chapter 2 limit and derivatives 1

The Tangent and Velocity ProblemsThe average velocity of the falling rock over anytime interval[t1, t2] is the change ∆y in the distance fallen divided by the length ∆t ofthe time interv

2 LIMITS AND DERIVATIVES

2.1 The Tangent and Velocity Problems

2.2 The Limit of a Function

2.3 Calculating Limits Using the Limit Laws

2.4 The Precise Definition of a Limit (do it yourself) 2.5 Continuity

2.6 Limits at Infinity; Horizontal Asymptotes

2.7 Derivatives and Rates of Change

2.8 Higher-Order Derivatives

2.1 The Tangent and Velocity Problems

I THE TANGENT PROBLEM

Find an equation of the tangent line to the curve y = x2 at the point (1, 1)

2.1 The Tangent and Velocity Problems

P(1,1) Q(x, x2 )

2.1 The Tangent and Velocity Problems

We say that the slope of the tangent line m is the limit of theslopes of the secant lines, and write

From the table, the slope of tangent line should be m = 2.

Then the equation of the tangent line is

y = 2(x – 1) + 1 = 2x – 1

PQ P

Q m

m



 lim

2.1 The Tangent and Velocity Problems

II THE VELOCITY PROBLEM

Let the distance of a rock falling down from a point near thesurface of the earth has the form:

y = 4.9t2 m

1) What is the average velocity of the falling rock during the first 5 s?

2) What is the average velocity from t = 5 to t = 6?

3) How fast is the rock falling at time t = 5?

2.1 The Tangent and Velocity Problems

The average velocity of the falling rock over anytime interval

[t1, t2] is the change ∆y in the distance fallen divided by the length ∆t of

the time interval

Average velocity = ∆y/∆t = [4.9(t2)2 – 4.9(t1)2]/(t2 – t1).

2.1 The Tangent and Velocity Problems

1) In the first 5 s (time interval [0, 5])

vave = [4.9(5)2 – 4.9(0)2]/(5 – 0) = 24.5 m/s

2) In the time interval [5, 6]

vave = [4.9(6)2 – 4.9(5)2]/(6 – 5) = 53.9 m/s

2.1 The Tangent and Velocity Problems

Suppose that the falling rock had a speedometer then what

would it show at time t = t0 (t0 is fixed)?

Average velocity over [t0, t0 + h] =

= ∆y/∆t = [4.9(t0 + h)2 – 4.9(t0)2]/h.

2.1 The Tangent and Velocity Problems

It appearsthat as we shorten

the time period, the

average velocity is

becoming closer to 49

m/s

The instantaneous velocity when t = 5 is defined to be the

limiting value of these average velocities over shorter and shorter time

periods that start at t = 5 Thus the instantaneous velocity at t = 5 is

2.1 The Tangent and Velocity Problems

There is a close connection between the tangent problem andthe problem of finding velocities

2.1 The Tangent and Velocity Problems

We find the limit of average velocity as h approaches zero to find out the instantaneous velocity at t = t0

∆y/∆t = (9.8t0h + 4.9h2)/h = 9.8t0 + 4.9h —› 9.8t0 as h —› 0

Hence 3) If t0 = 5, v = 49m/s.

2.2 The Limit of a Function

Def 1. Suppose f(x) is defined on some open interval containing a (except possibly at a itself) We say the limit of f(x) as x approaches a equals L and write

if the values of f(x) tend to get closer and closer to the number L as x gets closer and closer to the number a (from either side of a) but x ≠ a.

L x

2.2 The Limit of a Function

2.2 The Limit of a Function

(1) Guess the value of

1

1 lim 2

1 lim 2

1 1

x x





2.2 The Limit of a Function(2) Guess the value of

6

1 3

9 lim

 

2.2 The Limit of a Function

ONE-SIDED LIMITS

f(x) as x approaches a (or the limit of f(x) as x approaches a from the

left) is equal to L if we can make the value of f(x) arbitrarily close to L

by taking x to be sufficiently close to a and x less than a.

Def 3. If we require that x be greater than a, we get the hand limit of f(x) as x approaches a is equal to L and we write

right-L x

2.2 The Limit of a Function

Relationship between one-sided and two-sided limits

)

( lim )

( lim

) ( lim

L x

f x

f

L x

f

a x a

x

a x

2.2 The Limit of a FunctionExample. Find the left-hand and the right-hand limits of g(x) at x =

2 and x = 5 Does the limit of g(x) exists at these points?

2.2 The Limit of a Function

INFINITE LIMITS

infinity and write

if the value of f(x) can be

made as large as we want

by taking x sufficiently close

y = f(x)

y = f(x)

2.2 The Limit of a Function

negative infinity and write

if the value of f(x) can be

made arbitrarily large negative

by taking x sufficiently close to a.

2.2 The Limit of a Function

Similarly, we have definitions of infinity/negative infinity of f(x)

at the left/right side of a.

2.2 The Limit of a Function

2.2 The Limit of a Function

Def 6. For all these cases of infinite limits, the line x = a iscalled a vertical asymptote of the curve y = f(x).

Example. Find the vertical asymptotes of f(x) = tan x.

2.3 Calculating Limits, Limit Laws

Rules for Calculating Limits

If there exist and andlim f ( x ) k is a constant, then

.

a x a

x a

) ( lim )

( lim )

( ) ( lim

2 f x g x f x g x

a x a

x a



2.3 Calculating Limits, Limit Laws

0 )

( lim if

, ) ( lim

) ( lim )

(

)

( lim

x f x

g

x f

a x a

x

a x a

x

).

( lim )

( lim then

), ( )

( If

.

a x a

m n

2.3 Calculating Limits, Limit LawsExample 1. Find

)

( lim

2 g x

x f

x

2.3 Calculating Limits, Limit LawsExample 2. Find

2.3 Calculating Limits, Limit Laws

The Squeeze Theorem

Given f(x) ≤ g(x) ≤ h(x) for all x in some open interval containing a, except possibly at a itself If

then

L x

h x

f

a x a

L

2.3 Calculating Limits, Limit LawsExample

lim 2

0



2.4 The Precise Definition of a Limit

See pages 109 – 116 Do it yourself.

2.5 Continuity

Continuity at an interior point

Let y = f(x) be defined on [a, b] The function f(x) is continuous at

an interior point c in (a, b) if

).

( )

(

lim f x f cc



2.5 Continuity

If

1 f(x) is not defined at c or

2 does not exist or

3 exists but is not equal to f(c)

) (

lim f x

c

x

) (

2.5 ContinuityExample 1 At which numbers is the function discontinuous?

2.5 ContinuityExample 2 Where are each of the following functions discontinuous?

2

2 )

( (a)

x f

0

1 )

(

x

x x

x f

2 2

2 )

( (c)

2

x

x x

x

x x

f

2.5 Continuity

These kinds of discontinuity are called removable

2.5 Continuity

(b)

(d)

2.5 Continuity

Right and left continuity

We say that f(x) is right continuous at c if

We say that f(x) is left continuous at c if

).

( )

2.5 Continuity

Continuity at an endpoint

We say that f(x) is continuous at the left endpoint a of its domain if it

is right continuous there, and f(x) is continuous at the right endpoint b

of its domain if it is left continuous there

Continuity on an interval

2.5 Continuity

Example 1. State that whether the following function is continuous, discontinuous or one side continuous at each of the points: – 6, – 4, – 1, 2, 4

2.5 Continuity

Theorem 1 The following functions are continuous at every

number in their domains:

1 Polynomials

2 Rational functions & Root functions

3 Trigonometric & Inverse Trig functions

4 Exponential and logarithmic functions

2.5 Continuity

Theorem 2 Combining continuous functions

If the functions f and g are both defined on an interval containing c and both are continuous at c, the following functions are also continuous at c:

1. f + g and f – g;

2. f g and f /g (provided g(c) ≠ 0).

3. kg (k is any number) and (f(x)) 1/n (provided f(c) > 0 if n is even).

2.5 Continuity

Theorem 3 Composites of functions

If f is continuous at b and , then

In particular, if g(x) is continuous at c, then

b x

2.5 ContinuityExample Evaluate

1

2.5 Continuity

Theorem 1 (The Max-Min Theorem) If f(x) is continuous

on [a, b], then there exist numbers x1 and x2 in [a, b] such that for all x

in [a, b],

f(x1) ≤ f(x) ≤ f(x2),

CONTINUOUS FUNCTIONS ON CLOSED, FINITE INTERVALS

2.5 Continuity

Theorem 2 (The Intermediate-Value Theorem) If f(x) is

continuous on [a, b] and if N is a number between f(a) and f(b), then there exists a number c in [a, b] such that f(c) = N.

f(b) N f(b)

N

2.6 Limits at Infinity

f(x), as x approaches infinity, is L and

write

if the value of f(x) can be made arbitrarily

close to L by taking x sufficiently large.

L x

f



 ( ) lim

2.6 Limits at Infinity

limit of f(x), as x approaches

negative infinity, is L and write

if the value of f(x) can be made

arbitrarily close to L by taking x

sufficiently large negative

L x

f

3 2

x

1 2

2

5 lim 3

( lim ,

) ( lim ,

) (

x

lim )

d

, 5

lim )

c

, 2 3

lim )

b

, 2 3

lim )

a

3

3 4

3 4

2 3

2 3

x

x x

x

x x

x x

x x x x

2.6 Limits at Infinity

Precise Definitions , see pages 137 – 140.

2.7 Derivatives and Rates of Change

Non-vertical tangent lines

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 this limit exists

m a

x

a f x

(lim

2.7 Derivatives and Rates of Change

h

a f h

a f m

h

)()

(lim

2.7 Derivatives and Rates of Change

C

P

L

C P

2.7 Derivatives and Rates of Change

Vertical tangents

If f is continuous at P(a, f(a)) and if either

Then the vertical line x = a is tangent line to the graph y = f(x) at P.

a f h

a f h

a f

h h

) ( )

( lim or

) ( )

( lim

0 0

2.7 Derivatives and Rates of Change

Example 1. Find the tangent line to the curve y = x1/3 at the point(0, 0)

y = x1/3

2.7 Derivatives and Rates of Change

If the limit of the quotient (Newton quotient)

fails to exist, the graph y = f(x) has no tangent line at P.

h

a f h

a

f (  )  ( )

Example 2. Find the tangent line

to the curve y = x2/3 at the point (0, 0)

2.7 Derivatives and Rates of Change

Velocity

Suppose an object moves along a straight line according to an

equation of motion s = f(t), where s is the displacement of the object from the origin at time t The function f that describes the motion is

called the position function of the object In the time interval from t

= a to t = a + h the change in position is f(a+h) – f(a).

2.7 Derivatives and Rates of Change

1 The average velocity over this time interval is

average velocity = displacement/time

2 The velocity (instantaneous velocity) v(a) at time t = a is the limit of

average velocity

h

a f h

a

f a

v

h

) ( )

( lim )

a

f (  )  ( )



2.7 Derivatives and Rates of Change

a

f a

f

h

) ( )

( lim )

2.7 Derivatives and Rates of Change

An equivalent way of stating the definition of the derivative is

a x

a f x

f a

(lim)

('

The tangent line to y = f(x) at the point P(a, f(a)) is the line through P whose slope is equal to f’(a).

2.7 Derivatives and Rates of Change

2.7 Derivatives and Rates of Change

The difference quotient

is called the average rate of change of y with respect to x over the interval [x1, x2]

1 2

1

2) ( )(

x x

x f x

f x

2.7 Derivatives and Rates of Change

The limit of average rates of change is called the

(instantaneous) rate of change of y with respect to x at x = x1

rate of change at x1 =

1 2

1 2

0

)()

(lim

lim

1

x f x

f x

y

x x

2.7 Derivatives and Rates of Change

If we sketch the curve y = f(x), then the rate of change is the slope of the tangent to this curve at the point where x = a.

When the derivative is large(and therefore the curve is steep, as

at P), the y-values change rapidly.

When the derivative is small

(the curve is flat, as at Q) and the

y-values change slowly

P

Q

2.7 Derivatives and Rates of Change

Example 1. (p 148) A manufacturer produces bolts of a fabric

with a fixed width The cost of producing x yards of this fabric is C =

f(x) dollars

(a) What is the meaning of the derivative f’(x)? What are its units?

(b) In practical terms, what does it mean to say that f’(1000) = 9?

(c) Which do you think is greater f’(50), or f’(500)? What about

f’(5000)?

2.7 Derivatives and Rates of Change

Solution

(a) f’(x) is the rate of change of production cost with respect to the

number of yards produced (called marginal cost in economics)

Units: dollars per yard

(b) After 1000 yards of fabric have been manufactured, the rate atwhich the production cost is increasing is $9/yard (or the cost ofmanufacturing the 1001st yard increases about $9 – for well behavedfunction)

2.7 Derivatives and Rates of Change

(c) The rate at which the production cost is increasing (per yard) is

probably lower when x = 500 than when x = 50 (the cost of making the

500th yard is less than the cost of the 50th yard) because ofeconomies of scale

But, as production expands, the resulting large-scale operationmight become inefficient and there might be overtime costs Thus it ispossible that the rate of increase of costs will eventually start to rise

2.7 Derivatives and Rates of Change

Example 2 (p 149) Let D(t) be the

US national debt at time t The table

gives approximate values of this

function by providing end of year

estimates, in billions of dollars, from

1980 to 2005 Interpret and estimate

2.7 Derivatives and Rates of Change

Solution. D’(1990) means the rate of change of dept with respect to

)(lim)

D D

2.8 The Derivative as a Function

Given any number x for which the limit

exists, we assign to x the number f’(x).

So we can regard f’ as a new function, called the derivative

of f and defined by the above equation

The domain of f’ = {x | f’(x) exists}

)(')

()

(lim

h

x f h

x f



2.8 The Derivative as a Function

Example. Use the graph

of f(x) to sketch the graph of

2.8 The Derivative as a Function

Leibniz Notation

Some common alternative notations for f’(x):

The symbols D and d/dx are called differentiation operators

(Toán tử vi phân)

).

( )

( ' )

(

dx

d dx

dy y

y

2.8 The Derivative as a Function

The value of the derivative at a particular number a can be

expressed in some ways:

)

()

(')

(

dx

d dx

dy y

y D

a x a

x a

x x

a x

2.8 The Derivative as a Function

(1) f(x) = x2 – 2x (2) f(x) = |x|

2.8 The Derivative as a Function

How Can a Function Fail to Be Differentiable?

(a) f has a “corner” or “kink” in it,

(b) f has a discontinuity,

(c) f has vertical tangent.

2.8 The Derivative as a Function

Higher – Order Derivatives

The derivative f’(x) of a function y = f(x) is itself a function If it

is differentiable at x, we can calculate its derivative, which we can call

the second derivative of f and denote by f”(x), thus

)

()

()

()

(

"

2 2

2

x f D y

D x

f dx

d x

f dx

d dx

d dx

y d x

2.8 The Derivative as a Function

Similarly, we can consider the third-, fourth-, and in general

nth-order derivatives The nth derivative of y = f(x) is

Example Find y (n) of y = 1/(1 + x)

).

( )

( )

() ( )

(

x f D y

D x

f dx

d dx

y

d x

106 đến 107

trang 150 đến trang 153.

56, 57 từ trang 162 đến trang 165.

PETROVIETNAM UNIVERSITY

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