Stuff You MUST Know Cold For AP Calc Exam

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).