Answer:
32/225 ≈ 0.1422
Step-by-step explanation:
If you consider "3-digit" numbers to be between 100 and 999, inclusive, there are 128 such numbers divisible by 7. The probability of choosing one at random is ...
128/900 = 32/225 = 0.1422...(repeating)
__
If you consider all non-negative integers less than 1000 to be "3-digit numbers," then the probability is ...
142/1000 = 0.142 (exactly)
Answer:
![y''(-1) =8](https://tex.z-dn.net/?f=y%27%27%28-1%29%20%3D8)
General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Equality Properties
<u>Algebra I</u>
<u>Calculus</u>
Implicit Differentiation
The derivative of a constant is equal to 0
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Product Rule: ![\frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%20%5Bf%28x%29g%28x%29%5D%3Df%27%28x%29g%28x%29%20%2B%20g%27%28x%29f%28x%29)
Chain Rule: ![\frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28g%28x%29%29%5D%20%3Df%27%28g%28x%29%29%20%5Ccdot%20g%27%28x%29)
Quotient Rule: ![\frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%20%5B%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D%20%5D%3D%5Cfrac%7Bg%28x%29f%27%28x%29-g%27%28x%29f%28x%29%7D%7Bg%5E2%28x%29%7D)
Step-by-step explanation:
<u>Step 1: Define</u>
-xy - 2y = -4
Rate of change of the tangent line at point (-1, 4)
<u>Step 2: Differentiate Pt. 1</u>
<em>Find 1st Derivative</em>
- Implicit Differentiation [Product Rule/Basic Power Rule]:
![-y - xy' - 2y' = 0](https://tex.z-dn.net/?f=-y%20-%20xy%27%20-%202y%27%20%3D%200)
- [Algebra] Isolate <em>y'</em> terms:
![-xy' - 2y' = y](https://tex.z-dn.net/?f=-xy%27%20-%202y%27%20%3D%20y)
- [Algebra] Factor <em>y'</em>:
![y'(-x - 2) = y](https://tex.z-dn.net/?f=y%27%28-x%20-%202%29%20%3D%20y)
- [Algebra] Isolate <em>y'</em>:
![y' = \frac{y}{-x-2}](https://tex.z-dn.net/?f=y%27%20%3D%20%5Cfrac%7By%7D%7B-x-2%7D)
- [Algebra] Rewrite:
![y' = \frac{-y}{x+2}](https://tex.z-dn.net/?f=y%27%20%3D%20%5Cfrac%7B-y%7D%7Bx%2B2%7D)
<u>Step 3: Find </u><em><u>y</u></em>
- Define equation:
![-xy - 2y = -4](https://tex.z-dn.net/?f=-xy%20-%202y%20%3D%20-4)
- Factor <em>y</em>:
![y(-x - 2) = -4](https://tex.z-dn.net/?f=y%28-x%20-%202%29%20%3D%20-4)
- Isolate <em>y</em>:
![y = \frac{-4}{-x-2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-4%7D%7B-x-2%7D)
- Simplify:
![y = \frac{4}{x+2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B4%7D%7Bx%2B2%7D)
<u>Step 4: Rewrite 1st Derivative</u>
- [Algebra] Substitute in <em>y</em>:
![y' = \frac{-\frac{4}{x+2} }{x+2}](https://tex.z-dn.net/?f=y%27%20%3D%20%5Cfrac%7B-%5Cfrac%7B4%7D%7Bx%2B2%7D%20%7D%7Bx%2B2%7D)
- [Algebra] Simplify:
![y' = \frac{-4}{(x+2)^2}](https://tex.z-dn.net/?f=y%27%20%3D%20%5Cfrac%7B-4%7D%7B%28x%2B2%29%5E2%7D)
<u>Step 5: Differentiate Pt. 2</u>
<em>Find 2nd Derivative</em>
- Differentiate [Quotient Rule/Basic Power Rule]:
![y'' = \frac{0(x+2)^2 - 8 \cdot 2(x + 2) \cdot 1}{[(x + 2)^2]^2}](https://tex.z-dn.net/?f=y%27%27%20%3D%20%5Cfrac%7B0%28x%2B2%29%5E2%20-%208%20%5Ccdot%202%28x%20%2B%202%29%20%5Ccdot%201%7D%7B%5B%28x%20%2B%202%29%5E2%5D%5E2%7D)
- [Derivative] Simplify:
![y'' = \frac{8}{(x+2)^3}](https://tex.z-dn.net/?f=y%27%27%20%3D%20%5Cfrac%7B8%7D%7B%28x%2B2%29%5E3%7D)
<u>Step 6: Find Slope at Given Point</u>
- [Algebra] Substitute in <em>x</em>:
![y''(-1) = \frac{8}{(-1+2)^3}](https://tex.z-dn.net/?f=y%27%27%28-1%29%20%3D%20%5Cfrac%7B8%7D%7B%28-1%2B2%29%5E3%7D)
- [Algebra] Evaluate:
![y''(-1) =8](https://tex.z-dn.net/?f=y%27%27%28-1%29%20%3D8)
Well,
As we can see, the only difference is that the parentheses have moved.
This is an example of the associative property. It is specifically of multiplication, because products are used in this case.
Just as a test, let's see whether they are really equal.
Following PEMDAS, we get:
(2*4)7 = 2(7*4)
(8)7 = 2(28)
56 = 56
They are equivalent.
Answer:
Step-by-step explanation:
21. (24)(2)/6+4=48
=48/6+4
=8+4
=12
Answer:
<em>when </em><em>x </em><em>=</em><em> </em><em>5</em><em> </em>
<em>then </em><em> </em><em>the </em><em>value </em><em>of</em><em>. </em><em>(</em><em>3</em><em>x</em><em>+</em><em>2</em><em>)</em>
<em>(</em><em>3</em><em>*</em><em>5</em><em>)</em><em> </em><em>+</em><em> </em><em>2</em>
<em> </em><em> </em>
<em>1</em><em>5</em><em> </em><em>+</em><em> </em><em>2</em>
<em>=</em><em>. </em><em>1</em><em>7</em>
<em>her </em><em>mistake </em><em>was </em><em>that </em><em>she </em><em>doesn't</em><em> </em><em>multiplied</em><em> </em><em>5</em><em> </em><em>and </em><em>3</em><em> </em><em> </em>