1 / x^2\right)}=\lim _{x \rightarrow 0^{+}}(-x)=0$$

If a body of mass m falling from rest under the action of gravity encounters an air resistance proportional to the square of the velocity, then the body's velocity t sec into the fall satisfies the differential equation

$$m \frac{d v}{d t}=m g-k v^2$$

where $k$ is a constant that depends on the body's aerodynamic properties and the density of the air. (We assume that the fall is short enough so that the variation in the air's density will not affect the outcome significantly.) a. Show that

$$v=\sqrt{\frac{m g}{k}} \tanh \left(\sqrt{\frac{g k}{m}} t\right)$$

satisfies the differential equation and the initial condition that $v=0$ when $t=0$. b. Find the body's limiting velocity, $\lim _{t \rightarrow \infty} v$. c. For a 160-lb skydiver ( $m g=160$ ), with time in seconds and distance in feet, a typical value for $k$ is 0.005 . What is the diver's limiting velocity?

The region in the first quadrant bounded by the coordinate axes, the line y=3, and the curve $x=2 / \sqrt{y+1}$ is revolved about the y -axis to generate a solid. Find the volume of the solid.

Evaluate the integral. $\int_1^2 \frac{8 d x}{x^2-2 x+2}$

Which one is correct, and which one is wrong? Give reasons for your answers. a. $\lim _{x \rightarrow 0} \frac{x^2-2 x}{x^2-\sin x}=\lim _{x \rightarrow 0} \frac{2 x-2}{2 x-\cos x}$

$$=\lim _{x \rightarrow 0} \frac{2}{2+\sin x}=\frac{2}{2+0}=1$$

b. $\lim _{x \rightarrow 0} \frac{x^2-2 x}{x^2-\sin x}=\lim _{x \rightarrow 0} \frac{2 x-2}{2 x-\cos x}=\frac{-2}{0-1}=2$

Find the derivative of y with respect to the given independent variable:$y=\log _{3} r \cdot \log _{9} r$

For the function g(x) = x^2-2x-4 ln(x):

(a) Find the open intervals on which the function is increasing and decreasing. (b) Identify the function's local and absolute extreme values, if any, saying where they occur.

Derive the formula $\sinh ^{-1} x=\ln \left(x+\sqrt{x^2+1}\right)$ for all real $x$. Explain in your derivation why the plus sign is used with the square root instead of the minus sign.

Evaluate the integral.

$$\int \frac{d y}{y^2+6 y+10}$$

Evaluate the integral.

$$\int_{-1}^1 \frac{3 d v}{4 v^2+4 v+4}$$