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3 years ago

1. Calculate the rate of change for the following data:

Mathematics

1 answer:

expeople1 [14]3 years ago

4 0

the slope goes by several names

• average rate of change

• rate of change

• deltaY over deltaX

• Δy over Δx

• rise over run

• gradient

• constant of proportionality

however, is the same cat wearing different costumes.

well, to get it all we need is two points, say hmmm let's use (-1 , -7) and (7 , 25)

$(\stackrel{x_1}{-1}~,~\stackrel{y_1}{-7})\qquad (\stackrel{x_2}{7}~,~\stackrel{y_2}{25}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{25}-\stackrel{y1}{(-7)}}}{\underset{run} {\underset{x_2}{7}-\underset{x_1}{(-1)}}}\implies \cfrac{25+7}{7+1}\implies \cfrac{32}{8}\implies 4$

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Step-by-step explanation:

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A particle moves along line segments from the origin to the points (2, 0, 0), (2, 3, 1), (0, 3, 1), and back to the origin under

sweet [91]

Answer: Work done by the particle is 78 N.

Step-by-step explanation:

If C is the path the particle follows, then work done is $\int_{C}^{}F.dx$ .

According to the question the force $F=z^{2}i+5xyj+4y^2k$ and all four points are in the plane $z=\frac{1}{4}y$ .

therefore, if S is the at surface with boundary C, so that S is the portion of the plane $z=\frac{1}{4}y$ over the rectangle D=[0,2]\times [0,4].

Now,

$curlF=\begin{vmatrix}i &j &k \\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} &\frac{\partial }{\partial z} \\ z^{2}&5xy &4y^{2} \end{vmatrix}=8yi+2zj+5yk$

By the Stoke's theorem:

$\int_{C}^{}F.dx=\iint_{s}^{}curlF.dS\\\\=\iint_{D}^{}[-8y(0)-2z\left ( \frac{1}{4} \right )+5y]dA\\\\=\iint_{D}^{}\left ( -\frac{1}{2}z+5y \right )dA$

$=\iint_{D}^{}\left ( \frac{39}{8}y \right )dA\, \, \, \, \, \, \, \, \, Since\, \, z=\frac{1}{4}y\\\\=\int_{0}^{2}\int_{0}^{4}\frac{39}{8}y\, \, \, dy\, dx\\\\$

$\int_{0}^{2}\left [ \frac{39}{16}y^2 \right ]_{0}^{4}dx\\\\=\int_{0}^{2}39\, \, dx\\\\=39\times 2\\\\=78 \, N$

8 0

4 years ago

Help!!!!!<br> Suppose f(x) = x2. What is the graph of g(x) = f(3x)?

STALIN [3.7K]

Answer:

we conclude that g(x) = 9x² is the result of f(x) vertically stretched by multiplying each of its y-coordinates by 9.

Also, check the attached graph is matched with option B.

Hence, option (B) is the correct graph.

Step-by-step explanation:

Given

f(x) = x²

g(x) = f(3x)

g(x) = (3x)² = 9x²

Thus, the graph of g(x) = 9x² is shown below.

From the graph, it is clear that at x = 0, y = 0

So, the y-intercept is at (0, 0).

Clearly

f(x) = x²

g(x) = 9x²

Note that the vertex of the graph also lies at (0, 0), because it is clear that g(x) = 9x² is the result of f(x) vertically stretched by multiplying each of its y-coordinates by 9.

Since 9 > 0, it means g(x) = 9x² is vertically stretched by multiplying each of its y-coordinates by 9.

Therefore, we conclude that g(x) = 9x² is the result of f(x) vertically stretched by multiplying each of its y-coordinates by 9.

Also, check the attached graph is matched with option B.

Hence, option (B) is the correct graph.

7 0

3 years ago