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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 x

x

x

() ,

,

= if

= if =

Discontinuity

at =0

10

00

≠

fx

if x

fx

if x

x

() ,

()

=

=1,

<

Discontinuity

at =0

1

0

0

≥

−

fx xx

x

x

() ,

,

=if

= if =

Discontinuity

at =0

10

00

≠

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

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