Find K by evaluating Limit

We're given the implicit equation

x² + 2x + y² - 4y = 12

where "implicit" means that one of the variables is considered a function of the other. In this case, the variable y = y(x) depends on the value of the variable x - in other words, the possible values of y are implied by the value of x. We then say that this equation defines y as an implicit function of x.

We're also given the implicit derivative,

dy/dx = (-2x - 2)/(2y - 4)

which tells us the rate of change y as a function of both x and y. The value of dy/dx corresponds to the slope of the tangent line to the curve defined by the implicit equation above at some point (x, y).

The tangent line is horizontal when its slope is zero. This happens when

(-2x - 2)/(2y - 4) = 0

If y ≠ 2, then we can eliminate the denominator and we're left with

-2x - 2 = 0

Solve for x :

-2x = 2

x = -1

This tells us that the points on the curve with x-coordinate -1 have a tangent line to the curve that is horizontal.

The tangent line is vertical when its slope is undefined/infinite, or equivalently when the denominator of dy/dx is zero:

2y - 4 = 0

Solve for y :

2y = 4

y = 2

So any point on the curve with y-coordinate 2 will have a tangent line there that is vertical.

By completing the square in the implicit equation, we can easily identify where these points are located.

x² + 2x + y² - 4y = 12

x² + 2x + 1 + y² - 4y + 4 = 17

(x + 1)² + (y - 2)² = 17

This equation defines a circle centered at (-1, 2) with radius √17. When x = -1, we have

(y - 2)² = 17   ⇒   y = 2 ± √17

and when y = 2, we have

(x + 1)² = 17   ⇒   x = -1 ± √17

See the attached plot to see the circle and the tangents at these points.

Answer:

234

Step-by-step explanation:

Man I don't know

Answer:

N=20

Step-by-step explanation:

Angle 4 is on a straight line with angle 2 meaning that the angle 4 plus Angle 2 is 180 degrees:

    Angle 2 + Angle4 = 180

     72 + Angle 4 = 180

         Angle 4 = 108 degrees

Hope that helps!

This is equivalent to 13 kilojoules.

=======================================================

Work Shown:

m = 65 kg = mass

v = 20 m/s = velocity

KE = kinetic energy

KE = (1/2)*m*v^2

KE = (1/2)*65*(20)^2

KE = 13000 joules

This can be written as 13 kilojoules since "kilo" means "1 thousand"

The abbreviation of the unit "joules" is the letter J.

Before doing the question, let's recall the quadratic formula which is used to find the roots of a quadratic equation ;

For any quadratic equation of the form ax² + bx + c = 0 , it's roots are given by the quadratic formula as :

Where , D = b² - 4ac (Discriminant)

Now , simplifying the equation :

[tex]{:\implies \quad \sf -3\{(x)^{2}+2^{2}-2\times 2\times x\}+17=0\quad \qquad \{\because (a-b)^{2}=a^{2}+b^{2}-2ab\}}[/tex]

[tex]{:\implies \quad \sf -3(x^{2}+4-4x)+17=0}[/tex]

[tex]{:\implies \quad \sf -3x^{2}-12+12x+17=0}[/tex]

[tex]{:\implies \quad \sf -3x^{2}+12x+5=0}[/tex]

Dividing both sides by -1 will yield :

[tex]{:\implies \quad \sf 3x^{2}-12x-5=0}[/tex]

Now , it's in the form of the standard quadratic equation , where a = 3 , b = -12 , c = 5 , and D = (-12)² - 4 × 3 × -5 = 144+60 = 204

Now, By quadratic Formula ;

[tex]{:\implies \quad \sf x=\dfrac{-(-12)\pm \sqrt{204}}{2\times 3}}[/tex]

[tex]{:\implies \quad \sf x=\dfrac{12\pm 2\sqrt{51}}{2\times 3}}[/tex]

[tex]{:\implies \quad \sf x=\dfrac{\cancel{2}(6\pm \sqrt{51})}{\cancel{2}\times 3}}[/tex]

[tex]{:\implies \quad \bf \therefore \quad \underline{\underline{x=\dfrac{6\pm \sqrt{51}}{3}}}}[/tex]

Answer:

Answer:

15% off 212.5

30% off 175$

Step-by-step explanation:

15+15=30 so

250/100=2.5

2.5x30=75

250-75

Answer:

Step-by-step explanation:

area=π r²

=π×15²

=225 π cm²

Answer:

1quart 4cups =8quatcups

Step-by-step explanation:

thanks:)

Answer:

9

Step-by-step explanation:

2*4 + 1 = 9 cups

Answer:

-7

Step-by-step explanation:

-9x-3=60

add 3 to each side to cancel -3

-9x-3+3=60+3

simplify

-9x=63

divide by -9 to cancel the -9

-9x/-9=63/-9

simplify

x=-7

»f(x) = (x + 2)²

»f(5) = (5 + 2)²

»f(5) = 7²

»f(5) = 49✅

~Hello! I will try to help you!

•Given : -

•To Find : -

•Step-by-step explanation : -

[tex] \Rightarrow \sf f(x) = {x}^{2} + 4x + 4[/tex]

»Just substitute the value 5 into the function f(x) above, then:-

[tex] \Rightarrow \sf f(5) = {5}^{2} + 4(5) + 4 \\ \Rightarrow \sf f(5) = 25 + 20 + 4 \\ \Rightarrow \sf f(5) = \bf49[/tex]

With this it's done!

A translation followed by a reflection

Answer:The slope is 2 .

Step-by-step explanation: mark branlest if right!

Answer:

1/2

Step-by-step explanation:

In the attachment below

Answer:

D . (486 – 9 + 6) + (33 × 2)

Step-by-step explanation:

Which expression is equivalent to 486 – 9 + 6 + 33 × 2?

486 – 9 + 6 + 33 × 2

486 – 9 + 6 + 66

477 + 6 + 66

549

D . (486 – 9 + 6) + (33 × 2)

      477 + 6 + 66

               549

Hope this helps!

Answer:

$28.89

Step-by-step explanation:

[tex]398.5\times7.25[/tex]%

Calculate
[tex]398.5\times0.0725[/tex]

Calculate the product or quotient

[tex]28.89125[/tex]

I hope this helps you

:)

.......62% is the greatest......

Answer:

Option A is the correct answer.

Step-by-step explanation:

First, look at the second equation. Get s and r onto the same side.

For simplicity, I will use x for r and y for s

Multiply the first equation by 3, and multiply the second equation by -4.

3(2x+8y=−4)

−4(7x+6y=8)

Becomes:

6x+24y=−12

−28x−24y=−32

Add these equations to eliminate y:

−22x=−44

x=2=r

Now we substitute this back in for the original equation

(2)(2)+8y=−4

8y=-8

y=-1=s

Option A is the correct answer.

Answer:

Step-by-step explanation:

B = 180 - 50 - 35 = 115 degrees

By the law of sines,

[tex]c=\frac{13\sin 35^{\circ}}{\sin 115^{\crc}}[/tex]

And [tex]a=\frac{13\sin 50^{\circ}}{\sin 115^{\circ}}[/tex]

Answer:

B=95 a= 10 c= 7.5

Answer: A

Step-by-step explanation:

The triangles are similar by AA. Angles BPA and CPD are congruent because they are vertical angles, and angles ABP and CDP are congruent because all right angles are congruent.

Answer:

Step-by-step explanation:

The area of the larger rectangle is (4)(5)=20.

The area of the top left right triangle is (1/2)(2)(5)=5.

The area of the top right right triangle is (1/2)(2)(1)=1.

The area of the bottom right triangle is (1/2)(4)(4)=8.

So A = 20 - 5 - 1 - 8 = 15 - 9 = 6.

m = slope of the line

b = y-intercept of the line

b = y-intercept of line

m = slope of line

Hope this helps! Have a nice day! God bless and stay safe.

Answer:

B

Step-by-step explanation:

Simply choose an x coordinate and plug it into the equations from the choices given.

i.e. where x=1

y=2(1)^2 - 4(1)

y= 2-4

y= -2

it can then be seen that where x=1 on the graph, y= -2 aligns perfectly

Answer:

B

Step-by-step explanation:

In y = 2x^2-4x, we put y = 0 to represent the intersection points on the x-axis of the graph. (Since y = 0 is always on the x-axis)

For 2x^2-4x = 0, using the quadratic formula, we get x = 0 and x = 2.

This matches the int. points on the graph.

»f(x) = (9 + 2x)²

»f(2) = (9 + 2(2))²

»f(2) = (9 + 4)²

»f(2) = 13²

»f(2) = 169✅

[tex]\\ \rm\hookrightarrow f(x)=(9+2x)^2[/tex]

f(2):-

put 2 inplace of x

[tex]\\ \rm\hookrightarrow (9+2(2))^2[/tex]

[tex]\\ \rm\hookrightarrow (9+4)^2[/tex]

[tex]\\ \rm\hookrightarrow 13^2[/tex]

[tex]\\ \rm\hookrightarrow 169[/tex]

Step-by-step explanation:

14/11-15/11

14-15/11

-1/11

Answer:

3(4+5)

Step-by-step explanation:

just fill in the numbers for x and y

Answer:

3(x + y)

x = 4 y = 5

3(4 + 5)

12 + 15

= 27

Step-by-step explanation: