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ap_calculus_ab [2024/04/10 07:49] – mrdoughap_calculus_ab [2024/05/12 23:54] (current) – [AP Calc AB Study Guide] 172.88.72.108
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 ====== AP Calc AB Study Guide ====== ====== AP Calc AB Study Guide ======
 +  * Credit: Simplestudies.org 
 +  * This study guide has a lot of images so if u cant scroll to the very bottom without jittering just let it load for a bit 
 +  * Here is a Cheat Sheet/Shorter and compressed version of what's described. [[https://drive.google.com/file/d/1oTnZ5zSmNq0RWiAuZzW8q0ki5WLdhy70/view?usp=sharing|Final Notes for AB and BC]] 
 +  * Limit evaluation chart: [[https://drive.google.com/file/d/1bhdygywT-doVSjAEXM8BYZGj5CiAtQEV/view?usp=sharing|here]] 
 +  * Key words pdf [[https://drive.google.com/file/d/1MXi1LwqLF00C2uRe8h-i3E1SBnJdSmrk/view?usp=sharing|here]]
 ====== Unit 1 – Limits and Continuity ====== ====== Unit 1 – Limits and Continuity ======
  
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   * A limit refers to the y-value of a function   * A limit refers to the y-value of a function
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled.png}}]]  * The general limit exists when the right and left limits are the same/equal each other+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled.png}}]]   
 +* The general limit exists when the right and left limits are the same_equal each other
  
   * DNE = does not exist.   * DNE = does not exist.
  
-//Examples of estimating a limit numerically://+__Examples of estimating a limit numerically:__
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%201|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%201.png}}]]Example of using a graph to find a limit:\\ \\ +[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_1|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_1.png}}]]Example of using a graph to find a limit:\\ \\ 
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%202|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%202.png}}]]**When limits don’t exist:**\\ When the Left limit ≠ Right limit, then the limit is said to not exist.\\ +[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_2|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_2.png}}]]**When limits don’t exist:**\\ When the Left limit ≠ Right limit, then the limit is said to not exist.\\ 
  
   * In the picture below, you can tell that the two limits don’t equal each other, thus theanswer to this limit is DNE.   * In the picture below, you can tell that the two limits don’t equal each other, thus theanswer to this limit is DNE.
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%203|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%203.png}}]]**Unbounded Behavior:**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_3|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_3.png}}]]**Unbounded Behavior:**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%204|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%204.png}}]]**Evaluating Limits Analytically:**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_4|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_4.png}}]]**Evaluating Limits Analytically:**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%205|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%205.png}}]]**Limits Theorem:\\ Given:\\ ** Lim and Lim+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_5|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_5.png}}]]**Limits Theorem:\\ Given:\\ ** Lim and Lim
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%206|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%206.png}}]]**Limits at Infinity:**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_6|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_6.png}}]]**Limits at Infinity:**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%207|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%207.png}}]]  * If m<n, then the limit equals 0+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_7|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_7.png}}]]  * If m<n, then the limit equals 0
  
-  * If m=n, then the limit equals a/b+  * If m=n, then the limit equals a_b
  
   * If m>n, then the limit DNE   * If m>n, then the limit DNE
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 **Finding Vertical Asymptotes**\\ The only step you have to do is\\ **set the denominator equal to zero and solve.** **Finding Vertical Asymptotes**\\ The only step you have to do is\\ **set the denominator equal to zero and solve.**
  
-  * Example: [[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%208|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%208.png}}]] +  * Example: [[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_8|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_8.png}}]] 
     * (x+2)(x-2) = 0 → x = 2, -2     * (x+2)(x-2) = 0 → x = 2, -2
       * 2 is a removable hole while -2 is the non-removable vertical asymptote.       * 2 is a removable hole while -2 is the non-removable vertical asymptote.
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 Finding Horizontal Asymptotes\\ Use the\\ **two terms of the highest degree in the numerator and denominator** Finding Horizontal Asymptotes\\ Use the\\ **two terms of the highest degree in the numerator and denominator**
  
-  * Example [[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%209|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%209.png}}]] +  * Example [[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_9|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_9.png}}]] 
     * x and x2 are the two terms of the highest degree in the numerator and denominatorrespectively. After finding it, use the limits at infinity rule to determine the limit.     * x and x2 are the two terms of the highest degree in the numerator and denominatorrespectively. After finding it, use the limits at infinity rule to determine the limit.
  
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 Example #2: Example #2:
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2010|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2010.png}}]]  - For 0<t<60, must there be a time t when v(t) = -5?+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_10|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_10.png}}]]  - For 0<t<60, must there be a time t when v(t) = -5?
  
   - f(a) = f(0) = -20f(b) = f(60) =10   - f(a) = f(0) = -20f(b) = f(60) =10
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 **The Squeeze Theorem** **The Squeeze Theorem**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2011|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2011.png}}]]that means f(x) equals h(x) and g(x)+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_11|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_11.png}}]]that means f(x) equals h(x) and g(x)
  
-====== Unit 2 - Differentiation: Definition and Basic DerivativeRules ======+====== Unit 2 - Differentiation: Definition and Basic Derivative Rules ======
  
 **What is a derivative?** **What is a derivative?**
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   * The derivative of f(x) is denoted as f’(x) or   * The derivative of f(x) is denoted as f’(x) or
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2012|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2012.png}}]]**Derivatives as Limits**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_12|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_12.png}}]] 
 +**Derivatives as Limits**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2013|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2013.png}}]]**Steps to find derivatives as limits:**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_13|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_13.png}}]] 
 +**Steps to find derivatives as limits:**
  
   - Identify the form of the derivative first (look at the image above)… is it form a? b? c?   - Identify the form of the derivative first (look at the image above)… is it form a? b? c?
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 **Rules of Differentiation** **Rules of Differentiation**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2014|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2014.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2015|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2015.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2016|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2016.png}}]]**Derivatives of Trigonometric Functions:**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_14|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_14.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_15|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_15.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_16|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_16.png}}]]**Derivatives of Trigonometric Functions:**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2017|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2017.png}}]]  * HINT: If the original function starts with C, then the derivative is negative!+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_17|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_17.png}}]]   
 +* HINT: If the original function starts with C, then the derivative is negative!
     * Example: cosx, cotx, & cscx     * Example: cosx, cotx, & cscx
  
 **Derivative Rule for LN** **Derivative Rule for LN**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2018|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2018.png}}]]  * HINT: [Derive over copy]+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_18|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_18.png}}]]   
 +* HINT: [Derive over copy]
     * Example: h(x) = ln(2x^2 + 1)     * Example: h(x) = ln(2x^2 + 1)
       * First derive 2x^2 + 1. That would be 4x! And then put that over theoriginal function, which would be 2x^2 + 1.       * First derive 2x^2 + 1. That would be 4x! And then put that over theoriginal function, which would be 2x^2 + 1.
-      * Your answer would then be 4x%%/(%%2x^2 + 1)+      * Your answer would then be 4x%%_(%%2x^2 + 1)
  
 **Deriving Exponential Functions** **Deriving Exponential Functions**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2019|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2019.png}}]]**Continuity**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_19|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_19.png}}]] 
 + 
 +**Continuity**
  
 A function f is continuous at “c” if: A function f is continuous at “c” if:
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 Example: Example:
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2020|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2020.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2021|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2021.png}}]]====== Unit 3 - Differentiation: Composite, Implicit, and InverseFunctions ======+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_20|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_20.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_21|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_21.png}}]] 
 +====== Unit 3 - Differentiation: Composite, Implicit, and InverseFunctions ======
  
 **The Chain Rule** **The Chain Rule**
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 The chain rule helps us find the derivative of a composite function. For the formula, g’(x)\\ would be the chain.\\  The chain rule helps us find the derivative of a composite function. For the formula, g’(x)\\ would be the chain.\\ 
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2022|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2022.png}}]]Example:+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_22|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_22.png}}]]Example:
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2023|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2023.png}}]]**General Rule Power**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_23|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_23.png}}]]**General Rule Power**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2024|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2024.png}}]]We use the general rule power when finding the **derivative of a function that is raised to the\\ nth power\\ **. In the formula given, f’(x) is the chain.+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_24|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_24.png}}]]We use the general rule power when finding the **derivative of a function that is raised to the\\ nth power\\ **. In the formula given, f’(x) is the chain.
  
 **Implicit Differentiation** **Implicit Differentiation**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2025|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2025.png}}]]**Inverse Trig Functions: Differentiation**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_25|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_25.png}}]]**Inverse Trig Functions: Differentiation**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2026|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2026.png}}]]Example:+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_26|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_26.png}}]]Example:
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2027|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2027.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2028|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2028.png}}]]**Derivatives of Inverse Functions**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_27|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_27.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_28|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_28.png}}]]**Derivatives of Inverse Functions**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2029|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2029.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2030|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2030.png}}]]**Higher-Order Derivatives**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_29|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_29.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_30|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_30.png}}]]**Higher-Order Derivatives**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2031|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2031.png}}]]====== Unit 4 - Contextual Applications of Differentiation ======+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_31|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_31.png}}]] 
 +====== Unit 4 - Contextual Applications of Differentiation ======
  
 **Particle Motion** **Particle Motion**
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   * Example: s(t)=6t^3 -4t^2 → v(t)=18t^2 -8t → a(t)=36t-8   * Example: s(t)=6t^3 -4t^2 → v(t)=18t^2 -8t → a(t)=36t-8
  
-**Particle Moving Away/Toward the Origin(x-axis)**+**Particle Moving Away_Toward the Origin(x-axis)**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2032|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2032.png}}]]  * A particle is moving towards the origin when its position and velocity have opposite signs.+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_32|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_32.png}}]]  * A particle is moving towards the origin when its position and velocity have opposite signs.
  
   * A particle is moving away from the origin when its position and velocity have the samesigns,   * A particle is moving away from the origin when its position and velocity have the samesigns,
  
-**Particle Speeding Up/Slowing Down**+**Particle Speeding Up_Slowing Down**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2033|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2033.png}}]]  * A particle is speeding up (speed is increasing) if the velocity and acceleration have thesame signs at the point+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_33|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_33.png}}]]  * A particle is speeding up (speed is increasing) if the velocity and acceleration have thesame signs at the point
  
   * A particle is slowing down (speed is decreasing) if the velocity and acceleration haveopposite signs at the point.   * A particle is slowing down (speed is decreasing) if the velocity and acceleration haveopposite signs at the point.
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 **Related Rates: Algebraic** **Related Rates: Algebraic**
  
-Example: A point moves along the curve y = 2x^2 - 1 in which y decreases at the rate of 2 units\\ per second. What rate is x changing when x = -3/2?\\ +Example: A point moves along the curve y = 2x^2 - 1 in which y decreases at the rate of 2 units\\ per second. What rate is x changing when x = -3_2?\\ 
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2034|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2034.png}}]]**Related Rates: Circle**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_34|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_34.png}}]]**Related Rates: Circle**
  
-Example: The radius of a circle is increasing at a rate of 3cm/sec. How fast is the circumference\\ of the circle changing?\\ +Example: The radius of a circle is increasing at a rate of 3cm_sec. How fast is the circumference\\ of the circle changing?\\ 
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2035|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2035.png}}]]**Related Rates: Triangle**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_35|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_35.png}}]]**Related Rates: Triangle**
  
-Example: A 13ft ladder is leaning against a wall. If the top of the ladder slips down the wall at a\\ rate of 2 ft/s, how fast will the ladder be moving away from the wall when the top is 5ft above\\ the ground?\\ +Example: A 13ft ladder is leaning against a wall. If the top of the ladder slips down the wall at a\\ rate of 2 ft_s, how fast will the ladder be moving away from the wall when the top is 5ft above\\ the ground?\\ 
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2036|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2036.png}}]]**Related Rates: Cube**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_36|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_36.png}}]]**Related Rates: Cube**
  
-Example: The volume of a cube is increasing at a rate of 10cm^3/min. How fast is the surface area\\ increasing when the length of an edge is 30cm?\\ +Example: The volume of a cube is increasing at a rate of 10cm^3_min. How fast is the surface area\\ increasing when the length of an edge is 30cm?\\ 
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2037|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2037.png}}]]**Related Rates: Right Cylinder**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_37|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_37.png}}]]**Related Rates: Right Cylinder**
  
-Example: The radius of a right circular cylinder increases at the rate of 0.1cm/min and the height\\ decreases at the rate of 0.2 cm/mm. What is the rate of change of the volume of the cylinder, in\\ cm3/min, when the radius is 2cm and the height is 3cm?\\ +Example: The radius of a right circular cylinder increases at the rate of 0.1cm_min and the height\\ decreases at the rate of 0.2 cm_mm. What is the rate of change of the volume of the cylinder, in\\ cm3_min, when the radius is 2cm and the height is 3cm?\\ 
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2038|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2038.png}}]]**Related Rates: Sphere**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_38|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_38.png}}]]**Related Rates: Sphere**
  
-Example: As a balloon in the shape of a sphere is being blown up, the volume is increasing at a\\ rate of 4in3/s. At what rate is the radius increasing when the radius is 1 inch.\\ +Example: As a balloon in the shape of a sphere is being blown up, the volume is increasing at a\\ rate of 4in3_s. At what rate is the radius increasing when the radius is 1 inch.\\ 
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2039|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2039.png}}]]**Related Rates: Circumference**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_39|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_39.png}}]]**Related Rates: Circumference**
  
-Example: What is the value of the circumference of a circle at the instant when the radius is\\ increasing at 1/6 the rate the area is increasing?\\ +Example: What is the value of the circumference of a circle at the instant when the radius is\\ increasing at 1_6 the rate the area is increasing?\\ 
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2040|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2040.png}}]]**L’Hopital’s Rule**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_40|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_40.png}}]]**L’Hopital’s Rule**
  
-Let f and g be continuous and differentiable functions on an open interval (a,b). If the limit of\\ f(x) and g(x) as x approaches c produces the indeterminate form 0/0 or ∞/∞ then,\\ +Let f and g be continuous and differentiable functions on an open interval (a,b). If the limit of\\ f(x) and g(x) as x approaches c produces the indeterminate form 0_0 or ∞_∞ then,\\ 
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2041|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2041.png}}]]====== Unit 5 - Analytical Applications of Differentiation ======+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_41|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_41.png}}]]====== Unit 5 - Analytical Applications of Differentiation ======
  
 **Mean Value Theorem** **Mean Value Theorem**
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 If f(x) is a function that is **continuous on the closed intervals [a,b] and differentiable on the\\ open interval (a,b),\\ ** then there must exist a value **c between (a,b)** If f(x) is a function that is **continuous on the closed intervals [a,b] and differentiable on the\\ open interval (a,b),\\ ** then there must exist a value **c between (a,b)**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2042|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2042.png}}]]  * Example: Confirm f(x)=x^3 on [0,3] and find a value that satisfies this theorem+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_42|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_42.png}}]]  * Example: Confirm f(x)=x^3 on [0,3] and find a value that satisfies this theorem
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2043|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2043.png}}]]**Function Increasing or Decreasing**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_43|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_43.png}}]]**Function Increasing or Decreasing**
  
   * f(x) is increasing when f’(x) is positive   * f(x) is increasing when f’(x) is positive
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 **Extreme Value Theorem**\\ If f(x) is continuous on a closed interval [a,b], then f(x) has both a minimum and maximum on the interval.\\  **Extreme Value Theorem**\\ If f(x) is continuous on a closed interval [a,b], then f(x) has both a minimum and maximum on the interval.\\ 
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2044|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2044.png}}]](Absolute max on top and absolute min on bottom)+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_44|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_44.png}}]](Absolute max on top and absolute min on bottom)
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2045|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2045.png}}]]**First Derivative Test**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_45|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_45.png}}]]**First Derivative Test**
  
 For first derivative tests, **derive the function once and set it to 0**. After that, **find the zeros** and\\ plug them into a number line. Using your derived function, plug-in numbers before and after\\ your constant (the zeros of the function) to see if it becomes negative or positive, as shown\\ below.\\  For first derivative tests, **derive the function once and set it to 0**. After that, **find the zeros** and\\ plug them into a number line. Using your derived function, plug-in numbers before and after\\ your constant (the zeros of the function) to see if it becomes negative or positive, as shown\\ below.\\ 
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   * If it’s negative, constant, positive then it’s a **relative minimum**   * If it’s negative, constant, positive then it’s a **relative minimum**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2046|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2046.png}}]]**Concavity**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_46|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_46.png}}]]**Concavity**
  
   * The graph of f is concave up when f’(x) is increasing   * The graph of f is concave up when f’(x) is increasing
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   * If f’’(x) is negative then the graph of f is concave down   * If f’’(x) is negative then the graph of f is concave down
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2047|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2047.png}}]]  * Points of Inflection+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_47|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_47.png}}]]  * Points of Inflection
     * Occurs when f(x) changes concavity     * Occurs when f(x) changes concavity
     * Determined by a sign change for f’’(x)     * Determined by a sign change for f’’(x)
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 Example: Example:
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2048|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2048.png}}]]**Critical Numbers**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_48|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_48.png}}]]**Critical Numbers**
  
   * Critical numbers are **points on the graph of a function where there’s a change in direction.**   * Critical numbers are **points on the graph of a function where there’s a change in direction.**
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   * To **find critical numbers**, you use the **first derivative of the function and set it to zero.**   * To **find critical numbers**, you use the **first derivative of the function and set it to zero.**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2049|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2049.png}}]]====== Unit 6 - Integration and Accumulation of Change ======+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_49|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_49.png}}]]====== Unit 6 - Integration and Accumulation of Change ======
  
 **Riemann Sums** **Riemann Sums**
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 You use Riemann sums to find the actual area underneath the graph of f(x). You use Riemann sums to find the actual area underneath the graph of f(x).
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2050|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2050.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2051|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2051.png}}]]**Trapezoidal Reimann Sum**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_50|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_50.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_51|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_51.png}}]]**Trapezoidal Reimann Sum**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2052|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2052.png}}]]**Integration**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_52|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_52.png}}]]**Integration**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2053|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2053.png}}]]The area under the curve of derivatives of F from A to B is equal to the change in y-values of the\\ function F from A to B, given f is:\\ +[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_53|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_53.png}}]]The area under the curve of derivatives of F from A to B is equal to the change in y-values of the\\ function F from A to B, given f is:\\ 
  
   * Continuous in interval [a,b]   * Continuous in interval [a,b]
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   * Detonated by an integral sign: ∫   * Detonated by an integral sign: ∫
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2054|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2054.png}}]]  * f %%'(%%x) = integrand+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_54|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_54.png}}]]  * f %%'(%%x) = integrand
  
   * dx = variable of integration   * dx = variable of integration
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 **Reminder:** ALWAYS add +C when you’re solving for an INDEFINITE integral!\\ \\ **Reminder:** Differentiation and integration are inverses! **Reminder:** ALWAYS add +C when you’re solving for an INDEFINITE integral!\\ \\ **Reminder:** Differentiation and integration are inverses!
  
-**Basic Integration Rules (w/ examples)**+**Basic Integration Rules (w_ examples)**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2055|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2055.png}}]]**Antiderivative Trig Function**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_55|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_55.png}}]]**Antiderivative Trig Function**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2056|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2056.png}}]]C is still a constant when multiplied by 2 (a constant multiplied by a constant is still a constant)+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_56|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_56.png}}]]C is still a constant when multiplied by 2 (a constant multiplied by a constant is still a constant)
  
   * HINT: How I memorize antiderivatives by using derivatives of trigonometric functions   * HINT: How I memorize antiderivatives by using derivatives of trigonometric functions
-    * EX: d/dx sinx = cosx and for the antiderivative, you just switch the twotrigonometric functions and add +C since it’s an indefinite integration. +    * EX: d_dx sinx = cosx and for the antiderivative, you just switch the twotrigonometric functions and add +C since it’s an indefinite integration. 
-    * EX: d/dx cscx = -cscxcotx and for the antiderivative, just switch the twotrigonometric functions and add +c since it’s an indefinite integration. Also, if thederivative was negative, then the anti-derivative is also negative!+    * EX: d_dx cscx = -cscxcotx and for the antiderivative, just switch the twotrigonometric functions and add +c since it’s an indefinite integration. Also, if thederivative was negative, then the anti-derivative is also negative!
  
 **Integration by U-substitution** **Integration by U-substitution**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2057|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2057.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2058|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2058.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2059|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2059.png}}]]**Natural Log Function for Integration (Log rule for integration)**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_57|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_57.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_58|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_58.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_59|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_59.png}}]]**Natural Log Function for Integration (Log rule for integration)**
  
 Use this rule when ‘x’ becomes DNE Use this rule when ‘x’ becomes DNE
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2060|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2060.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2060|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2060.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2061|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2061.png}}]]**Integrals of the 6 Basic Trig Functions**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_60|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_60.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_60|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_60.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_61|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_61.png}}]]**Integrals of the 6 Basic Trig Functions**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2062|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2062.png}}]]  * **HINT:** For ∫ tan u du, I memorized it like this: ∫ tan u du = ∫ sin u/cos u du because of the trigonometric identities. After that, I just did u-substitution with cos u being u.+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_62|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_62.png}}]]  * **HINT:** For ∫ tan u du, I memorized it like this: ∫ tan u du = ∫ sin u_cos u du because of the trigonometric identities. After that, I just did u-substitution with cos u being u.
  
   * If you work it out, it looks like this:   * If you work it out, it looks like this:
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2063|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2063.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2064|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2064.png}}]]**Integration rules for “e”**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_63|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_63.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_64|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_64.png}}]]**Integration rules for “e”**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2065|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2065.png}}]]● With e, it’s just the same thing as regular u-substitution but with the additional ‘e’.+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_65|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_65.png}}]]● With e, it’s just the same thing as regular u-substitution but with the additional ‘e’.
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2066|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2066.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2067|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2067.png}}]]**Integration Rule for Exponential Functions**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_66|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_66.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_67|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_67.png}}]]**Integration Rule for Exponential Functions**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2068|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2068.png}}]]====== Unit 7 - Differential Equations ======+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_68|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_68.png}}]]====== Unit 7 - Differential Equations ======
  
 **Differential Equations (Separate the integral)** **Differential Equations (Separate the integral)**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2069|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2069.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2070|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2070.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2071|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2071.png}}]]**Differential Equation with Initial condition**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_69|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_69.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_70|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_70.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_71|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_71.png}}]]**Differential Equation with Initial condition**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2072|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2072.png}}]]**Slope Field**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_72|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_72.png}}]]**Slope Field**
  
-A visual depiction of a differential equation of dy/dx.+A visual depiction of a differential equation of dy_dx.
  
   * Example of what a slope field looks like   * Example of what a slope field looks like
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2073|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2073.png}}]]====== Unit 8 – Applications of Integration ======+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_73|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_73.png}}]]====== Unit 8 – Applications of Integration ======
  
 **Average Value** **Average Value**
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   * After that, **divide the answer by the length of the interval**   * After that, **divide the answer by the length of the interval**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2074|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2074.png}}]]Total Displacement+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_74|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_74.png}}]]Total Displacement
  
   * The difference between the starting position and ending position   * The difference between the starting position and ending position
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   * Formula:   * Formula:
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2075|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2075.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2076|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2076.png}}]]**Total Distance**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_75|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_75.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_76|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_76.png}}]]**Total Distance**
  
   * Total distance traveled by a particle is the sum of the amounts it displaces betweenthe start, all of the stop(s), and the end.   * Total distance traveled by a particle is the sum of the amounts it displaces betweenthe start, all of the stop(s), and the end.
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   * Formula:   * Formula:
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2077|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2077.png}}]][[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2078|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2078.png}}]]**Area of a Region Between Two Curves**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_77|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_77.png}}]][[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_78|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_78.png}}]]**Area of a Region Between Two Curves**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2079|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2079.png}}]]**The Disk Method**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_79|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_79.png}}]]**The Disk Method**
  
 If a region in the plane **revolves about a line, the resulting solid is a solid of revolution,** and\\ the line is called the\\ **axis of revolution.** The simplest solid is a right circular cylinder or disk,\\ which is formed by revolving a rectangle about an axis adjacent to one side of the rectangle.\\  If a region in the plane **revolves about a line, the resulting solid is a solid of revolution,** and\\ the line is called the\\ **axis of revolution.** The simplest solid is a right circular cylinder or disk,\\ which is formed by revolving a rectangle about an axis adjacent to one side of the rectangle.\\ 
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   * **Rotate Around x-axis**   * **Rotate Around x-axis**
     * The horizontal axis of revolution     * The horizontal axis of revolution
-  * [[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2080|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2080.png}}]] +  * [[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_80|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_80.png}}]] 
   * **Rotate Around y-axis**   * **Rotate Around y-axis**
     * The **vertical axis of revolution**     * The **vertical axis of revolution**
-  * [[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2081|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2081.png}}]] +  * [[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_81|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_81.png}}]] 
 **The Washer Method** **The Washer Method**
  
   * Horizontal Line of Rotation:   * Horizontal Line of Rotation:
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2082|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2082.png}}]]  * Vertical Line of Rotation+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_82|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_82.png}}]]  * Vertical Line of Rotation
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2083|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:Untitled%2083.png}}]]**The Washer Method Calculating Volume Using Integration**+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_83|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_83.png}}]]**The Washer Method Calculating Volume Using Integration**
  
 **Step One:** Draw a picture of your graph**→** **shade appropriate region** **Step One:** Draw a picture of your graph**→** **shade appropriate region**
Line 433: Line 446:
 **Step Three:** **Set up your Integral** **Step Three:** **Set up your Integral**
  
-[[AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:3ac721a0-33e0-4df3-a59e-bfceb06ee2cb|{{AP%20Calc%20AB%20Study%20Guide%20f289051ba2044b4ab01d25945296aa61:3ac721a0-33e0-4df3-a59e-bfceb06ee2cb.png}}]]**Step Four:** Simplify\\ \\ **Step Five:** Integrate Definite Integral+[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:3ac721a0-33e0-4df3-a59e-bfceb06ee2cb|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:3ac721a0-33e0-4df3-a59e-bfceb06ee2cb.png}}]]**Step Four:** Simplify\\ \\ **Step Five:** Integrate Definite Integral
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