Hello There!
The total number of different arrangements is 10,628.
Hope This Helps You!
Good Luck :)
- Hannah ❤
Answer:
See explanation.
General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
<u>Algebra I</u>
Functions
- Exponential Property [Rewrite]:

- Exponential Property [Root Rewrite]:
![\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csqrt%5Bn%5D%7Bx%7D%20%3D%20x%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D)
<u>Calculus</u>
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: ![\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Bcf%28x%29%5D%20%3D%20c%20%5Ccdot%20f%27%28x%29)
Derivative Property [Addition/Subtraction]: ![\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%20%2B%20g%28x%29%5D%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%5D%20%2B%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bg%28x%29%5D)
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Chain Rule]: ![\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28g%28x%29%29%5D%20%3Df%27%28g%28x%29%29%20%5Ccdot%20g%27%28x%29)
Step-by-step explanation:
We are given the following and are trying to find the second derivative at <em>x</em> = 2:


We can differentiate the 1st derivative to obtain the 2nd derivative. Let's start by rewriting the 1st derivative:

When we differentiate this, we must follow the Chain Rule: ![\displaystyle \frac{d^2y}{dx^2} = \frac{d}{dx} \Big[ 6(x^2 + 3y^2)^\big{\frac{1}{2}} \Big] \cdot \frac{d}{dx} \Big[ (x^2 + 3y^2) \Big]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5CBig%5B%206%28x%5E2%20%2B%203y%5E2%29%5E%5Cbig%7B%5Cfrac%7B1%7D%7B2%7D%7D%20%5CBig%5D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5CBig%5B%20%28x%5E2%20%2B%203y%5E2%29%20%5CBig%5D)
Use the Basic Power Rule:

We know that y' is the notation for the 1st derivative. Substitute in the 1st derivative equation:
![\displaystyle \frac{d^2y}{dx^2} = 3(x^2 + 3y^2)^\big{\frac{-1}{2}} \big[ 2x + 6y(6\sqrt{x^2 + 3y^2}) \big]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%20%3D%203%28x%5E2%20%2B%203y%5E2%29%5E%5Cbig%7B%5Cfrac%7B-1%7D%7B2%7D%7D%20%5Cbig%5B%202x%20%2B%206y%286%5Csqrt%7Bx%5E2%20%2B%203y%5E2%7D%29%20%5Cbig%5D)
Simplifying it, we have:
![\displaystyle \frac{d^2y}{dx^2} = 3(x^2 + 3y^2)^\big{\frac{-1}{2}} \big[ 2x + 36y\sqrt{x^2 + 3y^2} \big]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%20%3D%203%28x%5E2%20%2B%203y%5E2%29%5E%5Cbig%7B%5Cfrac%7B-1%7D%7B2%7D%7D%20%5Cbig%5B%202x%20%2B%2036y%5Csqrt%7Bx%5E2%20%2B%203y%5E2%7D%20%5Cbig%5D)
We can rewrite the 2nd derivative using exponential rules:
![\displaystyle \frac{d^2y}{dx^2} = \frac{3\big[ 2x + 36y\sqrt{x^2 + 3y^2} \big]}{\sqrt{x^2 + 3y^2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%20%3D%20%5Cfrac%7B3%5Cbig%5B%202x%20%2B%2036y%5Csqrt%7Bx%5E2%20%2B%203y%5E2%7D%20%5Cbig%5D%7D%7B%5Csqrt%7Bx%5E2%20%2B%203y%5E2%7D%7D)
To evaluate the 2nd derivative at <em>x</em> = 2, simply substitute in <em>x</em> = 2 and the value f(2) = 2 into it:
![\displaystyle \frac{d^2y}{dx^2} \bigg| \limits_{x = 2} = \frac{3\big[ 2(2) + 36(2)\sqrt{2^2 + 3(2)^2} \big]}{\sqrt{2^2 + 3(2)^2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%20%5Cbigg%7C%20%5Climits_%7Bx%20%3D%202%7D%20%3D%20%5Cfrac%7B3%5Cbig%5B%202%282%29%20%2B%2036%282%29%5Csqrt%7B2%5E2%20%2B%203%282%29%5E2%7D%20%5Cbig%5D%7D%7B%5Csqrt%7B2%5E2%20%2B%203%282%29%5E2%7D%7D)
When we evaluate this using order of operations, we should obtain our answer:

Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
Answer:
Divide the volume by the length of the prism to get the area of the face:
468/13 = 36 square yards.
The area of the face ( triangle) is 1/2 x base x height
Replace with what you have:
36 = 1/2 x 9 x g
Multiply both sides by 2 to remove the 1/2:
72 = 9 x g
Divide both sides by 9 to find g:
g = 72 /9
g = 8 yards.
The thickness of the board was decreased from 3 / 4 inches to 9 / 16 inches.
Therefore,

So the board's thickness in inches decreased by
Hello,
Answer A
x²+4x+3=(x+3)(x+1)
x^3+6x²+11x+6=(x+1)(x²+5x+6)
x²+5x+6=(x+2)(x+3)
so
x^3+6x²+11x+6=(x+1)(x+2)(x+3) (Horner's method)