Factor the expression. 48g2 – 22gh

Given f(x) = x² - 10x + 22, what is the range of f?

adoni [48]

Answer:

[-3, ∞)

Step-by-step explanation:

There are many ways to find the range but I will use the method I find the easiest.

First, find the derivative of the function.

f(x) = x² - 10x + 22

f'(x) = 2x - 10

Once you find the derivative, set the derivative equal to 0.

2x - 10 = 0

Solve for x.

2x = 10

x = 5

Great, you have the x value but we need the y value. To find it, plug the x value of 5 back into the original equation.

f(x) = x² - 10x + 22

f(5) = 5² - 10(5) + 22

= 25 - 50 +22

= -3

Since the function is that of a parabola, the value of x is the vertex and the y values continue going up to ∞.

This means the range is : [-3, ∞)

Another easy way is just graphing the function and then looking at the range. (I attached a graph of the function below).

Hope this helped!

6 0

3 years ago

Use the given transformation x=4u, y=3v to evaluate the integral. ∬r4x2 da, where r is the region bounded by the ellipse x216 y2

exis [7]

The Jacobian for this transformation is

$J = \begin{bmatrix} x_u & x_v \\ y_u & y_v \end{bmatrix} = \begin{bmatrix} 4 & 0 \\ 0 & 3 \end{bmatrix}$

with determinant $|J| = 12$ , hence the area element becomes

$dA = dx\,dy = 12 \, du\,dv$

Then the integral becomes

$\displaystyle \iint_{R'} 4x^2 \, dA = 768 \iint_R u^2 \, du \, dv$

where $R'$ is the unit circle,

$\dfrac{x^2}{16} + \dfrac{y^2}9 = \dfrac{(4u^2)}{16} + \dfrac{(3v)^2}9 = u^2 + v^2 = 1$

so that

$\displaystyle 768 \iint_R u^2 \, du \, dv = 768 \int_{-1}^1 \int_{-\sqrt{1-v^2}}^{\sqrt{1-v^2}} u^2 \, du \, dv$

Now you could evaluate the integral as-is, but it's really much easier to do if we convert to polar coordinates.

$\begin{cases} u = r\cos(\theta) \\ v = r\sin(\theta) \\ u^2+v^2 = r^2\\ du\,dv = r\,dr\,d\theta\end{cases}$

Then

$\displaystyle 768 \int_{-1}^1 \int_{-\sqrt{1-v^2}}^{\sqrt{1-v^2}} u^2\,du\,dv = 768 \int_0^{2\pi} \int_0^1 (r\cos(\theta))^2 r\,dr\,d\theta \\\\ ~~~~~~~~~~~~ = 768 \left(\int_0^{2\pi} \cos^2(\theta)\,d\theta\right) \left(\int_0^1 r^3\,dr\right) = \boxed{192\pi}$

3 0

2 years ago

Ursula Vanov can invest $5,000 in a 1-year CD at 3% compounded monthly or a 1-year CD compounded daily. Determine the amount at

Aliun [14]

Solving for the amount of maturity given that it is compounded monthly for 1 year with an interest of 3%, we have the formula and solution below:
A = P (1+r/n)^rn
A = $5,000 (1.040417)
A =$5202.085

For compounded daily, we have the solution below:
A = $5,000 (1.040443)
A = $5202.215

The difference in amount is shown below:
Difference = $5202.215 - $5202.085
Difference = $0.13

7 0

3 years ago

Which of these statements is true?

zlopas [31]

Which Statements?

You didn't put any Statements there

8 0

4 years ago

Which point is on a line that passes through point R and is perpendicular to line PQ?

aliya0001 [1]

Your answer would be (-4,-8)

5 0

3 years ago

Read 2 more answers

Factor the expression. 48g2 – 22gh – 15h2