Answer:
![\displaystyle \int { \Big( 4x^\bigg{\frac{1}{3}} - x^\bigg{\frac{-1}{3}} \Big) } \, dx = 3x^\bigg{\frac{4}{3}} - \frac{3x^\bigg{\frac{2}{3}}}{2} + C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%20%5CBig%28%204x%5E%5Cbigg%7B%5Cfrac%7B1%7D%7B3%7D%7D%20-%20x%5E%5Cbigg%7B%5Cfrac%7B-1%7D%7B3%7D%7D%20%5CBig%29%20%7D%20%5C%2C%20dx%20%3D%203x%5E%5Cbigg%7B%5Cfrac%7B4%7D%7B3%7D%7D%20-%20%5Cfrac%7B3x%5E%5Cbigg%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D%7B2%7D%20%2B%20C)
General Formulas and Concepts:
<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
Integration
- Integrals
- Indefinite Integrals
- Integration Constant C
Integration Rule [Reverse Power Rule]: ![\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7Bx%5En%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7Bx%5E%7Bn%20%2B%201%7D%7D%7Bn%20%2B%201%7D%20%2B%20C)
Integration Property [Multiplied Constant]: ![\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7Bcf%28x%29%7D%20%5C%2C%20dx%20%3D%20c%20%5Cint%20%7Bf%28x%29%7D%20%5C%2C%20dx)
Integration Property [Addition/Subtraction]: ![\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%5Bf%28x%29%20%5Cpm%20g%28x%29%5D%7D%20%5C%2C%20dx%20%3D%20%5Cint%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%5Cpm%20%5Cint%20%7Bg%28x%29%7D%20%5C%2C%20dx)
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify</em>
![\displaystyle \int { \Big( 4x^\bigg{\frac{1}{3}} - x^\bigg{\frac{-1}{3}} \Big) } \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%20%5CBig%28%204x%5E%5Cbigg%7B%5Cfrac%7B1%7D%7B3%7D%7D%20-%20x%5E%5Cbigg%7B%5Cfrac%7B-1%7D%7B3%7D%7D%20%5CBig%29%20%7D%20%5C%2C%20dx)
<u>Step 2: Integrate</u>
- [Integral] Rewrite [Integration Property - Addition/Subtraction]:
![\displaystyle \int { \Big( 4x^\bigg{\frac{1}{3}} - x^\bigg{\frac{-1}{3}} \Big) } \, dx = \int {4x^\bigg{\frac{1}{3}}} \, dx - \int {x^\bigg{\frac{-1}{3}}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%20%5CBig%28%204x%5E%5Cbigg%7B%5Cfrac%7B1%7D%7B3%7D%7D%20-%20x%5E%5Cbigg%7B%5Cfrac%7B-1%7D%7B3%7D%7D%20%5CBig%29%20%7D%20%5C%2C%20dx%20%3D%20%5Cint%20%7B4x%5E%5Cbigg%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D%20%5C%2C%20dx%20-%20%5Cint%20%7Bx%5E%5Cbigg%7B%5Cfrac%7B-1%7D%7B3%7D%7D%7D%20%5C%2C%20dx)
- [1st Integral] Rewrite [Integration Property - Multiplied Constant]:
![\displaystyle \int { \Big( 4x^\bigg{\frac{1}{3}} - x^\bigg{\frac{-1}{3}} \Big) } \, dx = 4\int {x^\bigg{\frac{1}{3}}} \, dx - \int {x^\bigg{\frac{-1}{3}}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%20%5CBig%28%204x%5E%5Cbigg%7B%5Cfrac%7B1%7D%7B3%7D%7D%20-%20x%5E%5Cbigg%7B%5Cfrac%7B-1%7D%7B3%7D%7D%20%5CBig%29%20%7D%20%5C%2C%20dx%20%3D%204%5Cint%20%7Bx%5E%5Cbigg%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D%20%5C%2C%20dx%20-%20%5Cint%20%7Bx%5E%5Cbigg%7B%5Cfrac%7B-1%7D%7B3%7D%7D%7D%20%5C%2C%20dx)
- [Integrals] Reverse Power Rule:
![\displaystyle \int { \Big( 4x^\bigg{\frac{1}{3}} - x^\bigg{\frac{-1}{3}} \Big) } \, dx = 4 \bigg( \frac{3x^\bigg{\frac{4}{3}}}{4} \bigg) - \frac{3x^\bigg{\frac{2}{3}}}{2} + C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%20%5CBig%28%204x%5E%5Cbigg%7B%5Cfrac%7B1%7D%7B3%7D%7D%20-%20x%5E%5Cbigg%7B%5Cfrac%7B-1%7D%7B3%7D%7D%20%5CBig%29%20%7D%20%5C%2C%20dx%20%3D%204%20%5Cbigg%28%20%5Cfrac%7B3x%5E%5Cbigg%7B%5Cfrac%7B4%7D%7B3%7D%7D%7D%7B4%7D%20%5Cbigg%29%20-%20%5Cfrac%7B3x%5E%5Cbigg%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D%7B2%7D%20%2B%20C)
- Simplify:
![\displaystyle \int { \Big( 4x^\bigg{\frac{1}{3}} - x^\bigg{\frac{-1}{3}} \Big) } \, dx = 3x^\bigg{\frac{4}{3}} - \frac{3x^\bigg{\frac{2}{3}}}{2} + C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%20%5CBig%28%204x%5E%5Cbigg%7B%5Cfrac%7B1%7D%7B3%7D%7D%20-%20x%5E%5Cbigg%7B%5Cfrac%7B-1%7D%7B3%7D%7D%20%5CBig%29%20%7D%20%5C%2C%20dx%20%3D%203x%5E%5Cbigg%7B%5Cfrac%7B4%7D%7B3%7D%7D%20-%20%5Cfrac%7B3x%5E%5Cbigg%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D%7B2%7D%20%2B%20C)
<u>Step 3: Check</u>
<em>Differentiate the answer.</em>
- Rewrite [Derivative Property - Addition/Subtraction]:
![\displaystyle \frac{d}{dx} \bigg[ 3x^\bigg{\frac{4}{3}} \bigg] - \frac{d}{dx} \bigg[ \frac{3x^\bigg{\frac{2}{3}}}{2} \bigg] + \frac{d}{dx} \bigg[ C \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%203x%5E%5Cbigg%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Cbigg%5D%20-%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20%5Cfrac%7B3x%5E%5Cbigg%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D%7B2%7D%20%5Cbigg%5D%20%2B%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20C%20%5Cbigg%5D)
- Rewrite [Derivative Property - Multiplied Constant]:
![\displaystyle 3\frac{d}{dx} \bigg[ x^\bigg{\frac{4}{3}} \bigg] - \frac{3}{2}\frac{d}{dx} \bigg[ x^\bigg{\frac{2}{3}} \bigg] + \frac{d}{dx} \bigg[ C \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%203%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20x%5E%5Cbigg%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Cbigg%5D%20-%20%5Cfrac%7B3%7D%7B2%7D%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20x%5E%5Cbigg%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%5Cbigg%5D%20%2B%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20C%20%5Cbigg%5D)
- Basic Power Rule:
![\displaystyle 3 \bigg( \frac{4}{3}x^\bigg{\frac{1}{3}} \bigg) - \frac{3}{2} \bigg( \frac{2}{3}x^\bigg{\frac{-1}{3}} \bigg)](https://tex.z-dn.net/?f=%5Cdisplaystyle%203%20%5Cbigg%28%20%5Cfrac%7B4%7D%7B3%7Dx%5E%5Cbigg%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Cbigg%29%20-%20%5Cfrac%7B3%7D%7B2%7D%20%5Cbigg%28%20%5Cfrac%7B2%7D%7B3%7Dx%5E%5Cbigg%7B%5Cfrac%7B-1%7D%7B3%7D%7D%20%5Cbigg%29)
- Simplify:
![\displaystyle 4x^\bigg{\frac{1}{3}} - x^\bigg{\frac{-1}{3}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%204x%5E%5Cbigg%7B%5Cfrac%7B1%7D%7B3%7D%7D%20-%20x%5E%5Cbigg%7B%5Cfrac%7B-1%7D%7B3%7D%7D)
∴ we have found the answer.
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
Book: College Calculus 10e
Answer:
x=8
Step-by-step explanation:
-4 = 12 - 2x
when plus 12 goes to the left hand side it changes its sign so,
-4 -12 = -2x
- * - is plus so add 4 and 12 but keep the sign - because 12 is the greatest number and its sign is -
-16=-2x
when -2 comes to the left hand side it changes its from multiplication to division.So,
-16/-2=x
8=x
we will get positive 8 because minus and minus also gets cancel.
Answer:
Step-by-step explanation:
i dont know hdpmvalesvrgpto
Comment
You are going to have to learn something about f(2)
Step One
f(1) = 160
f(1 + 1) = - 2f(1)
f(2) = -2 * 160
f(2) = -320
Step Two
The easiest way is just to keep on going. This is recursive which means you use the last answer to get the next one.
f(3) = -2f(2)
f(3) = -2 * -320
f(3) = 640 Two minus' make a plus.
Step Three
Find f(4)
f(4) = - 2 f(3)
f(4) = -2 * 640
f(4) = - 1280 <<<<<<<<< answer
Alternate Method
f(n) = 160 * (-2) ^ (n - 1)
f(4) = 160 * (-2)^(4 - 1)
f(4) = 160 * (-2)^3
f(4) = 160 * (- 8)
f(4) = -1280