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Ken’s Cheat Sheet for AP Calculus Derivative Facts d (u n ) nu n 1 u ' dx d (sin u ) cos u u ' dx d (cot u ) csc 2 u u ' dx d u (e ) e u u ' dx d 1 ( Arc sin u ) u ' dx 1 u2 f '(c) lim x c f ( x ) f (c ) xc a dx ax C e u u ' eu C d N DN ' ND ' dx D D2 dy dy du Chain Rule dx du dx d (tan u ) sec 2 u u ' dx d (csc u ) csc u cot u u ' dx d 1 (ln u ) u ' dx u d 1 ( Arc sec u ) u ' dx | u | u2 1 d f g f g g f dx d f ( g ( x)) f ( g ( x)) g ( x) dx d (cos u ) sin u u ' dx d (sec u ) sec u tan u u ' dx d u (a ) a u u ' ln a dx d 1 ( Arc tan u ) u ' dx 1 u2 f ( x) lim h0 f ( x h) f ( x ) h x n 1 x dx n 1 C, n 1 au u a u ' C ln a sec u tan u u ' sec u C n sin u u ' dx cos u C cos u u ' sin u C csc u cot u u ' csc u C tan u u ' ln sec u C cot u u ' ln sin u C csc u u ' ln csc u cot u C ln x dx x ln x x C Fundamental Theorem of Calculus Part I If F ( x) or v u 1 u u ' ln u a u C u ' ln a a u C sec u u ' tan u C csc u u ' cot u C sec u u ' ln sec u tan u C 2 2 f (t )dt , then F '( x) f (v )v '- f (u )u '. d x f (t )dt f ( x) dx c Fundamental Theorem of Calculus Part II Avg .Value of A function = f x dx F (b) F (a), where F (x ) is the antiderivative of f ( x) b a 1 b f ( x)dx b a a f x dx F (b) F (a) NET change in F where F is any antiderivative of f . aka b a b V (t) dt S(b) S(a) displacement a b a b a f '( x)dx=f (b) f (a) English: The definite integral of velocity is displacement. V (t ) dt Distance Traveled English: The definite integral of Speed is Displacement. speed velocity t S (t ) S0 v( x)dx "Postion is determined by adding the original position to the displacement." 0 If Velocity and Acceleration have the same sign, then speed is increasing. Formal Definition of Continuity lim f ( x ) f (c ) If and Only If f is continuous at x = c. x c “When looking for a limit at x = c, substitution works if and only if the function is continuous at x = c” Average Rate of Change = Slope of a Secant Line Instantaneous Rate of Change = Slope of a Tangent Line For a limit to exist, the left and right hand limits must agree (be equal) V R r dx 2 Washer Method…. V R 2 r 2 dx.... “Look for extremes at critical numbers and endpoints.” If a function is differentiable at point x a , it is continuous at that point. The converse is false, in other words, continuity does not imply differentiability Intermediate Value Theorem: If a function is continuous on [a,b], then it takes on every value between f (a ) and f (b) . Extreme Value Theorem: “Every Closed Endpoint is an Extreme” If f is continuous over a closed interval, then f has a maximum and minimum value over that interval Mean Value Theorem (for derivatives) If f is continuous on “For every Secant line, there is a parallel Tangent” a, b and differentiable on a, b then at some point between a and b : f (b) f (a ) f ( c) ba Mean Value Theorem (for definite integrals) “The function passes through its average value” 1 b f x dx If f is continuous on a, b then at some point c in a, b , f c b a a Rolle’s Theorem “A well behaved function that is not one to one will have a horizontal tangent” If f is continuous on a, b and differentiable on a, b such that f ( a ) f (b) , then there is at least one number c in the open interval a, b such that f (c) 0 . L’Hôpital’s Rule If lim x a 0 f ( x) f ( x) f ( x) f ( x) or is of the form , and if lim exists, then lim . lim x a g ( x ) x a g ( x) x a g ( x) 0 g ( x) x 2 2x lim 0 x x3 3 Q: x3 2x lim x x2 8 2 x 2 3x 2 2 lim 2 5 x 10 x 5 x Use Ratio of Leading Terms “What do we do with derivatives?” A1: We set them equal to zero and sign test them. A2: We integrate them (find the area under them) Definition of Concavity If f ' x is increasing, then f ( x) is concave up. If f ' x is decreasing, then f ( x) is concave down. Rate of increase is INCREASING…CONCAVE UP Rate of increase is DECREASING….CONCAVE DOWN. 1st Derivative shows: maximum ( + to -) and minimum (-0 to +) values, increasing and decreasing intervals, slope of the tangent line to the curve, and velocity 2nd Derivative shows: inflection points, concavity, and acceleration 2nd Derivative Test for Extremes f (c) 0, and f ' '(c) 0, then f has a local maximum at x = c. f (c) 0, and f ' '(c) 0, then f has a local minimum x = c. nd 2 Derivative Test for Concavity If f ( x) 0 in a, b , then f is concave upward in a, b . If f ( x) 0 in a, b , then f is concave downward in a, b . st 1 Derivative Test for Increasing and Decreasing If If f (c) 0 then is INCREASING at x c. . If f (c) 0 then is DECREASING at x c. OR If f '( x) 0 on (a,b) then f is increasing on (a,b) Definition of Critical Number. ..... c is a critical number of f if and only if c is in the interior of the domain of f and f '(c) 0 or f '(c) fails to exist. Definition of Inflection Point c is an inflection point of f if and only if f changes concavity at x c and f has a tangent line at x c. Exponential Growth Formula: K “The rate of growth is proportional to the amount present.” ln( what happened ) how long it took dP kP "math" for exponential growth. dt When separating variables…Find C as soon as possible. P(t ) P0 e kt Separate…..Integrate…..Find C…….Get Y alone. The definite integral of the rate of change of f yields the net change in f over the given interval. The definite integral of the rate of change of f yields the accumulation of f over the given interval. Ex: The integral from a to b of the rate of change of Population yields the change in Population from a to b. Integral means: Area “under” the curve number of rectangles) Anti-derivative Accumulation Net change Summation (often an infinite Inverse points have reciprocal slopes