1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Arturiano [62]
2 years ago
6

Can someone please explain how to get the answer I've watched multiple videos on implicit derivatives and I still cant figure it

out.​

Mathematics
1 answer:
Arisa [49]2 years ago
7 0

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.

You might be interested in
A 70-foot wire attached to the top of a building is anchored to the ground 20 feet from the base of the building. Which value is
JulijaS [17]

Answer:

The the angle which the wire make with the ground is 1.280 radian .

Step-by-step explanation:

Given as :

The length of the wire attached to the top of building = OB = 70 foot

The distance of wire anchored from base of ground = OA = 20 feet

Let the angle made by wire en and ground = Ф

Now from ,  In Triangle AOB

Cos angle = \dfrac{\textrm Base}{\textrm Hypotenuse}

Or, Cos Ф =  \dfrac{\textrm OA}{\textrm OB}

or, Cos Ф =  \dfrac{\textrm 20}{\textrm 70}

Or, Cos Ф =  \dfrac{\textrm 2}{\textrm 7}

∴  Ф = Cos^{-1}(\frac{2}{7})

I.e Ф = 73.39°

Now in radian ,

∵ 180° = \pi radian

∴ 73.39° = \frac{\pi }{180} × 73.39°

             =  \frac{3.14 }{180} × 73.39° = 1.280 radian

Hence The the angle which the wire make with the ground is 1.280 radian . Answer

4 0
2 years ago
Read 2 more answers
Bob is a high school basketball player. he is a 70% free-throw shooter (his probability of making a free throw is 0.70). what is
olchik [2.2K]
Probability he doesn't make a free shot = 10/10 - 7/10 = 3/10
Doesn't first time, does second time, so...
3/10 x 7/10 = 21/100
6 0
3 years ago
In a sequence the first term is 7 and the common difference is 2 what is the fifth term
Nady [450]

7, 14, 28, 56, 112.

The 5th term is 112

6 0
3 years ago
Read 2 more answers
I could really use some help with this!!!
Juliette [100K]
With what do you need help with??
8 0
2 years ago
Complete the solution of the equation. find the value of y when x equals -8 <br> 8x+8y=-48
Korvikt [17]
<span>8x + 8y = -48
-------------------------
Plug in -8 for x
(8) (-8) + 8y = -48
-------------------------------
Simplify
</span><span><span><span>(8) </span><span>(<span>−8</span>) </span></span>+ <span>8y </span></span>= <span>−<span>48
</span></span><span><span>−64 </span>+ <span>8y </span></span>= <span>−<span>48
</span></span><span><span>8y </span>− 64 </span>= <span>−<span>48
</span></span>-----------------------------------------
Add 64 to each side
<span><span><span>8y </span>− 64 </span>+ 64 </span>= <span><span>−48 </span>+ 64
</span><span>8y </span>= <span>16
</span>---------------------------------------------------
Finally, divide each side by 8
8y ÷ 8 = 16 ÷ 8
y = <span>2
</span>----------------------------------------------------------------
y = 2 is your answer
4 0
3 years ago
Other questions:
  • What is the solution set to the system of equations
    12·1 answer
  • In the expression 5 n minus StartFraction 2 m over 7 EndFraction + three-fourths, what is the constant?
    14·2 answers
  • Evaluate:5/8 - (1/4)2
    7·2 answers
  • A baker has 24 sesame bagels and 36 plain bagels to put into boxes each box must have the same number of each type of bagel what
    9·1 answer
  • Solve 3x^2+19x +6 please help
    15·1 answer
  • There are 15 students on the chess team. Seven of the students are in the fourth grade.
    8·1 answer
  • Plz help <br>plz<br>plz<br>plz<br>plz​
    6·2 answers
  • SOMEONE HELP MEEEE PLEASE
    13·1 answer
  • Work out the length of x
    8·2 answers
  • What is the sum of 12 – 5i and –3 4i? –16 63i 9 – i 9 – 9i 15 – 9i
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!