Average rate of changeAverage rate of changeon [a,b]on [a,b]

( ) ( )f b

b aA

fC

aR

Average rate of changeAverage rate of changeon [a,b]on [a,b]

1. Find the average rate of change of f(x) = 1. Find the average rate of change of f(x) = xx33 + 1 on + 1 on

A) [1, 2],A) [1, 2], B) [-1, 1]B) [-1, 1]

( ) ( )f b

b aA

fC

aR

( ) ( )f b

b aA

fC

aR

3 32 1

2 1

(2 1) (1 1)

1

( ( )7

)AR

fC

f

Average rate of changeAverage rate of changeon [a,b]on [a,b]

1. Find the average rate of change of f(x) = 1. Find the average rate of change of f(x) = xx33 + 1 on + 1 on

B) [-1, 1]B) [-1, 1]

( ) ( )f b

b aA

fC

aR

( ) ( )1 1

1 ) 21

0

(

21A

fRC

f

Limit Algebraic RulesLimit Algebraic Rules

Rule 1 – Plug the x xalue Rule 1 – Plug the x xalue into the equationinto the equation

If you get a/b andIf you get a/b and b 0, the answer is a/bb 0, the answer is a/b b = 0 and a 0, the answer is d.n.e.b = 0 and a 0, the answer is d.n.e. you get 0/0 then go to higher rulesyou get 0/0 then go to higher rules

..

..

2

3

5lim

1x

x

x

..

2.02.0

0.10.1

2

3

5lim

1x

x

x

..

2

3

9lim

1x

x

x

..

0.00.0

0.10.1

2

3

9lim

1x

x

x

..

A.A. -3-3

B.B. 00

C.C. d.n.ed.n.e

2

3

12lim

3x

x

x

..

A.A. -3-3

B.B. 00

C.C. d.n.ed.n.e

2

3

12lim

3x

x

x

Limit Algebraic RulesLimit Algebraic Rules

Rule 2 – Factor and cancelRule 2 – Factor and cancel return to Rule 1return to Rule 1

Limit Algebraic RulesLimit Algebraic Rules

Rule 2 – Factor and cancelRule 2 – Factor and cancel return to Rule 1return to Rule 1

Diff of 2 squares (aDiff of 2 squares (a22 - b - b22) = (a + b)(a – b)) = (a + b)(a – b)

Limit Algebraic RulesLimit Algebraic Rules

Rule 2 – Factor and cancelRule 2 – Factor and cancel return to Rule 1return to Rule 1

SOAP (aSOAP (a33 + b + b33) = (a + b)(a) = (a + b)(a22 – ab + b – ab + b22))

Limit Algebraic RulesLimit Algebraic Rules

Rule 2 – Factor and cancelRule 2 – Factor and cancel return to Rule 1return to Rule 1

a xa x22 + bx + c Find two numbers whose + bx + c Find two numbers whose product is ac, but add to bproduct is ac, but add to b

Limit Algebraic RulesLimit Algebraic Rules

Rule 2 – Factor and cancelRule 2 – Factor and cancel return to Rule 1return to Rule 1

Limit Algebraic RulesLimit Algebraic Rules

Rule 2 – Factor and cancelRule 2 – Factor and cancel return to Rule 1return to Rule 1

a xa x22 + bx + c Find two + bx + c Find two

numbers whose product is ac, but add to bnumbers whose product is ac, but add to b

Limit Algebraic RulesLimit Algebraic Rules

Rule 2 – Factor and cancelRule 2 – Factor and cancel return to Rule 1return to Rule 1

a xa x22 + bx + c Find two + bx + c Find two

numbers whose product is ac, but add to bnumbers whose product is ac, but add to b 2 x2 x22 + 3x - 9 Find two + 3x - 9 Find two

numbers whose product is -18, but add to 3numbers whose product is -18, but add to 3

Limit Algebraic RulesLimit Algebraic Rules

2 x2 x22 + 3x - 9 Find two + 3x - 9 Find two

numbers whose product is numbers whose product is

-18, but add to 3-18, but add to 3

= 2 x= 2 x22 + 6x - 3x - 9 + 6x - 3x - 9

=2x(x + 3) – 3(x + 3)=2x(x + 3) – 3(x + 3)

= (2x – 3)(x + 3)= (2x – 3)(x + 3)

18 -118 -1 9 -29 -2 6 -36 -3 3 -63 -6 2 -92 -9 1 -181 -18

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

SOAP (aSOAP (a33 + b + b33) = (a + b)(a) = (a + b)(a22 – ab + b – ab + b22))

3

2l

8

2imx

x

x

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

SOAP (aSOAP (a33 + b + b33) = (a + b)(a) = (a + b)(a22 – ab + b – ab + b22))

3

2l

8

2imx

x

x

2

2(( 2)lim

2 1( )*x

xx

x

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

SOAP (aSOAP (a33 + b + b33) = (a + b)(a) = (a + b)(a22 – ab + b – ab + b22))

3

2l

8

2imx

x

x

2

2(li

( 2)

( 2 1*

)

)m

2 4x

x xx

x

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

SOAP (aSOAP (a33 + b + b33) = (a + b)(a) = (a + b)(a22 – ab + b – ab + b22))

3

2l

8

2imx

x

x

2

2(li

( 2)

( 2 1*

)

)m

2 4x

x xx

x

22 2(2) 4 12

..2

2( 2)( )

2i

2 4l mx

x x x

x

..

12.012.0

0.10.1

2

2( 2)( )

2i

2 4l mx

x x x

x

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

Difference of two squaresDifference of two squares

2

4 2

16

4limx

x

x x

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

Difference of two squaresDifference of two squares

2

4 2

16

4limx

x

x x

4

( 4)( 4)

( 4)*limx

x x

x x

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

Difference of two squaresDifference of two squares

2

4 2

16

4limx

x

x x

4

( 4)( 4)

( 4)i

*l mx

x x

x x

4 4

4

2

..

3

2

2

9li

4m

3x

x

x x

3

( 3)( 3l

)

( 3)( 1i

)mx

x x

x x

..

3.03.0

0.10.1

3

2

2

9li

4m

3x

x

x x

3

( 3)( 3) 63

( 3)( 1) 2limx

x x

x x

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

a xa x22 + bx + c Find two numbers whose + bx + c Find two numbers whose product is ac, but add to bproduct is ac, but add to b

2

2

9x 12x im

l

12

2x x

Limit Algebraic RulesLimit Algebraic Rules

9 x9 x22 + 12x - 12 Find two + 12x - 12 Find two

numbers whose product is numbers whose product is

-108, but add to 12-108, but add to 12

= 9 x= 9 x22 + 18x - 6x - 12 + 18x - 6x - 12

= 9x(x + 2) -6(x + 2)= 9x(x + 2) -6(x + 2)

= (9x – 6)(x + 2)= (9x – 6)(x + 2)

108 -1108 -1 54 -254 -2 36 -336 -3 27 -427 -4 18 -618 -6

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

a xa x22 + bx + c Find two numbers whose + bx + c Find two numbers whose product is ac, but add to bproduct is ac, but add to b

2

2

9x 12x im

l

12

2x x

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

a xa x22 + bx + c Find two numbers whose + bx + c Find two numbers whose product is ac, but add to bproduct is ac, but add to b

2

(9x-6)(x+2li

)

2m

x x

2

2

9x 12x im

l

12

2x x

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

a xa x22 + bx + c Find two numbers whose + bx + c Find two numbers whose product is ac, but add to bproduct is ac, but add to b

2

(9x-6)(x+2

2lim 24

)x x

2

2

9x 12x im

l

12

2x x

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

a xa x22 + bx + c Find two numbers whose + bx + c Find two numbers whose product is ac, but add to bproduct is ac, but add to b

2

(9x-6)(x+2

2lim 24

)x x

2

2

9x 12x im

l

12

2x x

New problemNew problem

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

a xa x22 + bx + c Find two numbers whose + bx + c Find two numbers whose product is ac, but add to bproduct is ac, but add to b

1

2

2

6 15

9i

9m

9l

x

x x

x 1

23(2 5 3l

)

9( 1)(i

1)m

x

x x

x x

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

a xa x22 + bx + c Find two numbers whose + bx + c Find two numbers whose product is ac, but add to bproduct is ac, but add to b

1

2

2

6 15

9i

9m

9l

x

x x

x 1

23(2 5 3l

)

9( 1)(i

1)m

x

x x

x x

1

23(2 3 2 3)

9l

( 1)( 1)im

x

x x x

x x

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

a xa x22 + bx + c Find two numbers whose + bx + c Find two numbers whose product is ac, but add to bproduct is ac, but add to b

1

2

2

6 15

9i

9m

9l

x

x x

x 1

23(2 3 2 3)

9l

( 1)( 1)im

x

x x x

x x

1

3[ (2 3) 1(2 3)]

9( 1)(l m

1)ix

x x x

x x

Rule 2 – Factor and cancel Rule 2 – Factor and cancel

a xa x22 + bx + c Find two numbers whose + bx + c Find two numbers whose product is ac, but add to bproduct is ac, but add to b

1

2

2

6 15

9i

9m

9l

x

x x

x 1

23(2 3 2 3)

9l

( 1)( 1)im

x

x x x

x x

_3(1) 10.166666

3(3)( 2) 6

1

3[ (2 3) (2 3)]

9l

1m

1i

( )( )

x

x x x

x x 1

3( 1)

9( 1)

(2li

( 1)m

3)x

x

x

x

x

Rule 3 – Conjugate Rule 3 – Conjugate

3

2 1l

3

3i

5m

x

x

x

3

2 15 3

3

2 15 3lim

2 15 3x

x

x

x

x

Rule 3 – Conjugate Rule 3 – Conjugate

3

2 1l

3

3i

5m

x

x

x

3

2 15 3

3

2 15 3lim

2 15 3x

x

x

x

x

3

2 15 9 1lim

13 2 5 3x

x

x x

Rule 3 – Conjugate Rule 3 – Conjugate

3

2 1l

3

3i

5m

x

x

x

3

2 15 3

3

2 15 3lim

2 15 3x

x

x

x

x

_20.3333

3 3

3

2 15 9 1lim

13 2 5 3x

x

x x

3

3 2

3lim

2 15 3x

x

x x

Rule 3 – Conjugate Rule 3 – Conjugate

3

7l m

2

3ix

x

x

3

7 7 2lim

7 2

2

3x

x

x

x

x

10.25

2 2

3

7 4l m

7 23

1ix

x

x x

3

1lim

7

3

3 2x

x

x x

. 4. 4

4.04.0

0.10.1 2

2

2 2

2 2lim

2 2x

x

x

x

x

2

2

2 4

2 2lim

1x

x

x

x

If lim f(x) = L and lim g(x) = M, thenIf lim f(x) = L and lim g(x) = M, then

lim(f(x) + g(x)) = L + Mlim(f(x) + g(x)) = L + M

lim(f(x) - g(x)) = L - Mlim(f(x) - g(x)) = L - M

lim(k f(x)) = k Llim(k f(x)) = k L

lim(f(x)g(x)) = L M lim(f(x)g(x)) = L M

lim(f(x) / g(x)) = L / Mlim(f(x) / g(x)) = L / M

lim [f(x)lim [f(x)nn]= L]= Lnn

limlim

f ( ) lim f ( )n nx x