average rate of change on [a,b]. 1. find the average rate of change of f(x) = x 3 + 1 on a) [1,...
TRANSCRIPT
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
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
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*
)
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2 4x
x xx
x
22 2(2) 4 12
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
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
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
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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
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9( 1)(i
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x
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1
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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
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1
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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
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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
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2
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x
x
3
7 7 2lim
7 2
2
3x
x
x
x
x
10.25
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3
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x
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3
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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