AP CALCULUS AB All you need to know in 170 in2 Curve sketching and analysis Differentiation Rules Approximate Methods for Integration dy Chain Rule Trapezoidal Rule (equally sized intervals) critical point: = 0 or is undefined d du dy dy du b dx [ f( u )] = f '( u ) OR = f( x ) dx = local minimum: dx dx dx du dx a b- a 1 1 dy d2 y [f ( x )+ f ( x ) + ... + f ( x ) + f ( x )] goes from – to + or > 0 n 20 1n- 1 2 n dx dx2 Product Rule local maximum: d du dv Riemann Sums (rectangular strips) dy d2 y (uv ) = v+ u OR u 'v + uv ' goes from + to – or < 0 dx dx dx b dx dx2 f( x )= f ( x ) w + ..... f ( x ) w a 1 1 n n point of inflection: concavity changes Quotient Rule Representative points could be left, right, d2 y goes from – to + or from + to – d u duv- u dv u' v- uv ' midpoint or inscribed. dx2 骣 dx dx 琪 = 2 OR 2 global (absolute) max/min: can be at a dx v v v Vertical asymptotes: denominator(only) = 0 local (relative) max/min or endpoint. Horizontal asymptotes: f(x) as x approaches . If degree on top is larger, no HA. Basic Derivatives If f(x) is differentiable at a, then it’s If degree on bottom is larger, HA is y = 0. d continuous at a If degrees match, HA is ratio of leading coefficients. ( xn) = nx n-1 dx (but the converse is not always true). d Average value of a function (sinx) = cos x The Fundamental Theorem of dx a.k.a. Calculus d Mean Value Theorem of Integrals cosx sin x b ( ) = - f( x ) dx= F ( b ) - F ( a ) dx a d If the function f(x) is continuous on [a, b] and (tanx) = sec2 x where F '( x )= f ( x ) the first derivative exists on the interval (a, b), dx and then there exists a number d (cotx) = - csc2 x d x x = c on (a, b) such that dx f( t) dt = f( x ) dx a 1 b d f( c )= f ( x ) dx (secx) = sec x tan x and furthermore b- a a dx d b( x ) This value f (c) is the average value of the d f( t ) dt = function on the interval [a, b]. (cscx) = - csc x cot x dx a( x ) dx d 1 f( b ( x )) b '( x )- f ( a ( x ))a'( x ) (ln x) = dx x Intermediate Value Theorem Solids of Revolution (and friends) If f(x) is continuous on [a, b], and N is a Disk Method: d x x b 2 (e) = e number between f(a) and f(b), then there V= p [ r( x )] dx dx exists at least one number x= c in the open a

d x x interval (a, b) such that Washer Method: (2) = 2 ln 2 b 2 2 dx f( c ) = N . V=p r( x ) - r ( x ) dx a ([ o] [ i ] ) Mean Value Theorem If rotating about y-axis, everything (including And not so basic… If the function f(x) is continuous on [a, b], limits) must be in terms of y. d -1 1 AND the first derivative exists on the If rotating about a remote axis, e.g., y = 7, radius is (sin x) = interval (a, b), then there is at least one dx 1- x2 7 – f(x). number x = c in (a, b) such that General volume equation (not rotated): d -1 cos-1 x = f( b )- f ( a ) b ( ) 2 f'( c ) = . V= Area( x ) dx dx 1- x b- a a d 1 (average value of slope = instantaneous Shape of slice could be square, rectangle, -1 semicircle, equilateral triangle, etc. (tan x) = 2 dx 1+ x value)

Extreme Value Theorem Distance, Velocity, and Acceleration Integration reverses the derivative d Ds rules, but remember: If f(x) is continuous on [a, b] there exists velocity = (position) v = dt ave *add C for indefinite integrals an absolute maximum value f(c) and an Dx 1 absolute minimum value f(d) at some d * x dx= ln x+ C acceleration = (velocity) numbers c and d on [a.b]. dt (ln x is only defined for x>0) (Remember – this can occur at the speed = v endpoints!) t f t f displacement = v dt distance = v dt to to So much to know!

Logarithm Rules Inverse trig functions Values of Trigonometric logxy= log x + log y -1 p p Functions for Common Angles a a a sin is defined on [- 2, 2 ]

x -1 cos is defined on [0,p ] θ sin θ cos θ tan θ loga= log ax - log a y y -1 p p 0 0 1 0 tan is defined on [- 2, 2 ] log xn = nlog x p 1 3 3 a a 6 2 Properties of Integrals 2 3 p 2 2 Odd/even functions 蝌[ f( u )� g ( u )] du f ( u ) g ( u ) 1 f(x) is odd if f(-x) = - f(x) 4 2 2 蝌k� f( u ) du k f ( u ) du (i.e., replacing x w/-x and y w/-y p 3 1 doesn’t change the original equation) a 3 f( u ) du = 0 ex: f(x) = 3x3 a 3 2 2 b a f( u ) du= - f ( u ) du p 蝌a b 1 0 undefined f(x) is even if f(-x) = f(x) b c c 2 (i.e., replacing x w/-x doesn’t change f( u ) du+ f ( u ) du = f ( u ) du 蝌a b a π 0 -1 0 the original equation) a 2 If f(u) is odd: f( u ) du = 0 ex: f(x) = x – 4|x| - a (x, y) for θ on the unit circle are the If f(u) is even: values of (cos θ, sin θ) a a f( u ) du= 2 f ( u ) du 蝌- a 0

Limits Exponential Growth and Decay: Trig Identities To evaluate limits, try: “The rate of growth of a population is Double Angle factoring, simplifying fractions, or proportional to the population” means sin 2x= 2sin x cos x multiplying by conjugates until you can dy cos2x= cos2 x - sin 2 x = 1 - 2sin 2 x use direct substitution: = ky Pythagorean 2 dx 4-x - ( x + 2)( x - 2) 2 2 lim= lim Solving differential equations using sinx+ cos x = 1 x��2x+2 x 2 x + 2 (others are easily derivable by dividing separation of variables: 2 2 dy by sin x or cos x) =lim - (x - 2) 2 2 y = kdx 1+ tanx = sec x x� 2 dy 2 2 = 4 = kdx cotx+ 1 = csc x y Reciprocal Definition of the Derivative: ln y= kx + C 1 secx= or cos x sec x = 1 cos x f( x+ h ) - f ( x ) y= ekx+ C f( x )= lim 1 h 0 h C kx cscx= or sin x csc x = 1 y= e e sin x f( x )- f ( a ) sinx 1 f( x )= lim y= Aekx tan x= or x a x- a cosx cot x (Use initial values such as y(0) = 6 to find Odd-Even the value of A.) |x+ 4 | - | x | sin(–x) = – sin x (odd) N ote: lim = 1 since this is cos(–x) = cos x (even) Newton’s Law of Cooling h 0 h dT = -k( T - T ) just the derivative of f(x) = |x| at a = 4. dt m Derivatives of Inverse Functions

-1 1 Linearization: ( f( x )) = Using a tangent line at a point a on f(x) to Special limits f( y ) approximate values of f(x) near a. 1 lim(sinx )= DNE (infinitely wiggly) y - y1 = m(x - x1) x 0 becomes f(x) is continuous if limf ( x )= f ( a ) sin x x a limx = 1 f(x) – f(a) = f ’(a)(x – a) x 0 limf ( x ) = L iff or x a lim(1+ 1 )x L(x) = f ’(a)(x – a) + f(a) x = e x 0 limf ( x )= L and lim f ( x ) = L If f(x) is concave up, L(x) < f(x). x a- x a + If f(x) is concave down, L(x)>f(x).