CALCULUS REFERENCE 3/18/03 10:49 AM Page SPARKCHARTSTM CALCULUS REFERENCE SPARK CHARTS TM THEORY SPARKCHARTS TM DERIVATIVES AND DIFFERENTIATION f (x+h)−f (x) h h→0 Definition: f � (x) = lim is continuous and differentiable on the interval and F � (x) = f (x) d f (x) ± g(x) = f � (x) ± g � (x) Sum and Difference: dx � � d cf (x) = cf � (x) Scalar Multiple: dx � � d f (x)g(x) = f � (x)g(x) + f (x)g � (x) Product: dx � Mnemonic: If f is “hi” and g is “ho,” then the product rule is “ho d hi plus hi d ho.” � � � (x)g � (x) f (x) d = f (x)g(x)−f (g(x)) g(x) dx Quotient: Mnemonic: “Ho d hi minus hi d ho over ho ho.” The Chain Rule • First formulation: (f ◦ g)� (x) = f � (g(x)) g � (x) dy du dy = du • Second formulation: dx dx dy dx 781586 638962 $5.95 CAN $3.95 Left-hand rectangle approximation: n−1 � Ln = ∆x f (xk ) dx dx y) dy = dx − 2y dx cos y + x d(cos , dx dx and then cos y − x(sin y)y � − 2yy � = Finally, solve for y � = cos y−3 x sin y+2y Mn = ∆x =0 Linear: d (mx dx + b) = m Powers: d (xn ) dx = nx n−1 (true for all real n �= 0) Polynomials: d (an xn dx + · · · + a2 x + a1 x + a0 ) = an nx Logarithmic • Base e: d (ln x) dx d (sin dx x x x) = cos x d (tan dx d (sec dx = x) = sec x x) = sec x tan x n−1 • Arbitrary base: d (ax ) dx Simpson’s Rule: Sn = f (x0 )+4f (x1 )+2f (x2 )+· · ·+2f (xn−2 )+4f (xn−1 )+f (xn ) � ∆x � � � � • Arbitrary base: d (log a dx • Cosine: • Cotangent: • Cosecant: d (cos dx d (cot dx d (csc dx • Definite integrals: concatenation: = a ln a x) = x) = − sin x x) = − csc x x) = − csc x cot x • Arccosine: d (cos −1 dx 1+x2 • Arccotangent: x) = − √1−x x) = d (cot −1 dx • Arcsecant: d (sec −1 dx x) = √1 x x2 −1 • Arccosecant: x) = − 1+x d (csc −1 dx x) = − x√x12 −1 INTEGRALS AND INTEGRATION � � b f (x) dx = − a p f (x) dx + a • Definite integrals: comparison: If f (x) ≤ g(x) on the interval [a, b], then x ln a d (tan −1 dx √ 1−x2 • Definite integrals: reversing the limits: x d (sin −1 dx � � a f (x) dx b b f (x) dx = p b a � f (x) dx ≤ � � b f (x) dx a b g(x) dx a � � � Substitution Rule—a.k.a u-substitutions: f g(x) g � (x) dx = f (u) du � � � � f g(x) g � (x) dx = F (g(x)) + C if f (x) dx = F (x) + C • � Integration by Parts Best used to integrate a product when one factor (u = f (x)) has a simple derivative and the other factor (dv = g � (x) dx) is easy to integrate � �• Indefinite Integrals: � � f (x)g � (x) dx = f (x)g(x) − f � (x)g(x) dx or u dv = uv − v du • Definite Integrals: � b b f (x)g � (x) dx = f (x)g(x)]a − a � b f � (x)g(x) dx a Trigonometric Substitutions: Used to integrate expressions of the form DEFINITE INTEGRAL The definite integral f (x0 ) + 2f (x1 ) + 2f (x2 ) + · · · + 2f (xn−1 ) + f (xn ) ∆x + · · · + 2a2 x + a1 • Arctangent: � � Trapezoidal Rule: Tn = � • Arcsine: x) = f k=0 xk + xk+1 • Constant multiples: cf (x) dx = c f (x) dx =e k=1 � � � Properties of Integrals � � � • Sums and differences: f (x) ± g(x) dx = f (x) dx ± g(x) dx Constants: d (c) dx d (ex ) dx n−1 � TECHNIQUES OF INTEGRATION COMMON DERIVATIVES Exponential • Base e: Right-hand rectangle approximation: n � f (xk ) Rn = ∆x k=0 = y � and solve for y � f (t) dt a of x Differentiate both sides of the equation with respect to x Use the chain rule Ex: x cos y − y = 3x Differentiate to first obtain x a APPROXIMATING DEFINITE INTEGRALS Implicit differentiation: Used for curves when it is difficult to express y as a function carefully whenever y appears Then, rewrite � Part 2: If f (x) is continuous on the interval [a, b] and F (x) is any antiderivative of f (x), � b f (x) dx = F (b) − F (a) then Midpoint Rule: Inverse Trigonometric Printed the USA f (x) dx = F (x) + C if F � (x) = f (x) Part 1: If f (x) is continuous on the interval [a, b], then the area function F (x) = � • Tangent: • Secant: 50395 ISBN 1-58663-896-3 � FUNDAMENTAL THEOREM OF CALCULUS DERIVATIVE RULES Trigonometric • Sine: Copyright © 2003 by SparkNotes LLC All rights reserved SparkCharts is a registered trademark of SparkNotes LLC A Barnes & Noble Publication 10 antiderivatives: b √ ±a2 ± x2 f (x) dx is the signed area between the function y = f (x) and the a x-axis from x = a to x = b • Formal definition: Let n be an integer and ∆x = For each k = 0, 2, , n − 1, n−1 � f (x∗k ) pick point x∗k in the interval [a + k∆x, a + (k + 1)∆x] The expression ∆x k=0 � b n−1 � f (x) dx is defined as lim ∆x f (x∗k ) is a Riemann sum The definite integral b−a n n→∞ a k=0 Expression Trig substitution Expression becomes Range of θ Pythagorean identity used � a2 − x2 x = a sin θ dx = a cos θ dθ a cos θ − π2 ≤ θ ≤ π2 (−a ≤ x ≤ a) − sin θ = cos θ � x2 − a2 x = a sec θ dx = a sec θ tan θ dθ a tan θ 0≤θ< π≤θ< sec θ − = tan θ � x2 + a2 x = a tan θ dx = a sec θ dθ a sec θ − π2 < θ < INDEFINITE INTEGRAL of f (x) if F � (x) = f (x) • Antiderivative: The function F (x) is an antiderivative � • Indefinite integral: The indefinite integral f (x) dx represents a family of π 3π π + tan θ = sec θ APPLICATIONS GEOMETRY Area: � a b� Volume of revolved solid (shell method): f (x) − g(x) dx is the area bounded by y = f (x), y = g(x), x = a and x = b � if f (x) ≥ g(x) on [a, b] Volume of revolved solid (disk method):π � a b� f (x) �2 dx is the volume of the solid swept out by the curve y = f (x) as it revolves around the x-axis on the interval [a, b] � b � �2 � �2 f (x) − g(x) dx is the volume of Volume of revolved solid (washer method): π a the solid swept out between y = f (x) and y = g(x) as they revolve around the x-axis on � b 2πxf (x) dx is the volume of the solid a obtained by revolving the region under the curve y = f (x) between x = a and x = b around the y-axis � b� � �2 + f � (x) dx is the length of the curve y = f (x) from x = a Arc length: to x = b Surface area: a � b 2πf (x) a � + (f � (x)) dx is the area of the surface swept out by revolving the function y = f (x) about the x-axis between x = a and x = b the interval [a, b] if f (x) ≥ g(x) CONTINUED ON OTHER SIDE This downloadable PDF copyright © 2004 by SparkNotes LLC SPARKCHARTS™ Calculus Reference page of CALCULUS REFERENCE 3/18/03 10:49 AM Page MOTION � change in x(p) Increasing p leads to decrease in revenue • Demand is unitary if E(p) = Percentage change in p leads to similar percentage change in x(p) Small change in p will not change revenue • Demand is inelastic if E(p) < Percentage change in p leads to smaller VELOCITY percentage change in x(p) Increasing p leads to increase in revenue � � • Demand function: p = D(x) gives price per unit (p) when x units available • Market equilibrium is x¯ units at price p¯ • PROBABILITY AND STATISTICS � b PS = p¯x ¯− a CONTINUOUS DISTRIBUTION FORMULAS • Probability that X is between a and b: P (a ≤ X ≤ b) = ab f (x) dx �∞ xf (x) dx • Expected value (a.k.a expectation or mean) of X : E(X ) = µX = −∞ �2 � �2 �∞ � 2 • Variance: Var(X ) = σX = −∞ x − E(x) f (x) dx = E(X ) − E(X ) � • Standard deviation: Var(X ) = σX �m � • Median m satisfies −∞ f (x) dx = m∞ f (x) dx = 12 • Cumulative density function (F (x) is the probability that X is at most x): � f (x) = −∞ • −∞ −∞ �� � x − E(X ) y − E(Y ) f (x, y) dx dy Normal distribution (or Bell curve) with mean µ and (x−µ)2 variance σ: f (x) = √ e− 2σ2 σ 2π • P (µ − σ ≤ X ≤ µ − σ) = 68.3% -2 -1 • Simple interest: P (t) = P0 (1 + r)t • Compound interest �mt � • Interest compounded m times a year: P (t) = P0 + mr • Interest compounded continuously: P (t) = P0 ert EFFECTIVE INTEREST RATES The effective (or true) interest rate, reff , is a rate which, if applied simply (without compounding) to a principal, will yield the same end amount after the same amount of time �m � • Interest compounded m times a year: reff = + mr − • Interest compounded continuously: reff = er − • Cost function C(x): cost of producing x units • Marginal cost: C � (x) : cost per unit when x units produced • Average cost function C(x) = C (x) x � • Marginal average cost: C (x) If the average cost is minimized, then average cost = marginal cost • If C �� (x) > 0, then to find the number of units (x) that minimizes average cost, solve = C � (x) REVENUE, PROFIT • Demand (or price) function p(x): price charged per unit if x units sold • Revenue (or sales) function: R(x) = xp(x) • Marginal revenue: R� (x) • Profit function: P (x) = R(x) − C(x) • Marginal profit function: P � (x) PRESENT VALUE OF FUTURE AMOUNT The present value (PV ) of an amount (A) t years in the future is the amount of principal that, if invested at r yearly interest, will yield A after t years �−mt � • Interest compounded m times a year: PV = A + mr • Interest compounded continuously: PV = Ae−rt PRESENT VALUE OF ANNUITIES AND PERPETUITIES Present value of amount P paid yearly (starting next year) for t years or in perpetuity: Interest compounded yearly � � • Annuity paid for t years: PV = Pr − (1+r) t • Perpetuity: PV = Pr Interest compounded continuously • Annuity paid for t years: PV = rPeff (1 − e−rt ) = erP−1 (1 − e−rt ) If profit is maximal, then marginal revenue = marginal cost • The number of units x maximizes profit if R� (x) = C � (x) and R�� (x) < C �� (x) PRICE ELASTICITY OF DEMAND • Perpetuity: PV = • Demand curve: x = x(p) is the number of units demanded at price p P reff = P er −1 EXPONENTIAL (MALTHUSIAN) GROWTH / RESTRICTED GROWTH (A.K.A LEARNING EXPONENTIAL DECAY MODEL CURVE) MODEL = rP • Solution: • P (t) = P0 ert If r > 0, this is exponential growth; if r < 0, exponential decay > INTEREST σ COST dP dt ∂g ∂p • P (t): the amount after t years • P0 = P (0): the original amount invested (the principal) • r: the yearly interest rate (the yearly percentage is 100r%) MICROECONOMICS In all the following models • P (t): size of the population at time t; • P0 = P (0), the size of the population at time t = 0; • r: coefficient of rate of growth x FINANCE 95% 68% • P (µ − 2σ ≤ X ≤ µ + 2σ) = 95.5% χ-square distribution: with mean ν and variance 2ν: x ν f (x) = ν � ν � x −1 e− 22Γ � • Gamma function: Γ(x) = 0∞ tx−1 e−t dt BIOLOGY SUBSTITUTE AND COMPLEMENTATRY COMMODITIES COMMON DISTRIBUTIONS C (x) x y = L(x) completely equitable distribution X and Y are two commodities with unit price p and q, respectively • The amount of X demanded is given by f (p, q) • The amount of Y demanded is given by g(p, q) > and X and Y are substitute commodites (Ex: pet mice and pet rats) if ∂f ∂q X and Y are complementary commodities (Ex: pet mice and mouse feed) ∂g ∂f if ∂q < and ∂p < Cov(X, Y ) σ Correlation: ρ(X, Y ) = X Y = � σX σY Var(X)Var(Y ) for x in y The quantity L is between and The closer L is to 1, the more equitable the income distribution f (y) dy g(x, y) dy �∞ �∞ � • Covariance: Cov(X, Y ) = σX Y = (¯ p − S(x)) dx • Joint probability density function g(x, y) chronicles distribution of X and Y Then �∞ x x dP dt P P0 r0 = r (A − P ) A $5.95 CAN −∞ p = D(x) $3.95 �x S(x) dx = � x¯ The Lorentz Curve L(x) is the fraction of income received by the poorest x fraction of the population Domain and range of L(x) is the interval [0, 1] Endpoints: L(0) = and L(1) = Curve is nondecreasing: L� (x) ≥ for all x L(x) ≤ x for all x • Coefficient of Inequality (a.k.a Gini Index): � � � L=2 x − f (x) dx • Probability density function f (x) of the random variable X satisfies: F (x) = P (X ≤ x) = LORENTZ CURVE X and Y are random variables f (x) ≥ for all x; �∞ −∞ f (x) dx = � x¯ ( x, p) • Producer surplus: f (x) dx consumer surplus producer surplus b−a p x) = S (¯ x).) (So p¯ = D (¯ Consumer � x¯surplus: � x¯ ¯ = (D(x) − p¯) dx CS = D(x) dx − p¯x 20593 38963 = p = S(x) • Supply function: p = S(x) gives price per unit – • Average value of f (x) between a and b is f p (p) when x units demanded + ACCELERATION Acceleration a(t) vs time t graph: • The (signed) area under the graph gives the change � t in velocity: a(τ ) dτ v(t) − v(0) = TM CONSUMER AND PRODUCER SURPLUS – v(τ ) dτ t • Formula relating elasticity and revenue: R� (p) = x(p) − E(p) SPARKCHARTS + Writer: Anna Medvedovsky Design: Dan O Williams Illustration: Matt Daniels, Dan O Williams Series Editor: Sarah Friedberg Velocity v(t) vs time t graph: • The slope of the graph is the accleration: v � (t) = a(t) • The (signed) area under the graph gives the displacement�(change in position): s(t) − s(0) = x (p) • Price elasticity of demand: E(p) = − px(p) • Demand is elastic if E(p) > Percentage change in p leads to larger percentage POSITION Position s(t) vs time t graph: • The slope of the graph is the velocity: s� (t) = v(t) • The concavity of the graph is the acceleration: s�� (t) = a(t) � = rP − P K • K : the carrying P0 > A P � k capacity P0 < A • Solution: t P (t) = P0 > k k < P0 < k P0 < k t K 1+ � K −P P0 � e−rt SPARKCHARTS™ Calculus Reference page of ... capacity P0 < A • Solution: t P (t) = P0 > k k < P0 < k P0 < k t K 1+ � K −P P0 � e−rt SPARKCHARTS Calculus Reference page of .. .CALCULUS REFERENCE 3/18/03 10:49 AM Page MOTION � change in x(p) Increasing p leads to decrease in revenue... AND PRODUCER SURPLUS – v(τ ) dτ t • Formula relating elasticity and revenue: R� (p) = x(p) − E(p) SPARKCHARTS + Writer: Anna Medvedovsky Design: Dan O Williams Illustration: Matt Daniels, Dan O