Four lines are represented by the equations below. $\ell: y=-x+1 \quad m: y=-x+7 \quad n: y=2 x+1 \quad p: y=2 x+7$ Graph the four lines in the coordinate plane.

Show that if $b>1 / 2$, there are three straight lines through (0, b ), each of which is normal to the curve $y=x^{2}$. How many such lines are there if $b=1 / 2$? if $b=1 / 2$?

Define Poisson's ratio.

Described the relationship of the two graphed lines.

A peacock mantis shrimp smashes its prey with a hammer-like appendage by storing energy in a springlike section of exoskeleton on its appendage. The spring has a force constant of $5.9 \times 10^4 \mathrm{~N} / \mathrm{m}$. What power does the shrimp generate if it unleashes the energy in $0.0018$ s? (The power per mass of the shrimp appendage is more than 20 times greater than that generated by muscle alone, showcasing the shrimp's ability to harness the potential energy of its spring.)

In the given exercise below, find the centroid of the region.

Region between y=x(a-x) and the x-axis for a>0

Explaining the clues you use, make inferences about the speaker's character in (a) lines $31$-$43$ of "My Last Duchess" and (b) lines $18$- $20$ of "Life in a Love."

The integral represents the volume of a solid. Sketch the region and axis of revolution that produce the solid. $\int_{0}^{1} 2 \pi x\left(x-x^{2}\right) d x$

You have a fixed length $L$ of wire and wish to make a single loop that will experience the maximum torque in a magnetic field. Is it better to make the coil circular or square, or doesn't it matter?

The power output (in watts) of a certain brand of wind turbine generators is estimated to be

$$P=f(R, V)=0.772 R^2 V^3$$

where $R$ is the radius (in meters) of a rotor blade and $V$ is the wind speed (in meters per second). Estimate the power output of a model of these generators if its radius is $30 \mathrm{~m}$ and the wind speed is $16 \mathrm{~m} / \mathrm{s}$.

_____ open water area of a lake or pond that is well lit and dominated by plankton.

The lines $y=2 x+3$ and $y=a x+5$ are parallel if a = _____.

Consider a concentric shaft, fixed axially and rotated inside the sleeve. Let the inner and outer cylinders have radii

$$r_i$$

and

$$r_0$$

, respectively, with total sleeve length L. Let the rotational rate be Ω (rad/s) and the applied torque be M. Using these parameters, derive a theoretical relation for the viscosity μ of the fluid between the cylinders.

How many straight lines that pass through the origin are tangent to y = cos x? Find (to 6 decimal places) the slopes of the two such lines that have the largest positive slopes.