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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 + 2dr −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 =⇒©© ddy = 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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