This should be your answer for your question
Answer:
$147
Step-by-step explanation:
Bridget gets: x
Stephen gets: 3x
Richard gets: 2(3x) = 6x
Total = 10x = 490
x = 49
Stephen gets: 3x = 3(49) = $147
Answer:
3 packages of pencils and 4 packages of crayons
Answer:
![f(x) = \frac{x^4}{12} - sin(x) + 3x + 4](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cfrac%7Bx%5E4%7D%7B12%7D%20-%20sin%28x%29%20%2B%203x%20%2B%204)
General Formulas and Concepts:
<u>Calculus</u>
- Antiderivatives
- Integration Constant C
- [Int Rule] Reverse Power Rule:
![\int {x^n} \, dx = \frac{x^{n+1}}{n+1} + C](https://tex.z-dn.net/?f=%5Cint%20%7Bx%5En%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7Bx%5E%7Bn%2B1%7D%7D%7Bn%2B1%7D%20%2B%20C)
- Integration Property 1:
![\int {cf(x)} \, dx = c\int {f(x)} \, dx](https://tex.z-dn.net/?f=%5Cint%20%7Bcf%28x%29%7D%20%5C%2C%20dx%20%3D%20c%5Cint%20%7Bf%28x%29%7D%20%5C%2C%20dx)
- Integration Property 2:
![\int {f(x)+g(x)} \, dx = \int {f(x)} \, dx + \int {g(x)} \, dx](https://tex.z-dn.net/?f=%5Cint%20%7Bf%28x%29%2Bg%28x%29%7D%20%5C%2C%20dx%20%3D%20%5Cint%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%2B%20%5Cint%20%7Bg%28x%29%7D%20%5C%2C%20dx)
Step-by-step explanation:
<u>Step 1: Define</u>
f"(x) = x² + sin(x)
Condition f'(0) = 2
Condition f(0) = 4
<u>Step 2: Integrate Pt. 1</u>
- Set up:
![f'(x) = \int {f"(x)} \, dx](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20%5Cint%20%7Bf%22%28x%29%7D%20%5C%2C%20dx)
- Substitute:
![f'(x) = \int [{x^2 + sin(x)}] \, dx](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20%5Cint%20%5B%7Bx%5E2%20%2B%20sin%28x%29%7D%5D%20%5C%2C%20dx)
- Rewrite [Int Property 2]:
![f'(x) = \int {x^2} \, dx + \int {sin(x)} \, dx](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20%5Cint%20%7Bx%5E2%7D%20%5C%2C%20dx%20%2B%20%5Cint%20%7Bsin%28x%29%7D%20%5C%2C%20dx)
- Integrate [Reverse Power Rule/Trig]:
![f'(x) = \frac{x^3}{3} - cos(x) + C](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20%5Cfrac%7Bx%5E3%7D%7B3%7D%20%20-%20cos%28x%29%20%2B%20C)
<u>Step 3: Find f'(x)</u>
<em>Use the given condition to find the differential equation.</em>
- Substitute:
![f'(0) = \frac{0^3}{3} - cos(0) + C](https://tex.z-dn.net/?f=f%27%280%29%20%3D%20%5Cfrac%7B0%5E3%7D%7B3%7D%20%20-%20cos%280%29%20%2B%20C)
- Substitute:
![2 = \frac{0^3}{3} - cos(0) + C](https://tex.z-dn.net/?f=2%20%3D%20%5Cfrac%7B0%5E3%7D%7B3%7D%20%20-%20cos%280%29%20%2B%20C)
- Evaluate:
![2 = 0 - 1 + C](https://tex.z-dn.net/?f=2%20%3D%200%20-%201%20%2B%20C)
- Solve:
![3 = C](https://tex.z-dn.net/?f=3%20%3D%20C)
- Define:
![f'(x) = \frac{x^3}{3} - cos(x) + 3](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20%5Cfrac%7Bx%5E3%7D%7B3%7D%20%20-%20cos%28x%29%20%2B%203)
<u>Step 4: Integrate Pt. 2</u>
- Set up:
![f(x) = \int {f'(x)} \, dx](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cint%20%7Bf%27%28x%29%7D%20%5C%2C%20dx)
- Substitute:
![f(x) = \int [{\frac{x^3}{3} - cos(x) + 3}] \, dx](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cint%20%5B%7B%5Cfrac%7Bx%5E3%7D%7B3%7D%20%20-%20cos%28x%29%20%2B%203%7D%5D%20%5C%2C%20dx)
- Rewrite [Int Property 2]:
![f(x) = \int {\frac{x^3}{3} } \, dx + \int {-cos(x)} \, dx + \int {3} \, dx](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cint%20%7B%5Cfrac%7Bx%5E3%7D%7B3%7D%20%7D%20%5C%2C%20dx%20%2B%20%5Cint%20%7B-cos%28x%29%7D%20%5C%2C%20dx%20%20%2B%20%5Cint%20%7B3%7D%20%5C%2C%20dx)
- Rewrite [Int Property 1]:
![f(x) = \frac{1}{3} \int {x^3} \, dx - \int {cos(x)} \, dx + \int {3} \, dx](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cfrac%7B1%7D%7B3%7D%20%5Cint%20%7Bx%5E3%7D%20%5C%2C%20dx%20-%20%5Cint%20%7Bcos%28x%29%7D%20%5C%2C%20dx%20%20%2B%20%5Cint%20%7B3%7D%20%5C%2C%20dx)
- Integrate {Reverse Power Rule/Trig]:
![f(x) = \frac{1}{3}(\frac{x^4}{4} ) - sin(x) + 3x + C](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cfrac%7B1%7D%7B3%7D%28%5Cfrac%7Bx%5E4%7D%7B4%7D%20%29%20-%20sin%28x%29%20%2B%203x%20%2B%20C)
- Simplify:
![f(x) = \frac{x^4}{12} - sin(x) + 3x + C](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cfrac%7Bx%5E4%7D%7B12%7D%20-%20sin%28x%29%20%2B%203x%20%2B%20C)
<u>Step 5: Find f(x)</u>
<em>Use the given condition to find the equation.</em>
- Substitute:
![f(0) = \frac{0^4}{12} - sin(0) + 3(0) + C](https://tex.z-dn.net/?f=f%280%29%20%3D%20%5Cfrac%7B0%5E4%7D%7B12%7D%20-%20sin%280%29%20%2B%203%280%29%20%2B%20C)
- Substitute:
![4 = \frac{0^4}{12} - sin(0) + 3(0) + C](https://tex.z-dn.net/?f=4%20%3D%20%5Cfrac%7B0%5E4%7D%7B12%7D%20-%20sin%280%29%20%2B%203%280%29%20%2B%20C)
- Evaluate:
![4 = 0 - 0 + 0+ C](https://tex.z-dn.net/?f=4%20%3D%200%20-%200%20%2B%200%2B%20C)
- Solve:
![4 = C](https://tex.z-dn.net/?f=4%20%3D%20C)
- Define:
![f(x) = \frac{x^4}{12} - sin(x) + 3x + 4](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cfrac%7Bx%5E4%7D%7B12%7D%20-%20sin%28x%29%20%2B%203x%20%2B%204)
The answer is the 3rd one