Are the curves $y=(x-C)^{2}$ level curves of a function $f(x, y)$? What property must a family of curves in a region of the xy -plane have to be the family of level curves of a function defined in the region?

A windmill takes a fraction of the wind kinetic energy out as power on a shaft. In what manner does the temperature and wind velocity influence the power?

A l-ft-long steel rod with a 0.5-in. diameter is stretched iii a tensile test. What is the required work to obtain a relative strain of 0.1%? The modulus of elasticity of steel is $30 \times 10^{6} \mathrm{lbf} / \mathrm{in}^{2}$.

An old gas turbine has an efficiency of 21 percent and develops a power output of 6000 kW. Determine the fuel consumption rate of this gas turbine, in L/min, if the fuel has a heating value of 42,000 kJ/kg and a density of $0.8 \mathrm{g} / \mathrm{cm}^{3}.$

Show that at any point, the tangent vectors to a helix $\mathbf{r}(t)=a \cos t \mathbf{i}+a \sin t \mathbf{j}+b t \mathbf{k}$, where a and b are real numbers, intersect the direction of the z-axis at a constant angle.

Identify and graph each polar equation. Convert to a rectangular equation if necessary. $r=2 \cos \theta$

Simplify each of the expressions below. Start by expanding the exponents, and then simplify your results. Look for patterns or possible shortcuts that will help you simplify more quickly. Be prepared to justify your patterns or shortcuts to the class. a. $y^{5} \cdot y^{2}$ b. $\frac{w^{5}}{w^{2}}$ c. $(x^{2})^{4}$ d. $x^{10} \cdot x^{12}$ e. $\frac{13 p^{4} q^{5}}{p^{2} q^{2}}$ f. $\left(\frac{x^{2}}{y}\right)^{3}$ g. $5 h \cdot 2 h^{24}$ h. $\frac{10 m^{30}}{2 m^{8}}$ i. $\left(3 k^{20}\right)^{4}$ j. $\frac{24 h g^{2}}{3 h g^{9}}$ k. $\left(\frac{m^{3}}{n^{10}}\right)^{4}$ l. $w^{4} \cdot p \cdot w^{3}$

When we double the compression ratio of an ideal Otto cycle, what happens to the maximum gas temperature and pressure when the state of the air at the beginning of the compression and the amount of heat addition remain the same? Use constant specific heats at room temperature.

Find the integrals. Check your answers by differentiation. $\int \sin ^{3} \alpha \cos \alpha d \alpha$

Find the equation of the line that passes through the points (-800, 200) and (-400, 300).

If an object falls from rest under constant gravitational acceleration, show that its average height during the time T of its fa ll is its height at time $T / \sqrt{3}$.

True or False. The parametric equations defining a curve are unique.

The water pressure in the mains of a city at a particular location is 270 kPa gage. Determine if this main can serve water to neighborhoods that are 25 m above this location.

Write the expression using exponents.

$$s \cdot(7) \cdot s \cdot(7) \cdot(7)=$$