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Reference Guide

Permachart - Marketing Reference Guide: Inflection Point, Inflection, Farad

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Fall 2015
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Department
BAD - Business Administration
Course Code
BAD 200
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All
Chapter
Permachart

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LIMITS
l e a r n r e f e r e n c e r e v i e w
Differential Calculus
Differential Calculus
FUNCTIONS
EXPLICIT FUNCTIONS
L’HÔPITAL’S RULE
INDETERMINATE FORMS
IMPLICIT FUNCTIONS
LIMIT LAWS
yfx ax b xc==+ −()tan
2
• For evaluation of y, the inner function g(x) is determined first and then
substituted into the outer function f(x)
• Reversal of the order is not permitted (fro instance, f(g) πg(f))
yfgx x gx x fx gx
yfgx xa gx xa fx gx
yfgx x gx x fx gx
y
=
[]
=− ⇒ = =
[]
=
[]
=−=− =
=
[]
=−=− =
[]
() ( ) () () ()
() sin( ) () () sin ()
() ln () ln () ()
//
/
22
11
1313
12
But: == is a of two functionsxxfx gx product=•sin ( ) ( )
lim ()
()
xa
fx
gx
=
0
0 or
lim ()
() lim ()
() lim
()
()
()
()
xa xa xa
fx
gx
fx
gx
fnx
gnx
→→ →
=
lim ( ) lim[ ] lim[ ( ) ( )] lim ( ) lim ( )
lim lim( ) limsin lim( )
ln ( )
/
xa xa
fx
xa xa xa
n
n
x
x
xx
x
fx e fx gx fx gx
nxe ax
xax
en
→→ → →
→∞ → →
=⋅=
+
+= = =
=
1111
0
1
00
( = integer)
lim ( ) ( )fxL fxL xa
xa
=
εδ
where , = arbitrarily small numbers
()ffx L x a
x
−< <−<
ε
xa
x
<−<
δ
0
fx
if x
fx
if x
x
() ,
()
=
=1,
<
Discontinuity
at =0
1
0
0
fx x
fx
x
() /
=−123
No discontinuity
in ( )
Discontinuity in
tangent at
lim
lim sin
xx
x
x
x
x
x
+
()
=
+
()
()
2
211
2
21
4
But: lim
x0
does not exist because 1/xincreases
indefinitely
does not exist because sin xoscillates
indefinitely between ± 1
y = Dependent variable
x = Independent variable
f(x) = Explicit function of x
a, b, c = Constants
Fxy x yx y(,) sin=−+=
3120
• Equation cannot be easily solved for one variable
• Operations are conducted on the implicit form directly
COMPOSITE FUNCTIONS: FUNCTIONS OF A FUNCTION
• The differentiable functions f(x) and g(x) vanish at x= aso that
f(a)/g(a) = 0/0, provided the limit on
the right either exists or is ±
and g’(x) 0 for x=a
lim ()
() lim ()
()
xa
fx
gx xa
f' x
g' x
=
• The limits to 0/0, /, , 0 ¥ , 00, 0, and 1 are obtained by
expressions of L’Hôpital’s Rule according to the following procedures
Type 1 Type 2
Type 3 Type 4
• Limit found by repeated
differentiation
• Use this identity to reduce to
0/0 or /
fx gx fx
gx
gx
fx
() () ()
()
()
()
⋅= =
11
lim ( ) ( )
xa
fx gx
⋅=0
Factor or find common
denominator to reduce to 0/0
or /
lim [ ( ) ( )]
xa
fx gx
−=
lim ( ) () ,,
xa
fxgx
=∞
000
1
Use following transformation
to reduce to 0•:
y= f(x)g(x) ÆIny=
g(x) In f(x) Æ0•
FREQUENTLY USED LIMITS
• Equation is explicitly
solved for one of the
variables
• Limits arise in defining derivatives and integrals, in
establishing asymptotes to curves, in the evaluation of
indeterminate forms, and in establishing convergence of
sequences and series
• If a function f(x) increasingly approaches a constant value
Las xÆaand thereafter its distance from Lremains
arbitrarily small no matter how close the approach of xto
a,Lis the limit of f(x); this is mathematically expressed as
follows:
• Used to derive expressions (such as derivatives, integrals)
Addition Law: Limit of a sum
equals the sum of the limits
Product Law: Limit of
a product equals product
of the limits
Quotient Law: Limit of
a quotient equals quotient
of the limits, provided
the limit of the denominator
is not zero
Reciprocal Law:
Substitution Law
Composite Functions:
Squeeze Law: If a function g(x) is
positioned between two functions
f(x) and h(x) with the common
limit L, it will tend to the same limit L
The Number e:
Continuity of Functions:
A function f(x) is continuous
in an interval [a,b] if it does not
experience a jump or tend to ± x(namely, a single limit exists
at each point of the interval, including the end points)
lim ( ) ( ) lim ( ) lim ( )
xa xa xa
fx gx fx gx
→→
±= ±
[]
lim ( ) ( ) lim ( ) lim ( )
xa xa xa
fx gx fx gx
→→
•= •
[]
lim ()
()
lim ( )
lim ( ) lim ( )
xa
xa
xa
xa
fx
gx
fx
gx gx
=≠0
If then provided lim ( ) , lim ()
xa xa
fx L fx L L
→→
==
11 0
If and
then
lim ( ) , lim ( ) ( )
lim ( ) lim ( ) ( )
xa xa
xa xa
gx L f x fL
fgx f gx fL
→→
→→
==
==
[]
If
then
lim ( ) lim ( ) ,
lim ( )
() () ()
xa xa
xa
fx hx L
gx L
fx gx hx
→→
==
=
≤≤
The limit of the expression (1+ )
1/h is given by
lim
h0
1/h ... = e = 2.71828
h
h
+=++++( ) !!!
11
1
1
1
2
1
3
lim ( ) ( )
xc
fx fc a c b
=≤ for
Cusp
TM
permacharts
© 1996-2012 Mindsource Technologies Inc.
DIFFERENTIAL CALCULUS • A-763-3 1www.permacharts.com

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Description
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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