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2 years ago

Solve the system of equations. x + 3y= -8 4x + 4y = 0

Mathematics

1 answer:

Evgesh-ka [11]2 years ago

6 0

Answer:

x = -3

y = 4

Step-by-step explanation:

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At what point does the curve have maximum curvature? Y = 4ex (x, y) = what happens to the curvature as x → ∞? Κ(x) approaches as

MAXImum [283]

<u>Answer-</u>

At $x= \frac{1}{2304e^4-16e^2}$ the curve has maximum curvature.

<u>Solution-</u>

The formula for curvature =

$K(x)=\frac{{y}''}{(1+({y}')^2)^{\frac{3}{2}}}$

Here,

$y=4e^{x}$

Then,

${y}' = 4e^{x} \ and \ {y}''=4e^{x}$

Putting the values,

$K(x)=\frac{{4e^{x}}}{(1+(4e^{x})^2)^{\frac{3}{2}}} = \frac{{4e^{x}}}{(1+16e^{2x})^{\frac{3}{2}}}$

Now, in order to get the max curvature value, we have to calculate the first derivative of this function and then to get where its value is max, we have to equate it to 0.

${k}'(x) = \frac{(1+16e^{2x})^{\frac{3}{2} } (4e^{x})-(4e^{x})(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x})}{(1+16e^{2x} )^{2}}$

Now, equating this to 0

$(1+16e^{2x})^{\frac{3}{2} } (4e^{x})-(4e^{x})(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x}) =0$

$\Rightarrow (1+16e^{2x})^{\frac{3}{2}}-(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x})$

$\Rightarrow (1+16e^{2x})^{\frac{3}{2}}=(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x})$

$\Rightarrow (1+16e^{2x})^{\frac{1}{2}}=48e^{2x}$

$\Rightarrow (1+16e^{2x})}=48^2e^{2x}=2304e^{2x}$

$\Rightarrow 2304e^{2x}-16e^{2x}-1=0$

Solving this eq,

we get $x= \frac{1}{2304e^4-16e^2}$

∴ At $x= \frac{1}{2304e^4-16e^2}$ the curvature is maximum.

6 0

2 years ago

Lf f(x) = 5x, what is f^-1(x)?

Nonamiya [84]

Answer: $f^{-1}(x)=\frac{x}{5}$

Step-by-step explanation:

By definition the domain of an inverse function $f^-1(x)$ is the range of f(x) and the range of the inverse function is equal to the domain of the principal function f(x).

If you have a function $f(x)=5x$ , then to find the inverse function, follow these steps:

1. Make $y=f(x)$

$f(x)=y=5x$

$y=5x$

2. Solve for the variable "x":

$x=\frac{y}{5}$

3. Exchange the variable "x" with the variable "y":

$y=\frac{x}{5}$

4. Exchange "y" with $f^{-1}(x)$ . Then the inverse function is:

$f^{-1}(x)=\frac{x}{5}$

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3 years ago

What is one third plus one seventh

Kryger [21]

To do this, change into common denominators. The common denominator is 21 so change the problem from 1/3 + 1/7 to 7/21 + 3/21. This gives you 10/21. It is already in simplest form.

8 0

3 years ago

If you roll a 6-sided die what is the probability of rolling a one?

Ainat [17]

Answer:

the probablity of rolling a one is 1/6

Step-by-step explanation:

4 0

2 years ago

Read 2 more answers

After transforming f(x) = 2x² + 4x + 3 into vertex form, the vertex is easily identifiable. Which ordered pair is the vertex?

vagabundo [1.1K]

Answer:

The correct answer to your question is A because your asking for the vertex of the equation.

Step-by-step explanation:

Vertex Algebraic Equations

6 0

3 years ago

Solve the system of equations. x + 3y= -8 4x + 4y = 0​

Solve the system of equations. x + 3y= -8 4x + 4y = 0