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
1. Where will the line y = -1/2 -3 cross the y-axis?
soldier1979 [14.2K]

1) The y- axis is where x = 0, so the line "y = -1/2x - 3" will cross the y-axis when x = 0. You plug in "x = 0" into the equation, and get y = 0 -3, so y = -3. That means the line " y = -1/2x - 3" will cross the y-axis ar (0, -3)

2) If there are 2 parallel lines, then there is no solution, because the definition of parallel lines is, " 2 lines in the same plane, that never meet." That makes the solution inconsistent.

7 0
2 years ago
Which graph represents the linear function y = x 3/4– 4?
ahrayia [7]

Answer:

The graph in the attached figure

Step-by-step explanation:

we have

y=\frac{3}{4}x-4

This is a linear equation (the graph is a line)

To identify the graph find out the intercepts

<u><em>Find out the y-intercept</em></u>

The y-intercept is the value of y when the value of x is equal to zero

For x=0

y=\frac{3}{4}(0)-4=-4

The y-intercept is the point (0,-4)

<u><em>Find out the x-intercept</em></u>

The x-intercept is the value of x when the value of y is equal to zero

For y=0

0=\frac{3}{4}x-4

\frac{3}{4}x=4

x=16/3=5.33

The x-intercept is the point (5.33,0)

therefore

The graph in the attached figure

3 0
2 years ago
( 3x + 5 ) ki power 2 expend it​
Pepsi [2]

This question is soooo easy

Step-by-step explanation:

(3x)^2+(5)^2+2.3x.5

9x^2+25+30x

4 0
3 years ago
Amelia stores her gardening supplies in two cube shaped boxes. The smaller box has a volume of 100 inches to the third power. Am
Solnce55 [7]

Answer:

1000000 for the smaller box.

I need more info for the second box

Step-by-step explanation:

6 0
2 years ago
Read 2 more answers
750x + 17y = 141.61
MAVERICK [17]
I’ll answer this in a second
7 0
3 years ago
Read 2 more answers
Other questions:
  • PLEASE someone help me ASAP Please do both questions!!!
    9·1 answer
  • 35 per cent of a number is what fraction of that number
    13·1 answer
  • A polynomial function with a leading coefficient
    5·1 answer
  • A customer at the mall is shopping for holiday gifts, and is considering having them wrapped. He can get them wrapped at the dep
    8·1 answer
  • PLEASE HELP ME PLEASE I REALLY NEED HELP
    15·1 answer
  • VHelena has five different flowers. She plans to give one flower to each of her five teachers in any order. She gives the first
    6·1 answer
  • Linda makes booklets using 2 sheets of paper. She has 17 sheets of paper. How many of these booklets. can she make?
    13·1 answer
  • 586.875 to 2 decimal places​
    15·1 answer
  • Is 8/24 = 21/63 a proportion
    14·2 answers
  • The tallest water slide is 49 meters tall. Phineas and Ferb want to build a slide that is 5 meters taller. They build the slide
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!