pumped up kicks
Step-by-step explanation:
pumped up kicks
Answer:
i think this is it
Step-by-step explanation:
the answer i got is just 55
Answer:
a) Micheal's present age = 8 years old
b) The sum of their ages in 6 years time = 63 years.
Step-by-step explanation:
From the above question, we know that
Rui feng is 3 years old
We are told that
• Michael is 5 years old than Rui feng
Micheal's present age is calculated as:
= 5 + Rui feng's age
= 5 + 3 = 8 years
• Vishal is thrice as old as Micheal
Vishal's present age is calculated as
= 3(Micheal age)
= 3(8)
= 24 years.
The sum of their ages in six years time
= ( 3 + 6) +( 8 + 6) + ( 24 + 6)
= 18 + 15 + 30
= 63 years
Therefore, a) Micheal's present age = 8 years old
b) The sum of their ages in 6 years time = 63 years.
Answer:
7000
Step-by-step explanation:
I think this is the answer
Answer:
<u />![\displaystyle \lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} = \boxed{ \frac{1}{4} }](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B%5Csqrt%7Bx%20%2B%201%7D%20-%202%7D%7Bx%20-%203%7D%20%3D%20%5Cboxed%7B%20%5Cfrac%7B1%7D%7B4%7D%20%7D)
General Formulas and Concepts:
<u>Calculus</u>
Limits
Limit Rule [Variable Direct Substitution]:
![\displaystyle \lim_{x \to c} x = c](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20c%7D%20x%20%3D%20c)
Special Limit Rule [L’Hopital’s Rule]:
![\displaystyle \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20c%7D%20%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D%20%3D%20%5Clim_%7Bx%20%5Cto%20c%7D%20%5Cfrac%7Bf%27%28x%29%7D%7Bg%27%28x%29%7D)
Differentiation
- Derivatives
- Derivative Notation
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)
Derivative Rule [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:
<u>Step 1: Define</u>
<em>Identify given limit</em>.
![\displaystyle \lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B%5Csqrt%7Bx%20%2B%201%7D%20-%202%7D%7Bx%20-%203%7D)
<u>Step 2: Find Limit</u>
Let's start out by <em>directly</em> evaluating the limit:
- [Limit] Apply Limit Rule [Variable Direct Substitution]:
![\displaystyle \lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} = \frac{\sqrt{3 + 1} - 2}{3 - 3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B%5Csqrt%7Bx%20%2B%201%7D%20-%202%7D%7Bx%20-%203%7D%20%3D%20%5Cfrac%7B%5Csqrt%7B3%20%2B%201%7D%20-%202%7D%7B3%20-%203%7D)
- Evaluate:
![\displaystyle \begin{aligned}\lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} & = \frac{\sqrt{3 + 1} - 2}{3 - 3} \\& = \frac{0}{0} \leftarrow \\\end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B%5Csqrt%7Bx%20%2B%201%7D%20-%202%7D%7Bx%20-%203%7D%20%26%20%3D%20%5Cfrac%7B%5Csqrt%7B3%20%2B%201%7D%20-%202%7D%7B3%20-%203%7D%20%5C%5C%26%20%3D%20%5Cfrac%7B0%7D%7B0%7D%20%5Cleftarrow%20%5C%5C%5Cend%7Baligned%7D)
When we do evaluate the limit directly, we end up with an indeterminant form. We can now use L' Hopital's Rule to simply the limit:
- [Limit] Apply Limit Rule [L' Hopital's Rule]:
![\displaystyle \begin{aligned}\lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} & = \lim_{x \to 3} \frac{(\sqrt{x + 1} - 2)'}{(x - 3)'} \\\end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B%5Csqrt%7Bx%20%2B%201%7D%20-%202%7D%7Bx%20-%203%7D%20%26%20%3D%20%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B%28%5Csqrt%7Bx%20%2B%201%7D%20-%202%29%27%7D%7B%28x%20-%203%29%27%7D%20%5C%5C%5Cend%7Baligned%7D)
- [Limit] Differentiate [Derivative Rules and Properties]:
![\displaystyle \begin{aligned}\lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} & = \lim_{x \to 3} \frac{(\sqrt{x + 1} - 2)'}{(x - 3)'} \\& = \lim_{x \to 3} \frac{1}{2\sqrt{x + 1}} \leftarrow \\\end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B%5Csqrt%7Bx%20%2B%201%7D%20-%202%7D%7Bx%20-%203%7D%20%26%20%3D%20%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B%28%5Csqrt%7Bx%20%2B%201%7D%20-%202%29%27%7D%7B%28x%20-%203%29%27%7D%20%5C%5C%26%20%3D%20%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%20%2B%201%7D%7D%20%5Cleftarrow%20%5C%5C%5Cend%7Baligned%7D)
- [Limit] Apply Limit Rule [Variable Direct Substitution]:
![\displaystyle \begin{aligned}\lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} & = \lim_{x \to 3} \frac{(\sqrt{x + 1} - 2)'}{(x - 3)'} \\& = \lim_{x \to 3} \frac{1}{2\sqrt{x + 1}} \\& = \frac{1}{2\sqrt{3 + 1}} \leftarrow \\\end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B%5Csqrt%7Bx%20%2B%201%7D%20-%202%7D%7Bx%20-%203%7D%20%26%20%3D%20%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B%28%5Csqrt%7Bx%20%2B%201%7D%20-%202%29%27%7D%7B%28x%20-%203%29%27%7D%20%5C%5C%26%20%3D%20%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%20%2B%201%7D%7D%20%5C%5C%26%20%3D%20%5Cfrac%7B1%7D%7B2%5Csqrt%7B3%20%2B%201%7D%7D%20%5Cleftarrow%20%5C%5C%5Cend%7Baligned%7D)
- Evaluate:
![\displaystyle \begin{aligned}\lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} & = \lim_{x \to 3} \frac{(\sqrt{x + 1} - 2)'}{(x - 3)'} \\& = \lim_{x \to 3} \frac{1}{2\sqrt{x + 1}} \\& = \frac{1}{2\sqrt{3 + 1}} \\& = \boxed{ \frac{1}{4} } \\\end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B%5Csqrt%7Bx%20%2B%201%7D%20-%202%7D%7Bx%20-%203%7D%20%26%20%3D%20%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B%28%5Csqrt%7Bx%20%2B%201%7D%20-%202%29%27%7D%7B%28x%20-%203%29%27%7D%20%5C%5C%26%20%3D%20%5Clim_%7Bx%20%5Cto%203%7D%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%20%2B%201%7D%7D%20%5C%5C%26%20%3D%20%5Cfrac%7B1%7D%7B2%5Csqrt%7B3%20%2B%201%7D%7D%20%5C%5C%26%20%3D%20%5Cboxed%7B%20%5Cfrac%7B1%7D%7B4%7D%20%7D%20%5C%5C%5Cend%7Baligned%7D)
∴ we have <em>evaluated</em> the given limit.
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Learn more about limits: brainly.com/question/27807253
Learn more about Calculus: brainly.com/question/27805589
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Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Limits