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Reference Guide

Differential Calculus - Reference Guides

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l e a r n • r e f e r e n c e • r e v i e w permacharts TM Differential Calculus Differential Calculus F UNCTIONS LIMITS EXPLICIT FUNCTIONS • Limits arise in defining derivatives and integrals, in establishing asymptotes to curves, in the evaluation of n yxf=axa(b )x=tc+− y = Dependent variable indeterminate forms, and in establishing convergence of x = Independent variable sequences and series • Equation is explicitly solved for one of the f(x) = Explicit function of x • If a function f(x) increasingly approaches a constant value variables a, b, c = Constants L as x Æ a and thereafter its distance from L remains arbitrarily small no matter how close the approach of x to IMPLICIT FUNCTIONS a, L is the limit of f(x); this is mathematically expressed as follows: F(y,)x= yx− + =isy 0 f lim f(x)L= ff(L)− 0 Instantaneous rate of change of y at P dx lim ∆y = lim ) (()+ −x CONCAVITY ∆x→∆x0∆x0 ∆x • A curve is concave upward at any point for which the second derivative d y/dx is2 = slope of tangent at P 2 2 ∆y dy positive and concave downward when d y/dx is negative lim = = derivative of f( )at P ∆x→0 ∆x dx y y y y Rising Falling Concave Up Concave Down DERIVATIVE NOTATION First Derivative dy ∆y ∆fx) dx = ∆x−0 =∆x −0 ∆x X ∆x = lim)fx(+ − fx dy > 0 dy < 0 dy dy ∆x−0 ∆x dx dx 2 > 0 2 < 0 dy d dx dx dx = yy©xf = ()) (dx f x INFLECTION POINTS dy = slope of tangent at P dy dx • If f(x) is differentiable anf©() = 2 , and=f'(a) π 0 , P dx xa= Second Derivative then x = a is an inflection point 2 ∆y© ∆f©() )( ()+ − f© x dy 2 = lim = lim lim • If both derivatives vanish: f”(a) = f’(a) = 0, then x = a will be an inflection point dx ∆x−∆0∆x ∆x 0 ∆x ∆x provided f”(x) changes sign as one passes from x > a to x < a dy2 2 d d dy  Y Maximum Minimum Inflection Y Inflection 2 ==" D=y = ()x = ()© x   dx dx dy dx  nth Derivative n ∆y()− 1 1 ()∆ fn − () d y =lim = ⋅lim = dx n ∆x−∆x−∆x ∆x a a a a X f()−1 ( + −( n − x lim dy dy2 dy dy dy dy dy dy ∆x−0 ∆x =0< 2 0 =0> 2 0 =0= 2 0 ≠0= 2 0 n dx dx dx dx dx dx dx dx d y = = = D yf () n () dx n RADIUS OF CURVATURE DIFFERENTIAL ARC LENGTH d ()− d dn − y  32/ / 32 12 1/2 = f () =  n −   2  2  dy 2  2 dr 2 dx dx dx  1 + dy   r2 + dr   ds = 1+    dx = +r    dθ   dx     dθ    dx    dθ   ρ =   =   Differentiable Functions 2 2 2 • A function whose derivative “exists” dy r2 + 2dr  −r dr (that is, does not attain ± •) is termed dx 2 d θ 2 dθ differentiable DIFFERENTIAL NOTATION EXCEPTIONS: THE CUSPS • Derivatives can be cross-multiplied to obtain • When two portions of a curve differentials have a common tangent and the dy =⇒ = dy y©dx dx point of contact is a sharp point 2 (cusp), maxima and minima may dy =⇒©© ddy  = y"dx occur when dy/dx ≠ 0 dx2 dx  d y () dn −1y ()n n = ⇒ d  n −1 = y dx dx dx  • The differential of a function y = f(x) is the SHORT TABLE OF DIFFERENTIALS product of the derivative of the function by the differential of the variable x += = dc = 0 = d(u)v udv vdu () d cu cdu ( ) d eu u e du Note however: dy ≠2y "x u vdu udv du n n −1 dy ≠ny dx) d v = 2 du v+= +du dv ()dInu= )(u = du nu du v 2 DIFFERENTIAL CALCULUS • A-763-3 w w w . permacharts.com © 1996-2012 Mindsource Technologies Inc. permachartsM l e a r n • r e f e r e n c e • r e v i e w
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