AP Calculus AB/BC Formula and Concept Cheat Sheet, Cheat Sheet of Calculus

Download AP Calculus AB/BC Formula and Concept Cheat Sheet and more Cheat Sheet Calculus in PDF only on Docsity! AP Calculus AB/BC Formula and Concept Cheat Sheet Limit of a Continuous Function If f(x) is a continuous function for all real numbers, then lim 𝑥→𝑐 𝑓(𝑥) = 𝑓(𝑐) Limits of Rational Functions A. If f(x) is a rational function given by 𝑓(𝑥) = 𝑝(𝑥) 𝑞(𝑥) ,such that 𝑝(𝑥) and 𝑞(𝑥) have no common factors, and c is a real number such that 𝑞(𝑐) = 0, then I. lim 𝑥→𝑐 𝑓(𝑥) does not exist II. lim 𝑥→𝑐 𝑓(𝑥) = ±∞ x = c is a vertical asymptote B. If f(x) is a rational function given by 𝑓(𝑥) = 𝑝(𝑥) 𝑞(𝑥) , such that reducing a common factor between 𝑝(𝑥) and 𝑞(𝑥) results in the agreeable function k(x), then lim 𝑥→𝑐 𝑓(𝑥) = lim 𝑥→𝑐 𝑝(𝑥) 𝑞(𝑥) = lim 𝑥→𝑐 𝑘(𝑥) = 𝑘(𝑐) Hole at the point (𝑐, 𝑘(𝑐)) Limits of a Function as x Approaches Infinity If f(x) is a rational function given by (𝑥) = 𝑝(𝑥) 𝑞(𝑥) , such that 𝑝(𝑥) and 𝑞(𝑥) are both polynomial functions, then A. If the degree of p(x) > q(x), lim 𝑥→∞ 𝑓(𝑥) = ∞ B. If the degree of p(x) < q(x), lim 𝑥→∞ 𝑓(𝑥) = 0 y = 0 is a horizontal asymptote C. If the degree of p(x) = q(x), lim 𝑥→∞ 𝑓(𝑥) = 𝑐, where c is the ratio of the leading coefficients. y = c is a horizontal asymptote Special Trig Limits A. lim 𝑥→0 sin 𝑎𝑥 𝑎𝑥 = 1 B. lim 𝑥→0 𝑎𝑥 sin 𝑎𝑥 = 1 C. lim 𝑥→0 1−cos 𝑎𝑥 𝑎𝑥 = 0 L’Hospital’s Rule If results lim 𝑥→𝑐 𝑓(𝑥) or lim 𝑥→∞ 𝑓(𝑥) results in an indeterminate form ( 0 0 , ∞ ∞ , ∞ − ∞ , 0 ∙ ∞ , 00 , 1∞ , ∞0) , and 𝑓(𝑥) = 𝑝(𝑥) 𝑞(𝑥) , then lim 𝑥→𝑐 𝑓(𝑥) = lim 𝑥→𝑐 𝑝(𝑥) 𝑞(𝑥) = lim 𝑥→𝑐 𝑝′(𝑥) 𝑞′(𝑥) and lim 𝑥→∞ 𝑓(𝑥) = lim 𝑥→∞ 𝑝(𝑥) 𝑞(𝑥) = lim 𝑥→∞ 𝑝′(𝑥) 𝑞′(𝑥) The Definition of Continuity A function 𝑓(𝑥) is continuous at c if I. lim 𝑥→𝑐 𝑓(𝑥) exists II. 𝑓(𝑐) exists III. lim 𝑥→𝑐 𝑓(𝑥) = 𝑓(𝑐) Types of Discontinuities Removable Discontinuities (Holes) I. lim 𝑥→𝑐 𝑓(𝑥) = 𝐿 (the limit exists) II. 𝑓(𝑐) is undefined Non-Removable Discontinuities (Jumps and Asymptotes) A. Jumps lim 𝑥→𝑐 𝑓(𝑥) = 𝐷𝑁𝐸 because lim 𝑥→𝑐− 𝑓(𝑥) ≠ lim 𝑥→𝑐+ 𝑓(𝑥) B. Asymptotes (Infinite Discontinuities) lim 𝑥→𝑐 𝑓(𝑥) = ±∞ Derivatives of Exponential and Logarithmic Functions Explicit and Implicit Differentiation A. Explicit Functions: Function y is written only in terms of the variable x (𝑦 = 𝑓(𝑥)). Apply derivatives rules normally. B. Implicit Differentiation: An expression representing the graph of a curve in terms of both variables x and y. I. Differentiate both sides of the equation with respect to x. (terms with x differentiate normally, terms with y are multiplied by 𝑑𝑦 𝑑𝑥 per the chain rule) II. Group all terms with 𝑑𝑦 𝑑𝑥 on one side of the equation and all other terms on the other side of the equation. III. Factor 𝑑𝑦 𝑑𝑥 and express 𝑑𝑦 𝑑𝑥 in terms of x and y. Tangent Lines and Normal Lines A. The equation of the tangent line at a point (𝑎, 𝑓(𝑎)): 𝑦 − 𝑓(𝑎) = 𝑓′(𝑎)(𝑥 − 𝑎) B. The equation of the normal line at a point (𝑎, 𝑓(𝑎)): 𝑦 − 𝑓(𝑎) = − 1 𝑓′(𝑎) (𝑥 − 𝑎) Mean Value Theorem for Derivatives If the function f is continuous on the close interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c between a and b such that 𝑓′(𝑐) = 𝑓(𝑏)−𝑓(𝑎) 𝑏−𝑎 The slope of the tangent line is equal to the slope of the secant line. Rolle’s Theorem (Special Case of Mean Value Theorem) If the function f is continuous on the close interval [a, b] and differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one number c between a and b such that 𝑓′(𝑐) = 𝑓(𝑏)−𝑓(𝑎) 𝑏−𝑎 = 0 Particle Motion A velocity function is found by taking the derivative of position. An acceleration function is found by taking the derivative of a velocity function. 𝑥(𝑡) Position 𝑥′(𝑡) = 𝑣(𝑡) Velocity * |𝑣(𝑡)| = 𝑠𝑝𝑒𝑒𝑑 𝑥′′(𝑡) = 𝑣′(𝑡) = 𝑎(𝑡) Accleration Rules: A. If velocity is positive, the particle is moving right or up. If velocity is negative, the particle is moving left or down. B. If velocity and acceleration have the same sign, the particle speed is increasing. If velocity and acceleration have opposite signs, speed is decreasing. C. If velocity is zero and the sign of velocity changes, the particle changes direction. Related Rates A. Identify the known variables, including their rates of change and the rate of change that is to be found. Construct an equation relating the quantities whose rates of change are known and the rate of change to be found. B. Implicitly differentiate both sides of the equation with respect to time. (Remember: DO NOT substitute the value of a variable that changes throughout the situation before you differentiate. If the value is constant, you can substitute it into the equation to simplify the derivative calculation). C. Substitute the known rates of change and the known values of the variables into the equation. Then solve for the required rate of change. *Keep in mind, the variables present can be related in different ways which often involves the use of similar geometric shapes, Pythagorean Theorem, etc. Extrema of a Function A. Absolute Extrema: An absolute maximum is the highest y – value of a function on a given interval or across the entire domain. An absolute minimum is the lowest y – value of a function on a given interval or across the entire domain. B. Relative Extrema I. Relative Maximum: The y-value of a function where the graph of the function changes from increasing to decreasing. Another way to define a relative maximum is the y-value where derivative of a function changes from positive to negative. II. Relative Minimum: The y-value of a function where the graph of the function changes from decreasing to increasing. Another way to define a relative maximum is the y-value where derivative of a function changes from negative to positive. Critical Value When f(c) is defined, if f ‘ (c) = 0 or f ‘ is undefined at x = c, the values of the x – coordinate at those points are called critical values. *If f(x) has a relative extrema at x = c, then c is a critical value of f. Extreme Value Theorem If the function f continuous on the closed interval [a, b], then the absolute extrema of the function f on the closed interval will occur at the endpoints or critical values of f. *After identifying critical values, create a table with endpoints and critical values. Calculate the y – value at each of these x values to identify the extrema. Antiderivatives If 𝐹′(𝑥) = 𝑓(𝑥) for all x, 𝐹(𝑥) is an antiderivative of f. ∫ 𝑓(𝑥) = 𝐹(𝑥) + 𝐶 * The antiderivative is also called the Indefinite Integral Basic Integration Rules Let k be a constant. Definite Integrals (The Fundamental Theorem of Calculus) A definite integral is an integral with upper and lower limits, a and b, respectively, that define a specific interval on the graph. A definite integral is used to find the area bounded by the curve and an axis on the specified interval (a, b). If 𝐹(𝑥) is the antiderivative of a continuous function 𝑓(𝑥), the evaluation of the definite integral to calculate the area on the specified interval (a, b) is the First Fundamental Theorem of Calculus: ∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎) 𝑏 𝑎 Integration Rules for Definite Integrals *This means that c is a value of x, lying between a and b Riemann Sum (Approximations) A Riemann Sum is the use of geometric shapes (rectangles and trapezoids) to approximate the area under a curve, therefore approximating the value of a definite integral. If the interval [a, b] is partitioned into n subintervals, then each subinterval, Δx, has a width: ∆𝑥 = 𝑏−𝑎 𝑛 . Therefore, you find the sum of the geometric shapes, which approximates the area by the following formulas: A. Right Riemann Sum 𝐴𝑟𝑒𝑎 ≈ ∆𝑥 [𝑓(𝑥0) + 𝑓(𝑥1) + 𝑓(𝑥2) + ⋯ + 𝑓(𝑥𝑛−1)] B. Left Riemann Sum 𝐴𝑟𝑒𝑎 ≈ ∆𝑥 [𝑓(𝑥1) + 𝑓(𝑥2) + 𝑓(𝑥3) + ⋯ + 𝑓(𝑥𝑛)] C. Midpoint Riemann Sum 𝐴𝑟𝑒𝑎 ≈ ∆𝑥 [𝑓(𝑥1/2) + 𝑓(𝑥3/2) + 𝑓(𝑥5/2) + ⋯ + 𝑓(𝑥(2𝑛−1)/2)] D. Trapezoidal Sum 𝐴𝑟𝑒𝑎 ≈ 1 2 ∆𝑥 [𝑓(𝑥0) + 2 𝑓(𝑥1) + 2 𝑓(𝑥2) + ⋯ + 2𝑓(𝑥𝑛−1) + 𝑓(𝑥𝑛)] Properties of Riemann Sums A. The area under the curve is under approximated when I.A Left Riemann sum is used on an increasing function. II. A Right Riemann sum is used on a decreasing function. III. A Trapezoidal sum is used on a concave down function. B. The area under the curve is over approximated when I.A Left Riemann sum is used on a decreasing function. II. A Right Riemann sum is used on an increasing function. III. A Trapezoidal sum is used on a concave up function. Riemann Sum (Limit Definition of Area) Let f be a continuous function on the interval [a, b]. The area of the region bounded by the graph of the function f and the x – axis (i.e. the value of the definite integral) can be found using ∫ 𝑓(𝑥)𝑑𝑥 𝑏 𝑎 = lim 𝑛→∞ ∑ 𝑓(𝑐𝑖) ∆𝑥 𝑛 𝑖=1 Where 𝑐𝑖 is either the left endpoint (𝑐𝑖 = 𝑎 + (𝑖 − 1)∆𝑥) or right endpoint (𝑐𝑖 = 𝑎 + 𝑖∆𝑥) and ∆𝑥 = (𝑏 − 𝑎)/𝑛. Average Value of a Function If a function f is continuous on the interval [a, b], the average value of that function f is given by 1 𝑏 − 𝑎 ∫ 𝑓(𝑥)𝑑𝑥 𝑏 𝑎 Second Fundamental Theorem of Calculus If a function f is continuous on the interval [a, b], let 𝑢 represent a function of x, then 𝐀. 𝑑 𝑑𝑥 [∫ 𝑓(𝑡)𝑑𝑡 𝑥 𝑎 ] = 𝑓(𝑥) 𝐁. 𝑑 𝑑𝑥 [∫ 𝑓(𝑡)𝑑𝑡 𝑏 𝑥 ] = −𝑓(𝑥) 𝐂. 𝑑 𝑑𝑥 [∫ 𝑓(𝑡)𝑑𝑡 𝑢(𝑥) 𝑎 ] = 𝑓(𝑢(𝑥)) ∙ 𝑢′(𝑥) Integration of Exponential and Logarithmic Formulas BC Only: Improper Integrals An improper integral is characterized by having a limits of integration that is infinite or the function f having an infinite discontinuity (asymptote) on the interval [a, b]. A. Infinite Upper Limit (continuous function) ∫ 𝑓(𝑥)𝑑𝑥 ∞ 𝑎 = lim 𝑏→∞ ∫ 𝑓(𝑥)𝑑𝑥 𝑏 𝑎 B. Infinite Lower Limit (continuous function) ∫ 𝑓(𝑥)𝑑𝑥 𝑏 −∞ = lim 𝑎→−∞ ∫ 𝑓(𝑥)𝑑𝑥 𝑏 𝑎 C. Both Infinite Limits (continuous function) ∫ 𝑓(𝑥)𝑑𝑥 ∞ −∞ = lim 𝑎→−∞ ∫ 𝑓(𝑥)𝑑𝑥 𝑐 𝑎 + lim 𝑏→∞ ∫ 𝑓(𝑥)𝑑𝑥 𝑏 𝑐 , where 𝑐 is an 𝑥 value anywhere on 𝑓. D. Infinite Discontinuity (Let x = k represent an infinite discontinuity on [a, b]) ∫ 𝑓(𝑥)𝑑𝑥 𝑏 𝑎 = lim 𝑥→𝑘− ∫ 𝑓(𝑥)𝑑𝑥 𝑘 𝑎 + lim 𝑥→𝑘+ ∫ 𝑓(𝑥)𝑑𝑥 𝑏 𝑘 BC Only: Arc Length (Length of a Curve) A. If the function 𝑦 = 𝑓(𝑥)is a differentiable function, then the length of the arc on [a, b] is ∫ √1 + [𝑓′(𝑥)]2 𝑏 𝑎 𝑑𝑥 B. If the function 𝑥 = 𝑓(𝑦)is a differentiable function, then the length of the arc on [a, b] is ∫ √1 + [𝑓′(𝑦)]2 𝑏 𝑎 𝑑𝑦 C. Parametric Arc Length: If a smooth curve is given by x(t) and y(t), then the arc length over the interval 𝑎 ≤ 𝑡 ≤ 𝑏 is ∫ √( 𝑑𝑥 𝑑𝑡 ) 2 + ( 𝑑𝑦 𝑑𝑡 ) 2𝑏 𝑎 𝑑𝑡 BC Only: Logistic Growth A population, P, that experiences a limit factor in the growth of the population based upon the available resources to support the population is said to experience logistic growth. A. Differential Equation: 𝑑𝑃 𝑑𝑡 = 𝑘𝑃 (1 − 𝑃 𝐿 ) B. General Solution: 𝑃(𝑡) = 𝐿 1+𝑏𝑒−𝑘𝑡 𝑃 = population 𝑘 = constant growth factor 𝐿 = carrying capacity 𝑡 = time, 𝑏 = constant (found with intital condition) Graph Exponential Growth and Decay When the rate of change of a variable y is directly proportional to the value of y, the function y = f(x) is said to grow/decay exponentially. A. Differential Equation for rate of change: 𝑑𝑦 𝑑𝑡 = 𝑘𝑦 B. General Solution: 𝑦 = 𝐶𝑒𝑘𝑡 I. If k > 0, then exponential growth occurs. II. If k < 0, then exponential decay occurs. Characteristics of Logistics I. The population is growing the fastest where 𝑃 = 𝐿 2 II. The point where 𝑃 = 𝐿 2 represents a point of inflection III. lim 𝑡→∞ 𝑃(𝑡) = 𝐿 Area Between Two Curves A. Let 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑔(𝑥)represent two functions such that 𝑓(𝑥) ≥ 𝑔(𝑥)(meaning the function f is always above the function g on the graph) for every x on the interval [a, b]. Area Between Curves = ∫ [𝑓(𝑥) − 𝑔(𝑥)] 𝑏 𝑎 𝑑𝑥 B. Let 𝑥 = 𝑓(𝑦) and 𝑥 = 𝑔(𝑦)represent two functions such that 𝑓(𝑦) ≥ 𝑔(𝑦)(meaning the function f is always to the right of the function g on the graph) for every y on the interval [a, b]. Area Between Curves = ∫ [𝑓(𝑦) − 𝑔(𝑦)] 𝑏 𝑎 𝑑𝑦 Volumes of a Solid of Revolution: Disk Method If a defined region, bounded by a differentiable function f, on a graph is rotated about a line, the resulting solid is called a solid of revolution and the line is called the axis of revolution. The disk method is used when the defined region boarders the axis of revolution over the entire interval [a, b] A. Revolving around the x – axis Volume = 𝜋 ∫ (𝑓(𝑥)) 2 𝑑𝑥 𝑏 𝑎 B. Revolving around the y – axis Volume = 𝜋 ∫ (𝑓(𝑦)) 2 𝑑𝑦 𝑏 𝑎 C. Revolving around a horizontal line y = k Volume = 𝜋 ∫ (𝑓(𝑥) − 𝑘)2𝑑𝑥 𝑏 𝑎 D. Revolving around a vertical line x = m Volume = 𝜋 ∫ (𝑓(𝑦) − 𝑚)2𝑑𝑦 𝑏 𝑎 Differential Equations A differential equation is an equation involving an unknown function and one or more of its derivatives 𝑑𝑦 𝑑𝑥 = 𝑓(𝑥, 𝑦) Usually expressed as a derivative equal to an expression in terms of x and/or y. To solve differential equations, use the technique of separation of variables. Given the differential equation 𝑑𝑦 𝑑𝑥 = 𝑥𝑦 (𝑥2+1) Step 1: Separate the variables, putting all y’s on one side, with dy in the numerator, and all x’s on the other side, with dx in the numerator. 1 𝑦 𝑑𝑦 = 𝑥 (𝑥2 + 1) 𝑑𝑥 Step 2: Integrate both sides of the equation. ln|𝑦| = 1 2 ln √𝑥2 + 1 + 𝐶 Step 3: Solve the equation for y. 𝑦 = 𝐶√𝑥2 + 1 Given the differential equation 𝑑𝑦 𝑑𝑥 = 2𝑥2 with the initial condition 𝑦(3) = 10. A. The general solution to a differential equation is left with the constant of integration, C, undefined. 𝑑𝑦 = 2𝑥2 𝑑𝑥 → ∫ 𝑑𝑦 = ∫ 2𝑥2 𝑑𝑥 → 𝑦 = 2 3 𝑥3 + 𝐶 B. The particular solution uses the given initial condition to calculate the value of C. 10 = 2 3 (3)3 + 𝐶 → 𝐶 = −8 → 𝑦 = 2 3 𝑥3 − 8 BC Only: Euler’s Method for Approximating the Solution of a Differential Equation Euler’s method uses a linear approximation with increments (steps), h, for approximating the solution to a given differential equation, 𝑑𝑦 𝑑𝑥 = 𝐹(𝑥, 𝑦), with a given initial value. Process: Initial value (𝑥0, 𝑦0) 𝑥1 = 𝑥0 + ℎ 𝑦1 = 𝑦0 + ℎ ∙ 𝐹(𝑥0, 𝑦0) 𝑥2 = 𝑥1 + ℎ 𝑦2 = 𝑦1 + ℎ ∙ 𝐹(𝑥1, 𝑦1) 𝑥3 = 𝑥2 + ℎ 𝑦3 = 𝑦2 + ℎ ∙ 𝐹(𝑥2, 𝑦2) * This process repeats until the desired y – value is given. Slope Field The derivative of a function gives the value of the slope of the function at each point (x, y). A slope field is a graphical representation of all of the possible solutions to a given differential equation. The slope field is generated by plugging in the coordinates of every point (x, y) into the differential equation and drawing a small segment of the tangent line at each point. Given the differential equation 𝑑𝑦 𝑑𝑥 = 𝑥 𝑦 𝑑𝑦 𝑑𝑥 | (0,0) = 0 0 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑑𝑦 𝑑𝑥 | (0,±1) = 0 𝑑𝑦 𝑑𝑥 | (1,2) = 1 2 BC Only: Testing for Convergence/Divergence of a Series  Sequence of Partial Sums Given the series ∑ 𝑎𝑛 = 𝑎1 + 𝑎2 + 𝑎3 + ⋯ The sequence of partial sums for the series is 𝑆1 = 𝑎1 𝑆2 = 𝑎1 + 𝑎2 𝑆3 = 𝑎1 + 𝑎2 + 𝑎3 … 𝑆𝑛 = 𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛 If lim 𝑛→∞ 𝑆𝑛 = 𝑆, then ∑ 𝑎𝑛 converges to 𝑆.  Nth Term If the terms of a sequence do not converge to 0, then the series must diverge. 𝐈. If lim 𝑛→∞ 𝑎𝑛 ≠ 0, then ∑ 𝑎𝑛 diverges. 𝐈𝐈. If lim 𝑛→∞ 𝑎𝑛 = 0, then the test is inconclusive.  P – Series The form of a p – series is ∑ 1 𝑛𝑝 𝐈. If 𝑝 > 1, then the series converges. 𝐈𝐈. If 𝑝 < 1, then the series diverges. *These are only three example points. You would do this for every point in the given region of the graph.  Geometric Series A geometric series is any series of the form ∑ 𝑎𝑟𝑛 ∞ 𝑛=0 𝐈. If |𝑟| < 1, then the series converges to 𝑎 1−𝑟 *Series must be indexed at n = 0 𝐈𝐈. If |𝑟| > 1, then the series diverges.  Telescoping Series A telescoping series is any series of the form ∑ 𝑎𝑛 − 𝑎𝑛+1 *Convergence and divergence is found using a sequence of partial sums *Partial decomposition may be used to break a single rational series into the difference of two series that form the telescoping series.  Integral If f is positive, continuous, and decreasing for 𝑥 ≥ 1, then ∑ 𝑎𝑛 and ∫ 𝑓(𝑥)𝑑𝑥 ∞ 1 ∞ 𝑛=1 either both converge or both diverge.  Alternating Series A series, containing both positive terms, negative terms, and 𝑎𝑛 > 0, of the form ∑(−1)𝑛 𝑎𝑛 ∞ 𝑛=1 or ∑(−1)𝑛+1 𝑎𝑛 ∞ 𝑛=1 The series’ converge if both of the following conditions are met I. 𝑎𝑛+1 ≤ 𝑎𝑛 for all n II. lim 𝑛→∞ 𝑎𝑛 = 0  Direct Comparison When comparing two series, if 𝑎𝑛 ≤ 𝑏𝑛 for all n, 𝐈. If ∑ 𝑎𝑛 diverges, then ∑ 𝑏𝑛 diverges. 𝐈𝐈. If ∑ 𝑏𝑛 converges, then ∑ 𝑎𝑛 converges. *The convergence or divergence of the series chosen for comparison should be known