If $f(x, y, z)=x^{2}-y^{3}+x y z$, and $x = 6t + 7, y = \sin 2 t, z=t^{2}$, verify the chain rule by finding df/dt in two different ways.

Answer the questions about the function $g(x)=\frac{x+2}{x-6}$. If x=4, what is g(x)? What is the corresponding point on the graph of g?

Express the function in the form $f \circ g \circ h.$ $R(x)=\sqrt{\sqrt{x}-1}$

If A, B, and C are constants and $f (x, y) = Ax + By + C$, show that $\lim _{(x, y) \rightarrow\left(x_{0}, y_{0}\right)} f(x, y)=f\left(x_{0}, y_{0}\right)=A x_{0}+B y_{0}+C$

A restaurant offers 4 different family meal deals. Each meal deal includes tacos and burritos as shown. a. Which quantities are proportional? B. If the restaurant used the pattern shown in the table to create a fifth meal deal, how many burritos would it include?

Let n be an even number. Verify the triangle inequality in $R^n$ for a = (1, 0, 1, 0, . . . , 0) and b = (0, 1, 0, 1, . . . , 1).

Use derivatives to find the critical points and inflection points. $f(x)=5 x-3 \ln x$

A capacitor is connected to a 12-V battery. If the plate separation is tripled and the capacitor remains connected to the battery, (a) by what factor does the charge on the capacitor change? (b) By what factor does the energy stored in the capacitor change? (c) By what factor does the electric field between the plates of the capacitor change?

If

$$\mathbf{T}(u, v)=\left[\begin{array}{rr} {2} & {3} \\ {-1} & {1} \end{array}\right]\left[\begin{array}{l} {u} \\ {v} \end{array}\right]$$

and $D^{*}$ is the parallelogram whose vertices are (0, 0), (1, 3), (−1, 2), and (0, 5), determine $D = T(D^{*}).$

The function defined by $f^{\prime}(x)=-0.1624 x+3.4909$ approximates marginal U.S. nonfarm productivity from 2000–2009. Productivity is measured as total output per hour compared to a measure of 100 for 2000, and x is the number of years since 2000. a. Give the function that describes total productivity in year x. b.Use your function from part a to find productivity at the end of 2008. In 2009, productivity actually measured 122.3. How does your value using the function compare with this?

When the elements of an RLC circuit are both magnitude-scaled and frequency-scaled, which quality is unaffected? (a) resistor (b) resonant frequency (c) bandwidth (d) quality factor

Assume a class named Dollars exists. Write the headers for member functions that overload the prefix and postfix ++ operators for that class.

Transform the given integral in Cartesian coordinates to one in polar coordinates and evaluate the polar integral. $\int_{-1}^{1} \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}} 3 d y d x$

Graph the surface whose spherical equation is $\rho= 1-\sin \varphi$