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qwertyuiopasdfghjklzxcvbnmqw ertyuiopasdfghjklzxcvbnmqwert yuiopasdfghjklzxcvbnmqwertyui opasdfghjklzxcvbnmqwertyuiopa Calculus BC Review Book sdfghjklzxcvbnmqwertyuiopasdf Chapters 8 through 10 ghjklzxcvbnmqwertyuiopasdfghj klzxcvbnmqwertyuiopasdfghjklz xcvbnmqwertyuiopasdfghjklzxcv bnmqwertyuiopasdfghjklzxcvbn mqwertyuiopasdfghjklzxcvbnmq wertyuiopasdfghjklzxcvbnmqwe rtyuiopasdfghjklzxcvbnmqwerty uiopasdfghjklzxcvbnmqwertyuio pasdfghjklzxcvbnmqwertyuiopas dfghjklzxcvbnmqwertyuiopasdfg hjklzxcvbnmqwertyuiopasdfghjk 5/20/2013 Candice Johnson & Destini Torres About the AUTHOR Destini Torres was born and raised in New York City. She is going to college in the fall of 2013. Candice Johnson was born in New York City and raised in the Bronx. She is going to college in the fall of 2013. TABLE of CONTENTS Chapter Eight Page 2 through three Integration techniques L’ Hopital’s Rule Improper Integrals Chapter Nine Page three through seven Infinite Series (Converges verses Diverges) Chapter Ten Page eight through ten Parametric Equations Polar coordinates 1 Chapter 8: IntegratIon technIques, l’hopItal’s rule, and improper integrals Basic Integration and Differentiation Rules Here is a list of integrals and derivatives that you should know: Using Integration by Parts to Solve Integrals To use integration by parts, follow the formula: Identify u, v, du and dv within the integral. Plug in the vales and evaluate. Example: u= x du= 1 dx dv= ex dx v= ex 2 Integrating Improper Integrals Important Steps: 1. 2. 3. 4. 5. Set a limit approaching to the upper end of the integral (for example, if the integral is from 0 to infinity, set your limit as t approaching infinity). Integrate your integrand as normal. Substitute your upper limit and t. Evaluate the limit for your answer. If you get a constant L, the integral converges. If your answer is infinity, the integral diverges. Example: L’ Hopital Rule 0 ∞ 0 ∞ It is quite simple. Now, anytime a limit has a fraction of and just take the derivative of the top function alone and the bottom function. (DO NOT DO THE QUOTIENT RULE) EXAMPLE#1 EXAMPLE#2 = = = = = 4(-1) = -4 = 1 ∞ =0 3 CHAPTER 9: Infinite series 9.1- Sequences Theorem 9.1: If { } is a sequence such that f(n)= f then or every positive integer n, Theorem 9.2: Properties of Limits of Sequences and c is any real number 1. 2. 3. 4. Theorem 9.4: Absolute Value Theorem If , then EXAMPLE 1, 4, 7, 10 The expression is 3n-2 . lim 3𝑛 − 2 = ∞ 𝑛→∞ DIVERGES 9.2- Convergent and Divergent Series Sn = a1 + a2 +…+ an If the sequence of partial sums {Sn} converges to S, then the series converges. (the nth partial sum). If {Sn} diverges, then the series diverges. (sum of the series) Theorem 9.6: Convergence of a Geometric Series Diverges: Converges: 0 EXAMPLE a= 3 r= 1/2 1 ≥1 = 3( 2 ) 1 4 Theorem 9.7: Properties of Infinite Series If and then… 1. 2. 3. Theorem 9.8 & Theorem 9.9: Limit of nth of a Convergent and a Divergent Series If converges, then If , then . . 9.3- Integral Test and P-series Test Theorem 9.10: Integral Test If f is positive, continuous, and decreasing for x ≥ 1 and either both converge or both diverge. Theorem 9.11: Convergence of p-series = 1. converges: p › 1 2. diverges: 0 ‹ p ≤ 1 5 EXAMPLE 9.4- Direct Comparison Test and Limit Comparison Test Theorem 9.12: Direct Comparison Test If converges, then resembles converges = If diverges, then diverges. =n≥1 ≤ DIVERGENT Theorem 9.13: Limit Comparison Test Suppose that › 0, › 0, and Where L is finite and positive, then the two series and either both converge or both diverge. 9.5 Alternating Series Test If one or none of the conditions are met, then the series DIVERGES! 9.6 Ratio Test 6 9.7 Taylor Polynomials and Approximations 9.10 Taylor and Maclaurin Series 7 CHAPTER 10: parametric equations and polar coordinates What are parametric functions? If f and g are continuous functions of t on an interval t, then the equations: 𝒙 = 𝒇(𝒕) 𝒂𝒏𝒅 𝒚 = 𝒈(𝒕) are called parametric functions. A set of parametric equations on a graph is known as a plane curve, or C. Examples: x=2t3+2 y=4t2 x=3t2+ 5t y=t4+7t2+9 How do you find the derivative of a parametric function? The slope of the curve C at point (x,y) is dy = dy/dt dx dx/dt when dx/dt ≠ 0 This means that we must take the derivative of x=f(t) and y=g(t) with respect to t. Examples: 1. What is the derivative of the set of parametric equations, x=2t3+2 and y=4t2 dy = dy/dt = 8t dx dx/dt 6t2 2. Find dy/dx for the set of parametric equations, x=3t+5 and y=t2 dy = dy/dt = 2t dx dx/dt 3 8 How do you find the second derivative of a parametric function? The second derivative of the curve C at point (x,y) is given by d2y = d/dt [dy/dx] dx2 dx/dt We must take the derivative of dy/dx and the derivative of x=f(t). Example: 1. Find d2y/dx2 for the set of parametric functions, x=3t+5 and y=t2 d2y = d/dt [dy/dx] = d/dt [2t/3] = 2/3 = 2 dx2 dx/dt 3 3 How do we find the tangent line to a parametric curve? 1. 2. 3. 4. Find dy/dx for the set of parametric equations Evaluate dy/dx at the given t value Find your x and y values by plugging t into your parametric equations separately Use the point slope formula or equation formula to find the tangent line Example: Find the equation of the tangent line at the point x=t2+ 1 and y=t3+6t+5 at t=2 1. dy = dy/dt = 3t2+6 dx dx/dt 2t 3. x=t2+1 x=22+1 x=5 y=t3+6t+5 y=23+6(2)+5 y=25 2. dy = 3(2)2 + 6 = 18 = 9 dx 2(2) 4 4. 2 y-y1= m(x-x1) y-25=9/2 (x-5) y=9x – 95 2 2 How do we find the arc length of a parametric curve? To find arc length, use the formula: s= dt Example: Find the arc length of the curve, x=t2 and y=6t on the interval, 0≤t≤1 S= dt = 6.109 9 How do we graph a polar equation? To graph a polar equation, use the POL mode on your calculator: Go to mode Pol enter (adjust the window to fit the graph) Example: r= 5cos(Ө) How do we find the area of a polar equation? The area of a polar graph is given by: Where α= alpha, β=beta, and r=polar equation Example: Find the area of the polar equation r=4sinӨ on the interval 0 ≤ Ө ≤ π 10 1. Ms. Zhao.com: “AP Calculus AB.” AP Calculus AB. N.p.,n.d. Web. 18 May 2013 2. Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus of a Single Variable. Boston: Houghton Mifflin, 2006. Print. 3. Paul’s Online Math Notes: “Paul’s Online Notes: Calculus 2- Parametric Equations and Polar Coordinates. “Paul’s Online Notes: Calculus 2- Parametric Equations and Polar Coordinates. N.p., n.d. Web. 21 May 2013. 11 12