MATHEMATICS HIGH SCHOOL

Find the equation of all tangent lines having slope of -1 that are tangent to the curve y=(9)/(x+1)

Answers

Answer 1

Answer:

Answer: y = -x + 5 and y = -x - 7 (see attached graph)

Step-by-step explanation:

y = $(9)/(x + 1)$

= 9(x + 1)⁻¹

Use the product rule to find the derivative

a = 9 a' = 0

b = (x + 1)⁻¹ b' = -(x + 1)⁻²

ab' + a'b

= 9[-(x + 1)⁻²] + 0[(x + 1)⁻¹ ]

= $(-9)/((x + 1)^(2))$

Set the derivative equal to the desired slope of -1 to solve for x

-1 = $(-9)/((x + 1)^(2))$

-(x + 1)² = -9

(x + 1)² = 9

√(x + 1)² = √9

x + 1 = +/- 3

x + 1 = 3 x + 1 = -3

x = 2 x = -4

Plug those values into the original equation to solve for y:

y = $(9)/(x + 1)$

= $(9)/(2 + 1)$

= 3

(2, 3)

y = $(9)/(x + 1)$

= $(9)/(-4 + 1)$

= -3

(-4, -3)

Next, plug in the given slope (-1) and the coordinates above into the Point-Slope formula y - y₁ = m(x - x₁) to find the equations:

m = -1, (x₁ y₁) = (2, 3) m = -1, (x₁ y₁) = (-4, -3)

y - 3 = -1(x - 2) y + 3 = -1(x + 4)

y - 3 = -x + 2 y + 3 = -x - 4

y = -x + 5 y = -x - 7

Answer 2

Answer:

Answer:

$f(x)=\frac9{x+1}\n f'(x)=-\frac9{(x+1)^2}\n f'(x)=-1\ \iff\ -\frac9{(x+1)^2}=-1\ \to \ \frac9{(x+1)^2}=1\ \to \ (x+1)^2=9\n |x+1|=3\ \to \ x+1=3\ \vee\ x+1=-3\n x_1=2\ \vee\ x_2=-4\n f(x_1)=f(2)=\frac9{2+1}=3\n f(x_2)=f(-4)=\frac9{-4+1}=-3$

First tangent line:

$y=f'(x_1)\cdot (x-x_1)+f(x_1)\ \to \ y=-1(x-2)+3\ \to \ y=-x+5$

Second tangent line:

$y=f'(x_2)\cdot (x-x_2)+f(x_2)\ \to \ y=-1(x+4)-3\ \to \ y=-x-7$

Notice: slope of -1 means that both $f'(x_1), \ f'(x_2)$ are equal to -1, so $f'(x_1)=-1 \ and \ f'(x_2)=-1$

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How do you solve x+2/3=11//12

Suzanne wants to put a fence around Her Square Garden. If the garden covers an area of 121m2, how many meters of fencing does she need? Please help thank you!

Answers

So, since it is square, we are going to take the square root of 121 which is 11. And since fencing goes around something, that means we have to find the perimeter. So since all four sides are the same length, we can just multiply 11 by 4 to get 44 meters of fencing

Answer:

Step-by-step explanation:

$\sqrt121 = 11$

and there is 4 sides to a square so you multiply 11 * 4

resulting 44

Order the following numbers from least to greatest.
Put the least value on the left.

Answers

Answer:

-2, -5/2, 1.7

Step-by-step explanation:

since in negative larger numbers are smaller, a negative fraction is larger than a negative whole number

Answer:

Step-by-step explanation:

- 5/2 | -2 | 1.7

i just got it correct doing that

Solve the equation 4x^2-17x-15=0

Answers

$4x^2-17x-15=0\n 4x^2-20x+3x-15=0\n 4x(x-5)+3(x-5)=0\n (4x+3)(x-5)=0\n x=-(3)/(4) \vee x=5$

4x^2 - 17x - 15 = 0
(4x + 3)(x-5)
4x + 3 = 0
4x = -3
x = -3/4

x - 5 = 0
x = 5

Graph the line bu locating any two ordered pairs that satisfy the equation

Y=-3x

Answers

I did it in my calculator and I found it online.

Please help, math question

Answers

B is correct because when you factor it out you are trying to get something that equals that. So 4xx3=12x+2x from x x 2 then the 4x and x mutiply to be 4x^2 then the two and three mutiply to be six. Hope o helped.

What is the x-intercept of the line with the equation 4x + 2y = 12?A. (0, 6)
B. (6, 0)
C. (0, 3)
D. (3, 0)

Answers

x intercept is where the line crosses the x axis or where y=0
set y=0

4x+2(0)=12
4x+0=12
4x=12
divide by 4
x=3

(x,y)
(3,0)

D is answer

Answer:

Option (d) is correct.

The x- intercept of the line with the given equation of 4x + 2y = 12 is (3,0)

Step-by-step explanation:

Given : Equation of line 4x + 2y = 12

We have to find the x- intercept of the line with the given equation of 4x + 2y = 12 and choose the correct option from the given equations.

Consider the given equation 4x + 2y = 12

x intercept is defined as a point where the line cuts y axis that is where point y is 0.

Put y = 0 , we have,

4x + 2(0) = 12

4x = 12

Divide both side by 4, we get,

x = 3

Thus, the x intercept of the line with the given equation of 4x + 2y = 12 is (3,0)

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