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ap_calculus_bc [2024/04/10 20:14] – mrdoughap_calculus_bc [2024/05/12 23:55] (current) – [AP Calculus BC Study Guide] 172.88.72.108
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 ======AP Calculus BC Study Guide====== ======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]] 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]]   * 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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 Example: Example:
  
-[[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}}] +[[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 ======+====== Unit 3 - Differentiation: Composite, Implicit, and InverseFunctions ======
  
 **The Chain Rule** **The Chain Rule**
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   * Example of what a slope field looks like   * Example of what a slope field looks like
  
-[[AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_73|{{AP_Calc_AB_Study_Guide_f289051ba2044b4ab01d25945296aa61:Untitled_73.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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 [[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 [[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
 +
 +======Unit 9:  Defining and Differentiating Parametric Equations======
 +  * **Parametric** functions: a pair of functions that allow you to find the coordinates of both x
 +and y, usually written in terms of ‘t’
 +    * Parametric equations are usually written in terms of ‘t’.
 +  * Calculating the derivative of parametric functions:
 +{{:pasted:20240410-222418.png}}
 +{{:pasted:20240410-222424.png}}
 +  * Calculating the second derivative of parametric equations:
 +{{:pasted:20240410-222441.png}}
 +    * Using the example from above, to find the second derivative of that parametric function, take the derivative of $\frac{dy}{dx} \rightarrow \frac{d^2y}{dx^2}=\frac{(3/2)}{2t}=\frac{3}{4t}$
 +  * Finding the arc length of parametric curves:
 +    * {{:pasted:20240410-222754.png}}
 +      * From t=𝛼 to t=𝛽
 +    * Ex. (from Princeton book): Find the length of the curve defined by x=sin(t) and
 +y=cos(t), from t=0 to t=$\pi$
 +      * 1) find (dx/dt) and (dy/dt) → $\frac{dx}{dt}=cos(t)$ and $\frac{dy}{dt}=-sin(t)$
 +      * 2) plug into the formula →
 +      * {{:pasted:20240410-222957.png}}
 +  * **Vectors**
 +    * Vectors are quantities that have both a magnitude and direction
 +    * Vector functions are similar to parametric functions because they describe the
 +motion of a particle on a plane
 +    * Vector functions have similar relationships between position, velocity, and
 +acceleration as we’ve seen earlier.
 +      * If a vector function P = position of a particle, then the derivative of P gives
 +you velocity, and the derivative of velocity gives you the particle’s
 +acceleration
 +        * Both x(t) and y(t) have to be differentiable for the vector function
 +to be differentiable as well
 +        * To find speed of a vector function, take the magnitude of the
 +velocity vector function
 +          * The magnitude of a vector function = $\sqrt{x^2+y^2}$
 +    * Denotation of vectors:
 +    * {{:pasted:20240410-223144.png}}
 +      * ‘i’ and ‘j’ are the unit vectors in the x and y directions
 +      * x(t) and y(t) are real valued functions of the variable ‘t’
 +    * Ex. (from Saxon Calc BC book): 
 +    * {{:pasted:20240410-223210.png}}
 +    * Integrating vector valued functions:
 +      * Similar to how you take the derivative of vector valued functions by
 +differentiating x(t) and y(t), and leaving the ‘i’ and ‘j’, to find the integral of vector-valued functions, integrate the real-valued functions and leave
 +the ‘i’ and ‘j’
 +      * Integrating a velocity vector gives you the displacement of the particle
 +over the interval of time, and integrating a speed vector gives you the total
 +distance travelled by the particle in that interval.
 +      * Ex.:
 +      * {{:pasted:20240410-223246.png}}
 +  * **Polar curves:**
 +    * Polar coordinates: used to describe the location of points on a polar curve/graph
 +      * x = rcos𝜃and y = rsin𝜃
 +        * *x and y are defined parametrically, in terms of ‘𝜃′
 +      * Differentiating in polar form:
 +        * To find $\frac{dy}{dx}$, use the equation for finding the slope of a parametric equation → $\frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta}$
 +        * {{:pasted:20240410-223450.png}}
 +     * **Finding the area of a single polar curve:**
 +     * {{:pasted:20240410-223518.png}}
 +       * {{:pasted:20240410-223537.png}}
 +         * *to find polar area, you will often need to know these trig
 +identities:
 +           * {{:pasted:20240410-223559.png}}
 +         * To find the limits of the integral, find the𝜃points where r consists of that region.
 +           * In this case, it takes from 0 to 2𝜋for “r” to be drawn completely, which is why those are the limits
 +     * Finding the area of a region bounded by 2 polar curves:
 +       * A = $\int_a^b \frac{1}{2}(r_2^2-r_1^2)d\theta$ with $r_2(\theta) > r_1(\theta)$ 
 +       * Ex. (from Princeton book): find the area inside the circle r = 4 and outside of the curve r = 4 - cos𝜃(the shaded region in the graph below)
 +       * {{:pasted:20240410-223811.png}}
 +       * 1) To find the limits of the integral, set the two equations equal to each other, and solve for 𝜃→ 4 = 4 - cos𝜃 = cos𝜃 = 0 at 𝜃= -𝜋/2 and 𝜋/2
 +       * 2) find which function is on the “outside”, and in this case, r = 4 is
 +         * {{:pasted:20240410-224911.png}}
 +       * 3) integrate
 +   * **Connecting polar, vector, and parametric functions:**
 +     * A function can be written in parametric, rectangular/Cartesian, or polar form
 +     * Ex. (on the next page):
 +     * {{:pasted:20240410-224947.png}}
 +======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}}
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