87; If he gave two thirds of his remaining cards to Elaine, then he had one third left. If one third of his remaining cards is 25, then 25 x 3 will give you what he had after trading with Tommy. When he finished trading with Tommy he had 75 cards. if he gave away 15 cards, but got three back, then he must have had 87 cards to begin with.
Check your answer:
87-15 = 72
72+3 = 75
2/3 of 75 = 50
75-50 = 25
Great!
Answer:
![\displaystyle y' = \frac{5x^2 + 3}{3(1 + x^2)^\bigg{\frac{2}{3}}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B5x%5E2%20%2B%203%7D%7B3%281%20%2B%20x%5E2%29%5E%5Cbigg%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D)
General Formulas and Concepts:
<u>Pre-Algebra</u>
<u>Algebra I</u>
- Functions
- Function Notation
- Exponential Rule [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>Algebra II</u>
- Logarithms and Natural Logs
- Logarithmic Property [Multiplying]:
![\displaystyle log(ab) = log(a) + log(b)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20log%28ab%29%20%3D%20log%28a%29%20%2B%20log%28b%29)
- Logarithmic Property [Exponential]:
![\displaystyle log(a^b) = b \cdot log(a)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20log%28a%5Eb%29%20%3D%20b%20%5Ccdot%20log%28a%29)
<u>Calculus</u>
Derivatives
Derivative Notation
Derivative Property [Multiplied Constant]:
Derivative Property [Addition/Subtraction]:
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)
Logarithmic Derivative: ![\displaystyle \frac{d}{dx} [lnu] = \frac{u'}{u}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Blnu%5D%20%3D%20%5Cfrac%7Bu%27%7D%7Bu%7D)
Implicit Differentiation
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify</em>
<em />
<em />
<em />
<u>Step 2: Rewrite</u>
- [Equality Property] ln both sides:
![\displaystyle lny = ln(x\sqrt[3]{1 + x^2})](https://tex.z-dn.net/?f=%5Cdisplaystyle%20lny%20%3D%20ln%28x%5Csqrt%5B3%5D%7B1%20%2B%20x%5E2%7D%29)
- Logarithmic Property [Multiplying]:
![\displaystyle lny = ln(x) + ln(\sqrt[3]{1 + x^2})](https://tex.z-dn.net/?f=%5Cdisplaystyle%20lny%20%3D%20ln%28x%29%20%2B%20ln%28%5Csqrt%5B3%5D%7B1%20%2B%20x%5E2%7D%29)
- Exponential Rule [Root Rewrite]:
![\displaystyle lny = ln(x) + ln \bigg[ (1 + x^2)^\bigg{\frac{1}{3}} \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20lny%20%3D%20ln%28x%29%20%2B%20ln%20%5Cbigg%5B%20%281%20%2B%20x%5E2%29%5E%5Cbigg%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Cbigg%5D)
- Logarithmic Property [Exponential]:
![\displaystyle lny = ln(x) + \frac{1}{3}ln(1 + x^2)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20lny%20%3D%20ln%28x%29%20%2B%20%5Cfrac%7B1%7D%7B3%7Dln%281%20%2B%20x%5E2%29)
<u>Step 3: Differentiate</u>
- ln Derivative [Implicit Differentiation]:
![\displaystyle \frac{d}{dx}[lny] = \frac{d}{dx} \bigg[ ln(x) + \frac{1}{3}ln(1 + x^2) \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Blny%5D%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20ln%28x%29%20%2B%20%5Cfrac%7B1%7D%7B3%7Dln%281%20%2B%20x%5E2%29%20%5Cbigg%5D)
- Rewrite [Derivative Property - Addition]:
![\displaystyle \frac{d}{dx}[lny] = \frac{d}{dx}[ln(x)] + \frac{d}{dx} \bigg[ \frac{1}{3}ln(1 + x^2) \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Blny%5D%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bln%28x%29%5D%20%2B%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7B3%7Dln%281%20%2B%20x%5E2%29%20%5Cbigg%5D)
- Rewrite [Derivative Property - Multiplied Constant]:
![\displaystyle \frac{d}{dx}[lny] = \frac{d}{dx}[ln(x)] + \frac{1}{3}\frac{d}{dx}[ln(1 + x^2)]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Blny%5D%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bln%28x%29%5D%20%2B%20%5Cfrac%7B1%7D%7B3%7D%5Cfrac%7Bd%7D%7Bdx%7D%5Bln%281%20%2B%20x%5E2%29%5D)
- ln Derivative [Chain Rule]:
![\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{1}{3} \bigg( \frac{1}{1 + x^2} \bigg) \cdot \frac{d}{dx}[(1 + x^2)]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7By%27%7D%7By%7D%20%3D%20%5Cfrac%7B1%7D%7Bx%7D%20%2B%20%5Cfrac%7B1%7D%7B3%7D%20%5Cbigg%28%20%5Cfrac%7B1%7D%7B1%20%2B%20x%5E2%7D%20%5Cbigg%29%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5B%281%20%2B%20x%5E2%29%5D)
- Rewrite [Derivative Property - Addition]:
![\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{1}{3} \bigg( \frac{1}{1 + x^2} \bigg) \cdot \bigg( \frac{d}{dx}[1] + \frac{d}{dx}[x^2] \bigg)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7By%27%7D%7By%7D%20%3D%20%5Cfrac%7B1%7D%7Bx%7D%20%2B%20%5Cfrac%7B1%7D%7B3%7D%20%5Cbigg%28%20%5Cfrac%7B1%7D%7B1%20%2B%20x%5E2%7D%20%5Cbigg%29%20%5Ccdot%20%5Cbigg%28%20%5Cfrac%7Bd%7D%7Bdx%7D%5B1%5D%20%2B%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bx%5E2%5D%20%5Cbigg%29)
- Basic Power Rule]:
![\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{1}{3} \bigg( \frac{1}{1 + x^2} \bigg) \cdot (2x^{2 - 1})](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7By%27%7D%7By%7D%20%3D%20%5Cfrac%7B1%7D%7Bx%7D%20%2B%20%5Cfrac%7B1%7D%7B3%7D%20%5Cbigg%28%20%5Cfrac%7B1%7D%7B1%20%2B%20x%5E2%7D%20%5Cbigg%29%20%5Ccdot%20%282x%5E%7B2%20-%201%7D%29)
- Simplify:
![\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{1}{3} \bigg( \frac{1}{1 + x^2} \bigg) \cdot 2x](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7By%27%7D%7By%7D%20%3D%20%5Cfrac%7B1%7D%7Bx%7D%20%2B%20%5Cfrac%7B1%7D%7B3%7D%20%5Cbigg%28%20%5Cfrac%7B1%7D%7B1%20%2B%20x%5E2%7D%20%5Cbigg%29%20%5Ccdot%202x)
- Multiply:
![\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{2x}{3(1 + x^2)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7By%27%7D%7By%7D%20%3D%20%5Cfrac%7B1%7D%7Bx%7D%20%2B%20%5Cfrac%7B2x%7D%7B3%281%20%2B%20x%5E2%29%7D)
- [Multiplication Property of Equality] Isolate <em>y'</em>:
![\displaystyle y' = y \bigg[ \frac{1}{x} + \frac{2x}{3(1 + x^2)} \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20y%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7Bx%7D%20%2B%20%5Cfrac%7B2x%7D%7B3%281%20%2B%20x%5E2%29%7D%20%5Cbigg%5D)
- Substitute in <em>y</em>:
![\displaystyle y' = x\sqrt[3]{1 + x^2} \bigg[ \frac{1}{x} + \frac{2x}{3(1 + x^2)} \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20x%5Csqrt%5B3%5D%7B1%20%2B%20x%5E2%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7Bx%7D%20%2B%20%5Cfrac%7B2x%7D%7B3%281%20%2B%20x%5E2%29%7D%20%5Cbigg%5D)
- [Brackets] Add:
![\displaystyle y' = x\sqrt[3]{1 + x^2} \bigg[ \frac{5x^2 + 3}{3x(1 + x^2)} \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20x%5Csqrt%5B3%5D%7B1%20%2B%20x%5E2%7D%20%5Cbigg%5B%20%5Cfrac%7B5x%5E2%20%2B%203%7D%7B3x%281%20%2B%20x%5E2%29%7D%20%5Cbigg%5D)
- Multiply:
![\displaystyle y' = \frac{(5x^2 + 3)\sqrt[3]{1 + x^2}}{3(1 + x^2)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B%285x%5E2%20%2B%203%29%5Csqrt%5B3%5D%7B1%20%2B%20x%5E2%7D%7D%7B3%281%20%2B%20x%5E2%29%7D)
- Simplify [Exponential Rule - Root Rewrite]:
![\displaystyle y' = \frac{5x^2 + 3}{3(1 + x^2)^\bigg{\frac{2}{3}}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B5x%5E2%20%2B%203%7D%7B3%281%20%2B%20x%5E2%29%5E%5Cbigg%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Implicit Differentiation
Book: College Calculus 10e
Y=10.55
Z=17.53
Y: tan37=y/14 ; 14(tan37)=10.55
Z: sin53=14/z ; 14/sin53=17.53
Let the required point be (a,b)
The distance of (a,b) from (7,-2) is
= ![\sqrt{(a-7)^2+(b+2)^2}](https://tex.z-dn.net/?f=%5Csqrt%7B%28a-7%29%5E2%2B%28b%2B2%29%5E2%7D)
But this distance needs to be betweem 50 & 60
So
![50](https://tex.z-dn.net/?f=50%3C%5Csqrt%7B%28a-7%29%5E2%2B%28b%2B2%29%5E2%7D%3C60)
Squaring all sides
2500 < (a-7)² + (b+2)² < 3600
Let a = 7
So we have
2500 < (b+2)² <3600
b+2 < 60 or b+2 > -60 => b <58 or b > -62
Also
b+2 >50 or b + 2 < -50 => b >48 or B < -52
Let us take one value of b < 58 say b = 50
So now we have the point as (7, 50)
The other point is (7,-2)
Distance between them
= ![\sqrt{(7-7)^2+(50+2)^2}= \sqrt{(52)^2}=52](https://tex.z-dn.net/?f=%5Csqrt%7B%287-7%29%5E2%2B%2850%2B2%29%5E2%7D%3D%20%5Csqrt%7B%2852%29%5E2%7D%3D52)
This is between 50 & 60
Hence one point which has a distance between 50 & 60 from the point (7,-2) is (7, 50)
Answer:
-17p+5
Step-by-step explanation:
Based on the given conditions, formulate: (--6p+1)+(4--11p)
Remove the parentheses: --6p+ 1 + 4-- 11p
Reorder and gather like terms:(--6p--11p)+( 1 + 4)
Collect coefficients for the like terms:(--6--11) x p +( 1 + 4)
Calculate the sum or difference:--17p+( 1 + 4)
Calculate the sum or difference:--17p + 5