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
erik [133]
2 years ago
11

56. How many tangent lines to the curve

x+1)" alt="y=x /(x+1)" align="absmiddle" class="latex-formula"> pass through the point (1,2)? At which points do these tangent lines touch the curve?
Mathematics
1 answer:
PIT_PIT [208]2 years ago
8 0

There are 2 tangent lines that pass through the point

y=\frac{1}{(-1+\sqrt{3)^2} } (x-1)+2

and

y=\frac{1}{(-1-\sqrt{3)^2} } (x-1)+2

Explanation:

Given:

y=\frac{x}{x+1}

The point-slope form of the equation of a line tells us that the form of the tangent lines must be:

y=m(x-1)+2 [1]

For the lines to be tangent to the curve, we must substitute the first derivative of the curve for m:

\frac{dy}{dx} =\frac{d(x)}{dx}(x+1)-x^\frac{d(x+1)}{dx} \\ \\

\frac{dy}{dx} =\frac{x+1-x}{(x+1)^2}

\frac{dy}{dx}= \frac{1}{(x+1)^2}

m=\frac{1}{(x+1)^2} [2]

Substitute equation [2] into equation [1]:

y=\frac{x-1}{(x+1)^2}+2 [1.1]

Because the line must touch the curve, we may substitute y=\frac{x}{x+1}:

\frac{x}{x+1}=\frac{x-1}{(x+1)^2}+2

Solve for x:

x(x+1)=(x-1)+2(x+1)^2

x^2+x=x-1+2x^2+4x+2

x^2+4x+1

x\frac{-4±\sqrt{4^2-4(1)(1)} }{2(1)}

x=-2 ± \sqrt{3}

x=-2 ± \sqrt{3}<em> </em>and x=-2-\sqrt{3}

There are 2 tangent lines.

y=\frac{1}{(-1+\sqrt{3)^2} } (x-1)+2

and

y=\frac{1}{(-1-\sqrt{3)^2} } (x-1)+2

You might be interested in
I really need help with this its past due
zloy xaker [14]
Sorry but grade math is this so I can have a better understanding
7 0
2 years ago
0.32 as a mixed number
torisob [31]

Answer:

Hi! The answer to your question is 0 8/25

Step-by-step explanation:

☆*: .。..。.:*☆☆*: .。..。.:*☆☆*: .。..。.:*☆☆*: .。..。.:*☆

☁Brainliest is greatly appreciated!☁

Hope this helps!!

- Brooklynn Deka

6 0
2 years ago
How many solutions does this equation have?<br><br> –5(6y + 7) = –17y + 7 − 10y
aalyn [17]

Answer:

y=-14

Step-by-step explanation:

3 0
3 years ago
Which of the following expressions is equivalent to -5.4 - (2.1 + 1.8)?
Leno4ka [110]
-5.4 - (2.1 + 1.8)=-9.3

So I would say A. Thanks for asking your question today and have a good day on Brainly.com! :)
4 0
3 years ago
−125%+20/28=<br> help plzz
wlad13 [49]
-125%+20/28
-125/100+20/28
ANSWER: -15/28
3 0
3 years ago
Other questions:
  • Hank scored 10% extra on his 30 point assignment how many extra points did he earn?
    9·1 answer
  • Determine the awnser to (-5) +4 and explain the steps using a number line​
    12·2 answers
  • Evaluate the expression when x = 5: 10 I7-xI<br> (A)-120<br> (B)20 <br> (C)-20<br> (D)123
    10·2 answers
  • Andrea is given AABC and told that a² + b2 = c2. She draws right triangle RTS with legs
    9·1 answer
  • Alaska has a coastline of about 6.64 thousand miles. Florida has about 1.35 thousand miles of coast. How many times more does Al
    13·2 answers
  • Help please. step by step explanation appreciated
    9·2 answers
  • Please help me out please
    15·2 answers
  • Plz help me with this! Thx in advance! <br> (I need to show work)
    6·1 answer
  • What is the value of k?<br> k = ?
    14·2 answers
  • Please help me. justify your answers!​
    8·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!