Calculus Chapter 2

8/8/2019 Calculus Chapter 2

1/66

Calculus

Mrs. Doughertys Class


2/66

drivers

Start your engines


3/66

3Big Calculus Topics

Limits

Derivatives Integrals


4/66

Chapter 2


5/66

2.1 Limits and continuity


6/66

Limits can be found

Graphically


7/66

Limits can be found

Graphically

Numerically


8/66

Limits can be found

Graphically

Numerically

By direct substitution


9/66

Limits can be found

Graphically

Numerically


By the informal definition


10/66

Limits can be found

Graphically

Numerically


By the informal definition

By the formald

efinition


11/66

Limits Informal Def.


12/66

Limits Informal Def.

Given real numbers c and L, if the values

f(x) of a function approach or equal L


13/66

Limits

Informal Def.


f(x) of a function approach or equal L as

the values of x approach ( but do not

equal c),


14/66

Limits

Informal Def.


f(x) of a function approach or equal L as the

values of x approach ( but do not equal c),

then fhas a limit L as x approaches c.


15/66

Limits

notation


16/66

LIFE IS GOOD


17/66

Theorem 1

Constant Function

f(x)=k

Identity Function

f(x)=x


18/66

Theorem 2

Limits of polynomial functions can be

found by direct substitution.


19/66

Properties of Limits


20/66


If lim f(x) = L 1 and lim g(x) = L2

x-> c x -> c


21/66



x-> c x -> c

Sum Rule:

lim [f(x) + g(x)]= lim f(x) +lim g(x)=L1 + L2


22/66



x-> c x -> c

Difference Rule:

lim [f(x) - g(x)]= L1 - L2


23/66



x-> c x -> c

Product Rule:

lim [f(x) * g(x)]= L1 * L2


24/66



x-> c x -> c

Constant multiple Rule:

lim c f(x) = c L1


25/66



x-> c x -> c

Quotient Rule:

lim [f(x) / g(x)]= L1 / L2 , L1=0 NOT


26/66


27/66

Right-hand and Left-hand Limits


28/66

Theorem 4

A function, f(x),

has a limit as x approaches c


29/66

Theorem 4

A function, f(x),

has a limit as x approaches c

if and only if

the right-hand and left-hand limits at c exist


30/66


31/66

Calculus 2.2

Continuity


32/66

Definition

f(x) is continuous at an interior point of

the domain if


33/66

Definition

f(x) is continuous at an interior point of

the domain if lim f(x) = f(c )

x->c


34/66

Definition

f(x) is continuous at an endpoint

of the domain if


35/66

A continuous function is

continuous at each point of its

domain.


36/66

Definition

Discontinuity

If a function is not continuous at a point c,

then c is called a point ofdiscontinuity.


37/66

Types of Discontinuities

Removable


38/66


Removable

Non-removable

A) jump


39/66


Removable

Non-removable

A) jump

B) oscillating


40/66


41/66

Test for Continuity


42/66

Test for Continuity

y=f(x) is continuous at x=c iff

1.


43/66

Test for Continuity


1. f(c) exists


44/66

Test for Continuity


1. f(c) exists

2. lim f(x) exists

x-> c


45/66

Test for Continuity


1. f(c) exists

2. lim f(x) exists

x -> c

3. f(c ) = lim f(x)

x -> c


46/66

Theorem 5

Properties of Continuous Functions

If f(x) and g(x) are continuous at c, then

1. f(x)+g(x)


47/66


48/66

Theorem 5



1. f(x)+g(x)

2. f(x) g(x)

3. f (x) g(x)


49/66

Theorem 5



1. f(x)+g(x)

2. f(x) g(x)

3. f (x) g(x)

4. k g(x)


50/66


51/66

Theorem 6

If f and g are continuous at c,

Then g f and f g are

continuous at c


52/66

Theorem 7

If f(x) is continuous on [a ,b],

then f(x) has an absolute

maximum,M, and an absoluteminimum,m, on [a ,b].


53/66

Intermediate Value Theorem

for continuous functions

A function that is continuous on

[a,b] takes on every value

between f(a) and f(b).


54/66

Calculus 2.3

The Sandwich Theorem


55/66

If g(x) < f(x) < h(x) for all x /=c

and lim g(x) = lim h(x) = L, then

lim f(x) = L.


56/66

Use sandwich theorem to find

lim sin x

x->0 x


57/66


58/66

Calculus 2.4

Limits Involving Infinity


59/66

Limits at + infinity

are also called end behavior models for

the function.


60/66

Definition

y=b is a horizontal asymptote of f(x) if


61/66


62/66

Case 2 degree of numerator =

degree ofdenominator


63/66

Case 3 degree of numerator >

degree ofdenominator


64/66

Theorem

Polynomial End Behavior Model


65/66

Calculus 2.6

The Formal Definition of a Limit


66/66

Now this is mathematics!!!

Calculus Chapter 2

Documents