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3 years ago

Using synthetic division, find (x^4 − 2) ÷ (x + 1).

1 answer:

6 0

-1 | 1 ... 0 ... 0 ... 0 ... -2
... | ...... -1 .. 1 ... -1 ... 1
-------------------------------
... | 1 ... -1 .. 1 ... -1 | -1

In other words,

$\dfrac{x^4-2}{x+1}=x^3-x^2+x-1-\dfrac1{x+1}$

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When do you know to use either the shell method or washer method and how do you know what to take it to respect to when it is ro

Marat540 [252]

Step-by-step explanation:

Let's first consider a rotation about a vertical axis (x=0 or x=a).

Using washer method, the volume is:

∫ᵧ₁ᵞ² π (x₂² − x₁²) dy

∫ᵧ₁ᵞ² π ((x₂ − a)² − (x₁ − a)²) dy

Using shell method, the volume is:

∫ₓ₁ˣ² 2π x (y₂ − y₁) dx

∫ₓ₁ˣ² 2π (x − a) (y₂ − y₁) dx

First, notice that with washer method, the differential dy is parallel to the axis of rotation. With shell method, the differential dx is perpendicular to the axis of rotation.

Also, notice that the limits of integration must match the differential.

Choosing the best method will depend on the function.

If x² can be written in terms of y, and integrated, then washer method is the one to use.

If y can be written in terms of x, and integrated, then shell method is the one to use.

In other words, use washer method if x = f(y) is simple to integrate. Use shell method if y = f(x) is simple to integrate.

If instead we rotate about a horizontal axis (y=0 or y=a), the volume with washer method is:

∫ₓ₁ˣ² π (y₂² − y₁²) dx

∫ₓ₁ˣ² π ((y₂ − a)² − (y₁ − a)²) dx

Using shell method, the volume is:

∫ᵧ₁ᵞ² 2π y (x₂ − x₁) dy

∫ᵧ₁ᵞ² 2π (y − a) (x₂ − x₁) dy

We choose washer method when y² = f(x) is easy to integrate.

We choose shell method when x = f(y) is easy to integrate.

So here's the takeaway:

If the axis of rotation is x = 0 or x = a, and x² = f(y) is easy to integrate, use washer method.

If the axis of rotation is y = 0 or y = a, and y² = f(x) is easy to integrate, use washer method.

Otherwise, use shell method.

7 0

3 years ago

Finding the average rate of change of g(z) =4x^2_5 between the points( -2,11) and (2,11

krok68 [10]

Answer:

The average rate of change of g(z) =4x^2_5 between the points( -2,11) and (2,11) is 0.

Step-by-step explanation:

Given a function y, the average rate of change S of $y=g(z)$ in an interval $(z_{s}, z_{f})$ will be given by the following equation:

$S = \frac{g(z_{f}) - g(z_{s})}{z_{f} - z_{s}}$

In this problem, we have that:

$g(z) = 4z^{2} - 5$

Between the points( -2,11) and (2,11).

So $z_{f} = 2, z_{s} = -2$

$g(z_{f}) = g(2) = 4*(2)^{2} - 5 = 11$

$g(z_{s}) = g(-2) = 4*(-2)^{2} - 5 = 11$

$S = \frac{g(z_{f}) - g(z_{s})}{z_{f} - z_{s}} = \frac{11 - 11}{2 - (-2)} = \frac{0}{4} = 0$

The average rate of change of g(z) =4x^2_5 between the points( -2,11) and (2,11) is 0.

3 0

3 years ago

What is the sum of 4+4+2

9966 [12]

The sum of 4+4+2 is 10

3 0

4 years ago

Find the length of the base of the following pyramid, given the height of the pyramid is 211 meters and the angle of elevation o

jek_recluse [69]

Answer:

use formula length × width ×height

Step-by-step explanation:

plug in what they have given you

6 0

4 years ago

Plsss help I don't know how to graph I will give brainiest!

Anastasy [175]

Answer: (2, 5)

Step-by-step explanation:

y = 2x + 1 y = 4x − 3

Eliminate the equal sides of each equation and combine.

2x + 1 = 4x − 3

Solve 2x + 1 = 4x − 3 for x.

Move all terms containing x to the left side of the equation.

Subtract 4x from both sides of the equation.

2x + 1 − 4x = −3

Subtract 4x from 2x.

−2x + 1 = −3

Move all terms not containing x to the right side of the equation.

Subtract 1 from both sides of the equation.

−2x = −3 − 1

Subtract 1 from −3.

−2x = −4

Divide each term by −2 and simplify.

x = 2

Evaluate y when x = 2.

Substitute 2 for x. y = 4 (2) − 3

Simplify 4 (2) − 3.

Multiply 4 by 2.

y = 8 − 3

Subtract 3 from 8.

y = 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

(2, 5)

The result can be shown in multiple forms.

Point Form:

(2, 5)

Equation Form:

x = 2, y = 5

4 0

3 years ago