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--- ap_calculus_bc [2024/04/10 20:50] – [Unit 9: Defining and Differentiating Parametric Equations] mrdough
+++ ap_calculus_bc [2024/05/12 23:55] (current) – [AP Calculus BC Study Guide] 172.88.72.108
@@ Line 1: / Line 1: @@
 ======AP Calculus BC 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
 Units 1-8 are the same as [[AP Calculus AB| AP Calculus AB Guide]]
   * 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 ======
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 ======Unit 10: Infinite Sequences and Series======
+  * Difference between sequences and series:
+    * $a_n$ = sequence
+    * S = $\Sigma_{n=1}^\infty a_n$ = series
+    * Sequences are lists of the terms, whereas series are lists of the addends of the series.
+      * Series: to find the second term of the series, you ADD $𝑎_1$ and $𝑎_2$ together
+        * This is known as partial sums → 2nd partial sum = $S_2 = a_1 + a_2$
+          * nth partial sum = the sum of the first n terms of the series
+      *** Convergence/Divergence:**
+        * A series converges only if the SEQUENCE OF PARTIAL SUMS (𝑆𝑛) converges to a finite number. If it doesn’t converge, then it diverges.
+          * {{:pasted:20240410-225330.png}}
+  * **Geometric series:**
+    * Defined as a series with a constant ratio between each term
+    * {{:pasted:20240410-225416.png}}
+    * If |r| < 1, then the geometric series converges to $|frac{a}{1-4}$
+      * {{:pasted:20240410-225457.png}}
+    * If |𝑟| > 1, 𝑜𝑟 = 1,then the series diverges
+      * {{:pasted:20240410-230213.png}}
+    * **The nth term test for divergence:**
+      * Used to determine whether or not a function DIVERGES; it CANNOT tell you if a function converges
+      * {{:pasted:20240410-230240.png}}
+      * {{:pasted:20240410-230249.png}}
+    * **Integral test for convergence:**
+      * {{:pasted:20240410-230305.png}}
+    * **Harmonic series and p series:**
+      * {{:pasted:20240410-230346.png}}
+      * {{:pasted:20240410-230356.png}}
+      * Alternating series for convergence:
+        * This is used to test a series that alternates from negative↔positive
+          * {{:pasted:20240410-230419.png}}
+        * {{:pasted:20240410-230450.png}}
+      * **Ratio Test for convergence:**
+      * {{:pasted:20240410-230518.png}}
+      * {{:pasted:20240410-230528.png}}
+      * Absolute convergence:
+        * If an alternating series converges after taking the absolute value of the series function, then that series is said to converge absolutely
+          * {{:pasted:20240410-230626.png}}
+        * {{:pasted:20240410-230632.png}}
+      * **Conditional convergence:**
+        * If an alternating series converges, and does NOT converge absolutely, then it’s said to converge conditionally
+        * An alternating series that conditionally converges must follow these conditions:
+          * {{:pasted:20240410-230713.png}}
+      * **Alternating Series Error Bound:**
+        * {{:pasted:20240410-230724.png}}
+        * {{:pasted:20240410-230745.png}}
+    * **Taylor polynomials:**
+      * {{:pasted:20240410-230802.png}}
+      * {{:pasted:20240410-230821.png}}
+    * **Lagrange Error Bound**
+      * {{:pasted:20240410-230842.png}}
+      * {{:pasted:20240410-230900.png}}
+    * **Radius and interval of convergence of power series:**
+    * {{:pasted:20240410-230929.png}}
+            * You have to check the endpoints of the interval to see if the function converges at the endpoints! If it does converge at an endpoint, place the interval in brackets, and if not, place the interval in parentheses.
+              * If it converges at 1 endpoint and not the other, denote it like this: [a,b) or (a,b], with the bracket indicating convergence at that endpoint
+          * **Radius of convergence: **
+            * Similar to how the radius of a circle describes the length from the outer edge of the circle to the centerpoint, the radius of convergence describes the distance between the midpoint and the endpoint of the interval of convergence.
+            * In the example above, the interval of convergence was [-1,1]. Thus, the radius of convergence is 1, since the midpoint of that interval is 0, and the distance between 0 (the midpoint) and 1 (the endpoint) is 1.
+    * **Representing functions as Power series:**
+      * Differentiation of power series
+        * {{:pasted:20240410-231109.png}}
+      * **Substitution of power series**
+      * {{:pasted:20240410-231136.png}}