Consider the points below. P(â1, 4, 1), Q(0, 6, 2), R(4, 3, â1)

a. Find a nonzero vector orthogonal to the plane through the points P, Q, and R.
b. Find the area of the triangle PQR.

Answer:

Step-by-step explanation:

[tex]f(x,y)=arctan(xy)\\\\\dfrac{ \partial f}{ \partial x}=\dfrac{y}{1+x^2y^2} \\\\\dfrac{ \partial f}{ \partial y}=\dfrac{x}{1+x^2y^2} \\\\\dfrac{ \partial ^2 f}{ \partial x^2}=\dfrac{-2xy^3}{(1+x^2y^2)^2} \\\\\dfrac{ \partial ^2 f}{ \partial y^2}=\dfrac{-2x^3y}{(1+x^2y^2)^2} \\\\\dfrac{ \partial ^2 f}{ \partial x\ \partial y}=\dfrac{1-x^2y^2}{(1+x^2y^2)^2} \\[/tex]

[tex]f(x,y)= f(1,1)+(x-1)\dfrac{\partial f}{\partial x} (1,1)+(y-1)\dfrac{\partial f}{\partial y} (1,1)+\dfrac{(x-1)^2}{2} \dfrac{\partial^2 f}{\partial x^2} (1,1)+(x-1)(y-1)\dfrac{\partial^2 f}{\partial x\ \partial y} (1,1)+\dfrac{(y-1)^2}{2} \dfrac{\partial^2 f}{\partial y^2} (1,1)+...\\\\=\dfrac{\pi}{4}+\dfrac{(x-1)}{2} +\dfrac{(y-1)}{2} -\dfrac{(x-1)^2}{4} +(x-1)(y-1)*\dfrac{0}{4}-\dfrac{(y-1)^2}{4} \\\\[/tex]

[tex]\boxed{f(x,y)=\dfrac{\pi}{4}+\dfrac{(x-1)}{2} +\dfrac{(y-1)}{2} -\dfrac{(x-1)^2}{4} -\dfrac{(y-1)^2}{4} }\\\\[/tex]

Answer:

They can all be simplified to 2/3

Answer:

12:14

Step-by-step explanation:

Answer:

False

Brainliest, please!

Step-by-step explanation:

+ + = +

- - = +

+ - = -

- + = -

Answer:

False

Step-by-step explanation:

The question doesn't specify whether you are adding positive or negative integers. If you have add 2 positive integers, your answer will be positive. If you add 2 negative integers, the answer will be negative. Here is an example;

1 + 1 = 2

-1 + (-1) = -1 - 1 = -2

Best of Luck!

Answer:

1,2,73,and 136

Step-by-step explanation:

hope this helps

mark me as brainliest plz

Answer:

[tex]y=\frac{1}{3} x+6[/tex]

Step-by-step explanation:

[tex](3,7)(-6,4)[/tex]

Step 1. Find the slope (by using the slope-formula)

m = slope

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]m=\frac{4-7}{-6-3}[/tex]

[tex]m=\frac{-3}{-9}[/tex]

[tex]m=\frac{3}{9}[/tex]

[tex]m=\frac{1}{3}[/tex]

Step 2. Write the equation (using the slope and the points)

Here's how to do it:

Slope-intercept Formula [tex]y=mx+b[/tex] whrere m = slope and b = y-intercept

Plug in the slope into the Slope-intercept Formula

[tex]y=\frac{1}{3} x+b[/tex]

Find the y-intercept (b) by using a point and substituting their x and y values

[tex]y=\frac{1}{3} x+b[/tex]

Point: (3, 7)

[tex]7=\frac{1}{3} (3)+b[/tex]

[tex]7=1+b[/tex]

[tex]b=7-1[/tex]

[tex]b=6[/tex]

Step 3. Write the equation in Slope-intercept form

[tex]y=mx+b[/tex]

[tex]y=\frac{1}{3} x+6[/tex]

Answer:

(3,15)

Step-by-step explanation:

Rule 1

5x = y

Rule 2

x+12 = y

Set them equal

5x = x+12

Subtract x from each side

5x-x = x+12-x

4x= 12

Divide by 4

4x/4 = 12/4

x=3

5x=y

5*3 = y

15 = y

The solution is (3,15)

Given rules

Equate and solve

Opt ion B is correct as no other contains 3 as x

When we have a given function f(x), the rate of change in a given value x₀ is given by:

[tex]\frac{df(x_0)}{dx} = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}[/tex]

We will find that the rate of change at t = 4s is:

r = -1/3

Now we can't do this if we do not have the function and we only have a text description of the graph, but what we can do is find the average rate of change.

For a function f(x), the average rate of change in an interval (a, b) such that a < b is given by:

[tex]r = \frac{f(b) - f(a)}{b - a}[/tex]

Something really nice is that the average rate of change is equal to the exact rate of change if f(x) is a linear equation.

If we want to find the rate of change at t = 4, then we need to find the smallest interval that contains t = 4.

Here we know that the graph passes through the points:

(3,1) to (6, 0)

Because of the statement "a line falling from (3, 1) to (6, 0)" we know that in this segment we have a line, which implies that the average rate of change will be equal to the exact rate of change.

Using the equation of the average rate of change we get:

[tex]r = \frac{d(6) - d(3)}{6 - 3} = \frac{0 - 1}{3} = - 1/3[/tex]

Then the rate of change in the distance at the time t = 4s is -1/3.

If you want to learn more, you can read:

https://brainly.com/question/18904995

Answer:

False

Step-by-step explanation:

63 x 42= 2646

216 x 16= 3456

have a great day

Answer:

So for the sequence to work, the 5,3=18 would have to be changed to 5,3=23 in which it now fits and works to solve for x which is 4. There was a pattern of the numbers 4 and 5 that surfaced as the difference between each pairings sum total. The difference between each sum was alternating between the numbers 4 and 5, which equal 9 when added together. Which also helped me to form a chart of sorts that makes sense. My answer is typed above, that is the most stable conclusion that I could come to to make sense of the pairings that were listed.

Answer: A (if i have well corrected)

Step-by-step explanation:

[tex]f(x)=(x-6)(x^2+8x+16)\ i\ suppose)\\\\f(x)=(x-6)(x+4)^2\\Sol=\{6;-4\}\\Answer \ A[/tex]

Answer:382.5

Step-by-step explanation: 45 x 8.5 = 382.5

Answer:

1/8

Step-by-step explanation:

There is a 1/2 probability for each of the three flips to result in the desired outcome. To get 3 desired outcomes in a row, we multiply each of the probabilities together

(1/2)³ = 1/8

Answer: it is D) 2/3

Answer:

A. 49,920

Step-by-step explanation:

Multiply the beginning value by 1 plus the percentage in decimal form.

Answer:

1:4

Step-by-step explanation:

multiply both sides by 10 to get whole numbers which will be 6:24. then divide by the GCF which is 6 and you get 1:4

Answer:

3²×5

Step-by-step explanation:

Product of its prime numbers its more like having the number in its simplest form by factorising it

Answer:

f(-3)= -26

Step-by-step explanation:

Answer:

Try 0.17? I hope it's right

Answer:

2 10/15 or 2 2/3 (simplified)

Step-by-step explanation:

To Multiply we need to get the same denominator for each fraction:

2 x 5 =10

3 x 5 =15

10/15

4/15 x 10/15 (or) 4 x 10 = 40/15

40/15 (also) = 2 10/15

2 2/3 (simplified version of 2 10/15)

Hope This Helps! :)

Answer:

[tex]0.05796 = 5.796 \times {10}^{ - 2} [/tex]

I hope I helped you^_^

Answer:

Step-by-step explanation:

Scientific notation is always written as a number between 1 and 9.999999.  The number after the decimal is shown by a power of 10.

so 0.05796 = 5.796 * 10^-2

which means that the decimal was moved to represent a number less than one. That's the point of the minus sign. The two comes from the number of places you have to shift the decimal to get a number between 1 and  10.

Answer: Choice A) translated 3 units down

Explanation:

x is the input while y = f(x) is the output. If we say f(x) - 3, then it's the same as saying y - 3. This means g(x) = f(x) - 3 subtracts 3 from each y coordinate of the points on f(x). This visually moves the entire f(x) curve down 3 units

Answer:

min | miles

———|———

1 | 3

2 | 6

3 | 9

4 | 12

5 | 15

6 | 18

Step-by-step explanation:

try putting this into a T-chart like I kind of did here or some other kind of other table.

Answer:

c

Step-by-step explanation:

The expression that gives my weight one year later is (180 pounds - x pounds). Hence the subtraction action that gives weight one year later.

Given that;

On my last birthday, I weighed 180 pounds.

And, One year later my weight decreased by x pounds.

Now the equation to represent the weighed after one year,

180 pounds - x pounds

Hence we can use the subtraction action that gives weight one year later.

Therefore, the correct option is C.

To learn more about subtraction visit:

https://brainly.com/question/17301989

#SPJ3

Answer:

x= _5 is the correct answer

Answer:

Area is 307.72 ft^2

Circumference is 43.96 ft

Step-by-step explanation:

Area is 2pir^2

Circumference is 2rpi

pi is about 3.14

A=2pi(7)^2

A= 98pi which is about 307.72 ft^2

C= 2(7)pi

C= 14pi which is about 43.96 ft

(I'm not sure whether they want the answer left in terms of pi or not)

Answer:

use the formula sn= n(a1+an)/2

Step-by-step explanation:

2982=28(228+an)/2

5964=28(228+an)

5964/28=228+an

213=228+an

an=-15(last term)

to find difference use formula

an = a+(n-1)d

-15=228+(28-1)d

-243=27d

d=-243/27

d=-9

arithmetic sequence  can be found be keep on subtracting 9 from 228

hence the arithmetic sequence is

228, 219, 210, 201, 192, 183, 174........-15

Answer:

[tex] \frac{dy}{dx} = - \frac{ {x}^{3} }{ {y}^{3} } [/tex]

Step-by-step explanation:

Differentiate both sides of the equation (consider y as a function of x).

[tex] \frac{d}{dx} ( {x}^{4} + {y}^{4} (x)) = \frac{d}{dx} (16)[/tex]

the derivative of a sum/difference is the sum/difference of derivatives.

[tex]( \frac{d}{dx} ( {x}^{4} + {y}^{4} (x))[/tex]

[tex] = ( \frac{d}{dx} ( {x}^{4} ) + \frac{d}{dx} ( {y}^{4} (x)))[/tex]

the function of y^4(x) is the composition of f(g(x)) of the two functions.

the chain rule:

[tex] \frac{d}{dx} (f(g(x))) = \frac{d}{du} (f(u)) \frac{d}{dx} (g(x))[/tex]

[tex] = ( \frac{d}{du} ( {u}^{4} ) \frac{d}{dx} (y(x))) + \frac{d}{dx} ( {x}^{4} )[/tex]

apply the power rule:

[tex](4 {u}^{3} ) \frac{d}{dx} (y(x)) + \frac{d}{dx} ( {x}^{4} )[/tex]

return to the old variable:

[tex]4(y(x) {)}^{3} \frac{d}{dx} (y(x)) + \frac{d}{dx} ( {x}^{4} )[/tex]

apply the power rule once again:

[tex]4 {y}^{3} (x) \frac{d}{dx} (y(x)) + (4 {x}^{3} )[/tex]

simplify:

[tex]4 {x}^{3} + 4 {y}^{3} (x) \frac{d}{dx} (y(x))[/tex]

[tex] = 4( {x}^{3} + {y}^{3} (x) \frac{d}{dx} (y(x)))[/tex]

[tex] = \frac{d}{dx} ( {x}^{4} + {y}^{4} ))[/tex]

[tex] = 4( {x}^{3} + {y}^{3} (x) \frac{d}{dx}(y(x))) [/tex]

differentiate the equation:

[tex]( \frac{d}{dx} (16)) = (0)[/tex]

[tex] = \frac{d}{dx} (16) = 0[/tex]

derivative:

[tex]4 {x}^{3} + 4 {y}^{3} \frac{dy}{dx} = 0[/tex]

[tex] \frac{dy}{dx} = - \frac{ {x}^{3} }{ {y}^{3} } [/tex]

Answer:

Only Option III, IV, VI are correct

Step-by-step explanation:

Domain is the set of values of x at which the function is defined. f(x) has defined values from x>-3 to x<=4, so the domain is (-3,4). Incorrect.

Range of the function is the set of all defined values of it. Here all values with y=f(x)>-3 and y<=3 are defined. So the range is (-3, 3}.

Decreasing and increasing parts of the funtion are shown in the graph when the y value goes up (positive slope) and vice versa.

The negative part of the function should be (3,-2) and (2,4] since (4, 5] is not defined and there is no indication for its negative-ness.

Answer: -6

Step-by-step explanation:

-18 - (-12)

= -6

none of them

it's actually-6.