In an electric circuit with two resistors of resistance $R_1$ and $R_2$ connected in parallel, the total resistance $R$ is given by the formula

$$\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}$$

Show that

$$R_1 \frac{\partial R}{\partial R_1}+R_2 \frac{\partial R}{\partial R_2}=R$$

FInd

$$\partial w / \partial x _ { i }$$

for i=1, 2, ..., n. w=cos

$$\left( x _ { 1 } + 2 x _ { 2 } + \dots + n x _ { n } \right)$$

Write the rational function given as a partial sum of partial fraction.

$$\frac{x^{4}-x^{2}+2}{x^{2}(x-1)}$$

Write the rational function given as a partial sum of partial fractions.

$$\frac{3 x^{2}+2 x-1}{x(x+1)}$$

The x- and y-components of a fluid moving in two dimensions are given by the following functions u and v. The speed of the fluid at (x, y) is s(x, y)=$\sqrt { u ( x , y ) ^ { 2 } + v ( x , y ) ^ { 2 } }$. Use the Chain Rule to find $\partial s / \partial x$ and $\partial s / \partial y$. u(x, y)=x(1-x)(1-2y) and v(x, y)=y(y-1)(1-2x); $0 \leq x \leq 1,0 \leq y \leq 1$

Draw the digestive features indicated on this anterior view of the abdomen of a cadaver, using the terms provided.
Gallbladder

Determine the solution of $\partial \rho / \partial t=\rho$ that satisfies $\rho(x, t)=1+\sin x$ along x = −2t.

Multiply $1.3 X 2.4$, showing all partial products. Illustrate with a rectangular array model showing the partial products.

Which of the following fees would likely be the highest. A.Atm Fee. B.Overdraft fee C.Monthly service fee. D.Account transfer fee

Suppose that the function f(x)=P(x)/Q(x) is a proper rational function. (a) For each factor of Q(x) of the form

$$( a x + b ) ^ { m }$$

, the partial fraction decomposition of f contains the following sum of m partial fractions:___. (b) For each factor of Q(x) of the form

$$\left( a x ^ { 2 } + b x + c \right) ^ { m }$$

, where

$$a x ^ { 2 } + b x + c$$

is an irreducible quadratic, the partial fraction decomposition of f contains the following sum of m partial fractions: ___

Air-track gliders with masses $300 \mathrm{~g}, 400 \mathrm{~g}$, and $200 \mathrm{~g}$ are lined up and held in place with lightweight springs compressed between them. All three are released at once. The $200 \mathrm{~g}$ glider flies off to the right while the $300 \mathrm{~g}$ glider goes left. Their positionversus-time graphs, as measured by motion detectors, are shown in Figure mentioned. What are the direction (right or left) and speed of the $400 \mathrm{~g}$ glider that was in the middle?

For a rectangular channel with a bottom width of $1.00 \mathrm{~m}$, calculate the flow area and hydraulic radius for depths ranging from $0.10 \mathrm{~m}$ to $2.0 \mathrm{~m}$. Plot a graph of the area and hydraulic radius versus depth.

Refer to the question.

Prepare a simple one-view sketch to show the use of a long break line.

Suppose $y=h(x)$ is a differentiable function of $x$ and that $f(x, y)=C$ for some constant $C$. Use the chain rule (with $x$ taking the role of $t$ ) to show that

$$\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} \frac{d y}{d x}=0$$

Conclude that the slope at each point $(x, y)$ on the level curve $F(x, y)=C$ is given by

$$\frac{d y}{d x}=-\frac{f_x}{f_y}$$