91Ó°ÊÓ

Use Theorem 12.33 to find the indicated derivatives in Exercises 27–30. Express your answers as functions of two variables.

$\frac{∂z}{∂r}whenz=(x^2+xy)e^y,x=r\cos \mathrm{θ}andy=r\sin \mathrm{θ}$

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.

$f(x,y,z)=x{y}^2{z}^3,P=(0,0,0),\mathbf{v}=⟨1,−2,−1âŸ\copyright$

Find the first-order partial derivatives for the functions in Exercises 27–36.

$f\left(x,y\right)=x^y$

The derivative when two variables are held fixed: Let

\begin{equation}f(x, y, z)=x^2 y^3 \sqrt{z}\end{equation}

. Find the rate of change of f in the (positive) z-direction when the values of x and y are constant. Find the rate of change of f in the (positive) y direction when the values of x and z are constant. Find the rate of change of f in the (positive) x direction when the values of y and z are constant.

Critical points: What is the definition of a critical point for a function of a single variable? How do we use critical points to locate the extrema of the function?

Second-Derivative Test: Review the second-derivative test for functions of a single variable. Explain how the test works, what conditions a function must satisfy to make the test useful, and when, if ever, the test might fail.

Reordering a Double Summation: Explain why

$\underset{i=1}{\overset{m}{∑}}\underset{j=1}{\overset{n}{∑}}f(i,j)=\underset{j=1}{\overset{n}{∑}}\underset{i=1}{\overset{m}{∑}}f(i,j).$

The gradient at a minimum: If a function of three variables, $$f(x, y,z)$$, is differentiable at a point $$(x_{0}, y_{0}, z_{0})$$ where the function has a minimum, what is $$\bigtriangledown f(x_{0}, y_{0}, z_{0})$$ ?

A kind of derivative for a function of three variables: Explain why the derivative of the function $$xe^{−4z} \sin{y}$$ is $$e^{−4z} \sin{y}$$ if $$x$$ is the variable and $$y$$ and $$z$$ are constants, and the derivative is $$xe^{−4z} \cos{y}$$ if $$y$$ is the variable and $$x$$ and $$z$$ are constants, and the derivative is $$-4xe^{−4z} \sin{y}$$ if $$z$$ is the variable and $$x$$ and $$y$$ are constants. What is the derivative if $$x$$, $$y$$, and $$z$$ are all constants?

How do you find the critical points of a function of two variables, $f\left(x,y\right)$? What is the significance of the critical points?