Basic Derivatives where u is a function of x, and a is a constant. ( ) d n x = nx n−1 dx d ( sin x ) = cos x dx d ( cos x ) = − sin x dx d ( tan x ) = sec2 x dx d ( cot x ) = − csc2 x dx d ( sec x ) = sec x tan x dx d ( csc x ) = − csc x cot x dx d 1 du ( ln u ) = dx u dx d 1 ( log a x ) = dx x ln a d u u du e =e dx dx d u( x) du u x a a ( ) ln a ⋅ = dx dx d 1 du sin −1 u = 2 dx dx 1− u ( ) ( ) ( ) d −1 cos −1 x = dx 1 − x2 d 1 du −1 tan= u ⋅ dx 1 + u 2 dx d −1 cot −1 x = dx 1 + x2 d du 1 sec −1 u = ⋅ 2 dx u u − 1 dx ( ) ( ) ( ) ( ) d −1 csc −1 x = dx x x2 − 1 ( ) Differentiation Rules Chain Rule d f ( g ( x )) = f '( g ( x )) g ' ( x ) dx d dv du (= uv) u +v dx dx dx du d u v − u dv Quotient Rule = dx 2 dx dx v v Product Rule Some handy INTEGRALS: ∫ tan = x dx ln sec x + C = − ln cos x + C ∫ sec x dx= ln sec x + tan x + C AP CALCULUS Stuff you MUST know Cold Curve sketching and analysis y = f(x) must be continuous at each: dy critical point: = 0 or undefined dx inc/dec. function f(x): f ’ > 0, f ’ < 0 concavity ≡ inc/dec slope: f “ > 0, f “< 0 local minimum: dy goes (–,0,+) or (–,DNE,+) or d 2 y >0 dx 2 dx local maximum: dy goes (+,0,–) or (+,DNE,–) or d 2 y <0 dx 2 dx point of inflection: f ” = 0 or DNE AND concavity changes * topic only on BC Approximation Methods for Integration Use Geometry formulas Rectangles - Left, Right and Middle Riemann Sums A = bh Trapezoids: A = ½ (b1 + b2)h (Effects of inc/dec & cocavity on approx.) Concave up: M under estimate, T over estimate Concave down: M over estimate, T under estimate Inc: L=under, R=over. Dec: L = over, R = under First Fundamental Th. of Calculus ∫ b a )dx f (b) − f (a ) f '( x= 2nd Fundamental Th. of Calculus d b( x) f (t )dt = dx ∫a ( x) d 2 y goes from (+ to –), (– to +), dx 2 Abs. Max/Min: eval. crit # & endpts. f (b( x)) b '( x) − f (a ( x)) a '( x) OR discuss “always inc or always dec.” Intermediate Value Theorem: If the function f(x) is continuous on [a, b], Solids of Revolution and friends for all k between f(a) and f(b), there exists at Disk Method b 2 least one number x= c in the open interval V = π ∫ [ R ( x) ] dx a (a, b) such that f (c) = k . Washer Method Extreme Value Theorem: b 2 2 If the function f(x) is continuous on [a,= b], V π ∫ [ R( x) ] − [ r ( x) ] dx a then there exists an absolute max and min volume by cross section (not rotated) on that interval. b Rolle’s Theorem: V = ∫ Area ( x) dx a If the function f(x) is continuous on [a, b], s2 AND differential on the interval (a, b), 3 Aeq . lat . ∆ = AND f(a) = f(b), then there is at least one 4 b 2 number x = c in (a, b) such that f '(c) = 0 *Arc Length = s ∫ 1 + [ f '( x) ] dx a Mean Value Theorem: m secant = m tangent *Surface Area If the function f(x) is continuous on [a, b], b 2 then AND differential on the interval (a, b), = SA 2π ∫ radius 1 + [ f '( x) ] dx a there is at least one number x = c in (a, b) such that f '(c) = f (b) − f (a ) . Distance, Velocity, and Acceleration b−a MVT of Integrals i.e. AVERAGE VALUE: velocity = d (position) If the function f(x) is continuous on [a, b] dt and differential on the interval (a, b), acceleration = d (velocity) then there exists at least one number dt x = c on (a, b) such that dx dy b *velocity vector = , f (c ) ( b − a ) = ∫a f ( x)dx dt dt Area rectangle = Net Area Integral speed = = v ( x ') 2 + ( y ') 2 * b 1 f (c ) = f ( x ) dx . ( (b − a ) ) ∫ a This value f(c) is the “average value” of the function on the interval [a, b]. Limit Strategies: Factor and cancel, Rationalize Numerator, a x u-sub,a HA rules: = lim x →∞ b bx 2 + c 2 HA −a ax = lim 2 x →−∞ b bx + c To find all HA: Take limit as x → both ± ∞ displacement = tf ∫t v dt o total distance = ∫ final time initial time tf ∫t v dt ( x ')2 + ( y ')2 dt * o Av. velocity = 1 b b − a ∫a 1 b v(t )dt = s(b) − s(a) b − a ∫a b−a [ x '(t )] + [ y '(t )] 2 2 dt * BC TOPICS and important TRIG identities and values: l’Hôpital’s Rule Slope of a Parametric equation Given x(t) and y(t), the slope is dy dy dy d 2 y Dt dt dt , = dx= dx dx dt dx 2 dt f (a) 0 ∞ If = = , or g (b) 0 ∞ f ( x) f '( x) then lim = lim x →a g ( x) x → a g '( x ) ( ) Euler’s Method If given that dy dx Polar Curve = f ( x, y ) For a polar curve r(θ), the and the AREA inside a “leaf” = solution passes through (xo, yo), xnew = xold + ∆x 2 The SLOPE of r(θ) at a given θ is dy dy = dx d dθ dθ = d dθ dx dθ Integration by Parts (u=ILATE=dv) uv − ∫ vdu f ( x= ) g '( x)dx f ( x) g ( x) − ∫ g ( x) f '( x)dx Ratio Test: converges if lim an +1 < 1 n →∞ n →∞ an If the limit equals 1, you know nothing so check the endpoints using another test. converges if lim n an < 1 n →∞ p–series Test: ln x dx = x ln x − x + C ≠0 diverges if lim an Root Test: Use integration by parts and let u = ln x r (θ ) sin θ r (θ ) cos θ nth Term: Integral of Ln ∫ r (θ ) dθ θ1 and θ2 are the “first” two times that r = 0. dy (∆x) + yold = ynew dx ( x , y ) old old ∫ θ dr determines inc/dec and relative max/mins. dθ Basically Algebra I: y = m(x – x1) + y1 ∫ udv= ∫ 2 1 2 θ 1 ∞ ∑ n =1 1 converges if p > 1 pn Alternating: converges if alt. & lim an = 0 Values of Trigonometric Functions for Common Angles θ sin θ cos θ tan θ 0 0 0° 1 1 π 3 3 6 2 2 3 π 2 2 1 4 2 2 π 1 3 3 3 2 2 π 2 π 1 0 “∞ ” 0 −1 0 Know both the inverse trig and the trig values. π Careful: tan 3π = − −1 but arctan ( −1) = 4 4 Trig Identities Double Angle sin 2 x = 2sin x cos x cos 2 x = cos 2 x − sin 2 x = 1 − 2sin 2 x Power Reduction (1 − cos 2 x ) sin 2 x = 2 (1 + cos 2 x ) cos 2 x = 2 n →∞ ∞ Taylor Series Geometric Series: If the function f is “smooth” (continuous and differentiable) at x = c, then it can be approximated by the nth degree polynomial ∑a r n=0 n 1 converges if |r|<1 Limit Comparison: If lim an exists and ≠ 0, n →∞ bn what one series does (C or D), so does the other. Direct Comparison: If 0 < an < bn and Pythagorean sin 2 x + cos 2 x = 1 (others are easily derivable by dividing by sin2x or cos2x) 1 + tan 2 x = sec 2 x cot 2 x + 1 = csc 2 x ∞ Reciprocal if ∑ a diverges, then ∑ bn diverges; f ''(c) f ( n ) (c ) 2 n + ( x − c) + + ( x − c) . 1 n =1 = sec x = or cos x sec x 1 2! n! ∞ ∞ cos x if ∑ bn converges then ∑ a converges. n 1 n =1 n =1 = csc x = or sin x csc x 1 sin x Maclaurin Series (Taylor Series about x = 0) Error Bound (Remainder) Odd-Even Alternating Series: x 2 x3 xn e x =1 + x + + + + + ... N sin(–x) = – sin x (odd) n 2! 3! n! If S= −1) an is the Nth partial sum of a ( ∑ N cos(–x) = cos x (even) n k =1 x 2 n ( −1) x2 x4 tan(–x) = –tan x (odd) cos x =1 − + − + + ... convergent alternating series, 2! 4! ( 2n )! RN ≤ aN +1 (the next term) Infinite Sums n 2 n +1 3 5 x ( −1) x x Lagrange Error Bound of a Taylor Series: sin x =x − + − + + ... Geometric Series: S∞ = a1 if |r|<1 Let c = # centered on and x = value you want to 3! 5! ( 2n + 1)! 1− r approximate. There exists a z between x and c, Telescoping Series: Expand & cancel 1 f n +1 ( z ) ∞ =1 + x + x 2 + x 3 + + x n + ... n +1 1 such that Rn ≤ x−c 1− x Special Series: = e1 ( n + 1)! 1 2 3 n n +1 f ( x) ≈ f (c) + f '(c)( x − c) + ∞ n n =1 ln(= x) ( x − 1) ( x − 1) ( x − 1) 1 − 2 + 3 − + ( x − 1) ( −1) n + ... Find interval of f n +1 ( z ) to find error interval. ∑ n! n =0