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Diano4ka-milaya [45]
3 years ago
13

A set of data is shown

Mathematics
1 answer:
nata0808 [166]3 years ago
3 0
I think it is B because it isn’t a solid line and it’s going up
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If f(x, y, z) = x sin(yz), (a) find the gradient of f and (b) find the directional derivative of f at (2, 4, 0) in the direction
valentina_108 [34]

Answer:

a) \nabla f(x,y,z) = \sin{yz}\mathbf{i} + xz\cos{yz}\mathbf{j} + xy \cos{yz}\mathbf{k}.

b) Du_{f}(2,4,0) = -\frac{8}{\sqrt{11}}

Step-by-step explanation:

Given a function f(x,y,z), this function has the following gradient:

\nabla f(x,y,z) = f_{x}(x,y,z)\mathbf{i} + f_{y}(x,y,z)\mathbf{j} + f_{z}(x,y,z)\mathbf{k}.

(a) find the gradient of f

We have that f(x,y,z) = x\sin{yz}. So

f_{x}(x,y,z) = \sin{yz}

f_{y}(x,y,z) = xz\cos{yz}

f_{z}(x,y,z) = xy \cos{yz}.

\nabla f(x,y,z) = f_{x}(x,y,z)\mathbf{i} + f_{y}(x,y,z)\mathbf{j} + f_{z}(x,y,z)\mathbf{k}.

\nabla f(x,y,z) = \sin{yz}\mathbf{i} + xz\cos{yz}\mathbf{j} + xy \cos{yz}\mathbf{k}

(b) find the directional derivative of f at (2, 4, 0) in the direction of v = i + 3j − k.

The directional derivate is the scalar product between the gradient at (2,4,0) and the unit vector of v.

We have that:

\nabla f(x,y,z) = \sin{yz}\mathbf{i} + xz\cos{yz}\mathbf{j} + xy \cos{yz}\mathbf{k}

\nabla f(2,4,0) = \sin{0}\mathbf{i} + 0\cos{0}\mathbf{j} + 8 \cos{0}\mathbf{k}.

\nabla f(2,4,0) = 0i+0j+8k=(0,0,8)

The vector is v = i + 3j - k = (1,3,-1)

To use v as an unitary vector, we divide each component of v by the norm of v.

|v| = \sqrt{1^{2} + 3^{2} + (-1)^{2}} = \sqrt{11}

So

v_{u} = (\frac{1}{\sqrt{11}}, \frac{3}{\sqrt{11}}, \frac{-1}{\sqrt{11}})

Now, we can calculate the scalar product that is the directional derivative.

Du_{f}(2,4,0) = (0,0,8).(\frac{1}{\sqrt{11}}, \frac{3}{\sqrt{11}}, \frac{-1}{\sqrt{11}}) = -\frac{8}{\sqrt{11}}

6 0
3 years ago
the ramirez family comes home from vacation to find their mailbox overflowing with mail. there are 6 more magazines than there a
nirvana33 [79]

Answer:

9 magazines

Step-by-step explanation:

To solve this problem you will want to work backwards. Start with the junk mail. There is 18 junk mail and if there are 1/3 as many bills as junk mail then you will need to multiply 18 * 1/3. This equals 6 bills. If there is twice as many bills as letters then you will need to divide bills(6) by 2 which equals 3. If there are 6 more magazines than letters then you need to add 6 to the letters (3). 3 + 6 = 9

4 0
2 years ago
Which number is rational?<br> –2.1010010001...<br> –0.8974512...<br> 1.2547569...<br> 5.3333333...
saveliy_v [14]

Answer:

5.3333333333

Step-by-step explanation:

A number is called a rational if it can be written in the form of p/q where p/q are integers and q≠0.

The terminating numbers are always a rational number.

The number 5.3333333... is a rational number.

8 0
3 years ago
Please help<br><br> -3/4(8n + 12) = 3(n-3)
Dennis_Churaev [7]

Step-by-step explanation:

-  \frac{3}{4} (8n + 12) = 3(n - 3)

- 3(8n + 12) = 12(n - 3)

\cancel{- 3}(8n + 12) =  \cancel{12}(n - 3)

- (8n + 12) = 4(n - 3)

- 8n - 12 = 4n - 12

- 8n - 4n =  - 12 + 12

- 12n = 0

n = 0

8 0
2 years ago
A desk is 2 meters long and 1 meter wide. The desk needs to be represented in a scale drawing of an office that uses a scale of
Strike441 [17]

Answer:

<em>Thus, the dimensions of the desk in the drawing are 4 cm long and 2 cm wide.</em>

Step-by-step explanation:

<u>Scaling</u>

The scale factor established to represent a desk 2 meters long and 1 meter wide is:

1 centimeter = 0.5 meter

We need to convert the real dimensions to the scaled dimensions. To complete the task, it's a good idea to multiply the real dimensions by the ratio:

\displaystyle \frac{1\ cm}{0.5\ m}

to get the scaled dimensions.

The length of 2 meters is scaled to:

\displaystyle 2\ m\frac{1\ cm}{0.5\ m}=4\ cm

And the width of 1 meter is scaled to:

\displaystyle 1\ m\frac{1\ cm}{0.5\ m}=2 cm

Thus, the dimensions of the desk in the drawing are 4 cm long and 2 cm wide.

3 0
2 years ago
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