Consider the function f(x, y) = 4xy / x²+y², (x, y) ≠ (0, 0) {0, (x, y) = (0, 0) and the unit vector u = 1/√2 (i + j). Does the directional derivative of f at P(0, 0) in the direction of u exist? If f(0, 0) were defined as 2 instead of 0, would the directional derivative exist?

Find the directional derivative of $f$ at $P$ in the direction of a, where

$$f=x-y: P:(4,5), \mathbf{a}=2 \mathbf{i}+\mathbf{j}$$

Convert the second-order differential equation

$$\frac{d^2 y}{d t^2}+2 y=0$$

into a first-order system in terms of $y$ and $v$, where $v=d y / d t$.

Briefly describe the behavior of the solutions.

Consider the function $f(x, y)=x^2-y^2$ at the point (4,-3,7).

Determine the directional derivative $D_{u} f(4,-3)$ as a function of $\theta$, where $u=\cos \theta i+\sin \theta j .$ Use a graphing utility to graph the function on the interval $[0,2 \pi)$

Find the directional derivative of $f$ at $P$ in the direction of a, where

$$f=e^x \cos y, P:(2, \pi, 0), \mathbf{a}=2 \mathbf{i}+3 \mathbf{j}$$

Assume that f(x, y) is a differentiable function. Is the statement below true or false? Explain your answer.

There is a function with a point in its domain where $\|\operatorname{grad} f\|=0$ and where there is a nonzero directional derivative.

Level curves for barometric pressure (in millibars) are shown for 6:00 AM on November 10, 1998. A deep low with pressure 972 mb is moving over northeast Iowa. The distance along the red line from $K$ (Kearney, Nebraska) to $S$ (Sioux City, Iowa) is 300 km. Estimate the value of the directional derivative of the pressure function at Kearney in the direction of Sioux City. What are the units of the directional derivative?

Find the directional derivative of $f$ at $P$ in the direction of a, where

$$f=\ln \left(x^2+y^2\right), P:(4,0), \mathbf{a}=\mathbf{i}-\mathbf{j}$$

Consider v=3u. Is the directional derivative of a differentiable function f (x, y) in the direction of v at the point $\left(x_{0}, y_{0}\right)$ three times the directional derivative of f in the direction of u at the point $\left(x_{0}, y_{0}\right) ?$ Explain.

Consider the function

$$f(x, y)= \begin{cases}\frac{4 x y}{x^2+y^2}, & (x, y) \neq(0,0) \\ 0, & (x, y)=(0,0)\end{cases}$$

and the unit vector $\mathbf{u}=\frac{1}{\sqrt{2}}(\mathbf{i}+\mathbf{j})$ Does the directional derivative of $f$ at $P(0,0)$ in the direction of u exist? If $f(0,0)$ were defined as $2$ instead of $0$ , would the directional derivative exist?

Find the derivate ive of the function. h(t) = (t + 1)2/3 (2t2 - 1)3

Find values of a, b, c, and d such that $g(x)=a x^3+b x^2+c x+d$ has a local maximum at $(2,4)$ and a local minimum at $(0,0)$.

Indicate whether the statement is true or false. If (0,0) is a critical point of $f$ then $f$ has either a local maximum or local minimum at $(0,0)$.

Indicate whether the statement is true or false. The function f(x, y)=3 x y has a neither a local maximum or minimum at $(0,0)$.