In the sentence below, identify the underlined word by writing above it N if it is a noun or ADV if it is an adverb.

$\underline{\text{Tomorrow}}$ is the first day of spring.

Compare using <,=, or >. 7 _ -12

As discussed in the text, a function f (x, y) may have partial derivatives fx (a, b) and fy (a, b) yet fail to be differentiable at (a, b). Geometrically, if a function f (x, y) is differentiable at (a, b), then, aswe zoom in on the point (a, b, f (a, b)), the graph of z = f (x, y) will “flatten out” and look like the plane given by equation (4) in this section. For the functions f (x, y) given in Exercise, (a) calculate fx (a, b) and fy (a, b) at the indicated point (a, b) and write the equation for the plane given by formula (4) of this section, (b) use a computer to graph the equation z = f (x, y) together with the plane calculated in part (a). Zoom in near the point (a, b, f (a, b)) and discuss whether or not f (x, y) is differentiable at (a, b). (c) Give an analytic (i.e., nongraphical) argument for your answer in part (b). $f(x, y)=\sin x \cos y, \ (a, b)=\left(\frac{\pi}{6}, \frac{3 \pi}{4}\right)$

The equation of motion of a weight attached to a spring and a dashpot damping device is

$$x(t) = -\dfrac{1}{\sqrt{2}} e^{-4t}\text{ sinh } 2\sqrt{2}t$$

where $x(t)$, measured in feet, is the displacement from the equilibrium position of the spring system and $t$ is measured in seconds.
a. Find the initial position and the initial velocity of the weight.
b. Plot the graph of $x(t)$.

Choose the correct response to each question or statement.

Me llamo Gabriel.

a. Igualmente.

b. Encantado.

c. El gusto es mío.

Each of the following vectors is given in terms of its x- and y-components. Draw the vector, label an angle that specifies the vector's direction, then find the vector's magnitude and direction. a. $v _ { x } = 20 \mathrm { m } / \mathrm { s } , v _ { y } = 40 \mathrm { m } / \mathrm { s }$ b. $a _ { x } = 2.0 \mathrm { m } / \mathrm { s } ^ { 2 } , a _ { \mathrm { y } } = - 6.0 \mathrm { m } / \mathrm { s } ^ { 2 }$

Consider the system given by

$$\left\{\begin{array}{l} x^{\prime}=-b y+f(x, y) \\ y^{\prime}=b x+g(x, y) \end{array}\right.$$

with $f(0) = g(0) = 0$ and $f_{x}(0)=f_{y}(0)=g_{x}(0)=g_{y}(0)=0$ and the quantity $\mu=\frac{1}{16}\left[f_{x x x}+f_{x y y}+g_{x x y}+g_{y y y}\right] + +\frac{1}{16 b}\left[f_{x y}\left(f_{x x}+f_{y y}\right)-g_{x y}\left(g_{x x}+g_{y y}\right)-f_{x x} g_{x x}+f_{y y} g_{y y}\right]$, where the partial derivatives are all evaluated at (0; 0). That the system will undergo a supercritical Hopf bifurcation if $\mu<0$ and a subcritical Hopf bifurcation if $\mu>0$. Use this to show that the $x^{\prime}=y+r x, y^{\prime}=-x+r y-x^{2} y$ undergoes a supercritical Hopf bifurcation when r = 0.

On the line provided, rewrite the following sentence to correct the misplaced or dangling modifier. Example 1. Ms. Nguyen will finish grading the projects that we turned in during spring break. $\underline{\text{\color{#c34632}During spring break, Ms. Nguyen will finish grading the projects that we turned in.}}$

To cook a complete dinner for twenty guests, careful planning is the first step.

The position of a mass oscillating on a spring is given by $x=(7.8 \mathrm{~cm}) \cos [2 \pi t /(0.68 \mathrm{~s})]$. What is the frequency of this motion?

Compare using <, =, or >. |-7| _ |7|

Compare:

$$- 10 \bigcirc 10$$

Compare:

$$| - 10 | \bigcirc | 10 |$$

Consider the radial field $F=\frac { \mathbf { r } } { | \mathbf { r } | } = \frac { \langle x , y , z \rangle } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 1 / 2 } }$.

Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a) is finite. Evaluate the triple integral of the Divergence Theorem as an improper integral as follows. Integrate div F over the region between two spheres of radius a and $0 < \varepsilon < a$. Then let $\varepsilon \rightarrow 0 ^ { + }$ to obtain the flux computed in part (a).

Compare the decimals. 0.732 ____ 0.75