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katrin [286]
2 years ago
9

What are the intercept(s) of ƒ(x) = x2?

Mathematics
1 answer:
IgorC [24]2 years ago
4 0
There are no intercepts. (o,o)
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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
Kelly plans to put her graduation money into an account and leave it there for 4 years while she goes to college. she recieves $
Firlakuza [10]

Answer:

C) $877.50

Step-by-step explanation:

Time t = 4 years

Principal, p = $750

rate, r = 4.25% = 0.0425

Amount that will be in her account in four years =

Simple interest (prt/100) + principal

750 * 4.25* 4 / 100 + 750

127.5 + 750

= $877.50

7 0
3 years ago
$800 is deposited in an account
Sergeeva-Olga [200]

Answer:

The balance after four years is $1129.27

Step-by-step explanation:

The formula for compound interest, including principal sum, is A=P(1+\frac{r}{n})^{nt}

  • A = the future value of the investment/loan, including interest
  • P = the principal investment amount (the initial deposit or loan amount)
  • r = the annual interest rate (decimal)
  • n = the number of times that interest is compounded per unit t
  • t = the time the money is invested or borrowed for

∵ $800 is deposited in an account

∴ P = 800

∵ The account pays 9% annual interest

∴ r = 9% = 9 ÷ 100 = 0.09

∵ The interest is compounded annually

∴ n = 1

∵ The time is 4 years

∴ t = 4

- Substitute the values of P, r, n, and t in the formula above

∵ A=800(1+\frac{0.09}{1})^{(1)(4)}

∴ A=800(1.09)^{4}

∴ A = 1129.265

∴ The balance after four years is $1129.27

8 0
3 years ago
The volume of a cube is 441cm3.<br> Work out the length of its side rounded to 1 DP.
Rainbow [258]

Answer:

volume of cube=441cm³

l³=441cm³

l=

\sqrt[3]{441}

length=7.6cm

3 0
3 years ago
Is there another way to write JK? Why or why not?
daser333 [38]

Answer:

What do u mean

. I'm confused

4 0
3 years ago
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