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Tangent lines Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope provided that the limit exists. Remark. If the limit does not exist, then the curve does not have a tangent line at P(a,f(a)). h a f h a f a x a f x f m h a x ) ( ) ( lim ) ( ) ( lim 0

# Tangent lines Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope

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### Text of Tangent lines Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x)... Tangent lines Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x) at the point P(a,f(a))

is the line through P with slope

provided that the limit exists. Remark. If the limit does not exist, then the curve does

not have a tangent line at P(a,f(a)).

h

afhaf

ax

afxfm

hax

)()(lim

)()(lim

0 Tangent lines Ex. Find an equation of the tangent line to the hyperbola

y=3/x at the point (3,1).

Sol. Since the limit

an equation of the tangent line is

or simplifies to

)3(3

11 xy

0

( ) ( ) ( ) ( ) 1lim lim

3x a h

f x f a f a h f am

x a h

.063 yx Velocities Recall: instantaneous velocity is limit of average velocity Suppose the displacement of a motion is given by the

function f(t), then the instantaneous velocity of the motion at time t=a is

Ex. The displacement of free fall motion is given by

find the velocity at t=5. Sol. The velocity is

h

afhafv

h

)()(lim

0

,5.0)( 2gtts

49555.0)5(5.0

lim)5()5(

lim22

00

g

h

ghg

h

shsv

hh Rates of change Let The difference

quotient

is called the average rate of change of y with respect to x. Instantaneous rate of change =

Ex. The dependence of temperature T with time t is given by the function T(t)=t3-t+1. What is the rate of change of temperature with respective to time at t=2?

Sol. The rate of change is

).()(, 000 xfxxfyxxx

x

xfxxf

x

y

)()( 00

x

yx

0

lim

0 0

(2 ) (2)lim lim 11.t t

T T t T

t t Definition of derivative Definition The derivative of a function f at a number a,

denoted by is

if the limit exists.

Similarly, we can define left-hand derivative and right-

hand derivative

exists if and only if both and exist and

they are the same.

),(af

h

afhafaf

h

)()(lim)(

0

)(af ).(af

)(af )(af )(af Example Ex. Find the derivative given

Sol. Since does not exist,

the derivative does not exist.

),0(f

.00

01

cos)(x

xx

xxf

hh

fhfhh

1coslim

)0()0(lim

00

)0(f Example Ex. Determine the existence of of f(x)=|x|.

Sol. Since

does not exist.

)0(f

).0(1lim)0()0(

lim)0(

,1lim)0()0(

lim)0(

00

00

fh

h

h

fhff

h

h

h

fhff

hh

hh

)0(f Continuity and derivative Theorem If exists, then f(x) is continuous at x0.

Proof.

Remark. The continuity does not imply the existence of derivative.

For example,

)( 0xf

0limlim)(limlim0000

x

fx

x

fxf

xxxx

.00

01

cos)(x

xx

xxf Interpretation of derivative The slope of the tangent line to y=f(x) at P(a,f(a)), is the

derivative of f(x) at a,

The derivative is the rate of change of y=f(x) with respect to x at x=a. It measures how fast y is changing with x at a.

).(af

)(af Derivative as a function Recall that the derivative of a function f at a number a is

given by the limit:

Let the above number a vary in the domain. Replacing a by variable x, the above definition becomes

If for any number x in the domain of f, the derivative

exists, we can regard as a function which assigns to x.

h

afhafaf

h

)()(lim)(

0

)(xf h

xfhxfxf

h

)()(lim)(

0

)(xf f Remark Some other limit forms

0 0

( ) ( )( ) lim lim

h x

f x h f x yf x

h x ExampleFind the derivative function of

Sol. Let a be any number, by definition,

Letting a vary, we get the derivative function

.)( nxxf

.)(lim

lim)()(

lim)(

11221

nnnnn

ax

nn

axax

naaxaaxx

ax

ax

ax

afxfaf

.)( 1 nnxxf Other notations for derivative If we use y=f(x) for the function f, then the following notat

ions can be used for the derivative:

D and d/dx are called differentiation operators.

A function f is called differentiable at a if exists. f is differentiable on [a,b] means f is differentiable in (a,b) and both and exist.

)()()()( xfDxDfxfdx

d

dx

df

dx

dyyxf x

)(af

)(af )(bf

axax dx

dy

dx

dyaf

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