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3 years ago

Consider the following function.

Mathematics

1 answer:

goldfiish [28.3K]3 years ago

6 0

We have that
f(x) = (3/2)*x + 4

case 1) for x=-8
f(-8) = (3/2)*(-8) + 4--------> -8
point (-8,-8)

case 2) for x=-4
f(-4) = (3/2)*(-4) + 4--------> -2
point (-4,-2)

case 3) for x=-2
f(-2) = (3/2)*(-2) + 4--------> 1
point (-2,1)

case 4) for x=0
f(0) = (3/2)*(0) + 4--------> 4
point (0,4)

case 5) for x=2
f(2) = (3/2)*(2) + 4--------> 7
point (2,7)

case 6) for x=4
f(4) = (3/2)*(4) + 4--------> 10
point (4,10)

using a graph tool
see the attached figure

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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.

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

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

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

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

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3 years ago

A store purchased a DVD for $12.00 and sold it to a customer for 50% more than the purchase price. The customer was charged a 7%

olga55 [171]

Answer:

$19.26 or $19.

Step-by-step explanation:

Sold DVD for 50% more of the purchase price of $12, so:

12 + 50/100 * (12) = 12 + 6 = 18.

The customer will pay 7% tax on $18.

18 * (1.07) = $19.26

4 0

3 years ago

Read 2 more answers

The normal body temperature of a camel is 37°C. This temperature varies by up to 3°C throughout the day. Write and solve an abso

uysha [10]

The normal body temperature : to = 37° C
An inequality is:
| t - to | ≤ 3
| t - 37 | ≤ 3
t - 37 ≤ 3 AND ( - t - 37 ) ≤ 3
t ≤ 3 + 37 - t + 37 ≤ 3
t ≤ 40 - t ≤ 3 - 37
 - t ≤ - 34 / *( - 1 )
 t ≥ 34
Answer t ∈ [ 34°C, 40° C ].