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Tangent LinesTangent Lines
1.1. Equation of linesEquation of lines
2.2. Equation of secant linesEquation of secant lines
3.3. Equation of tangent linesEquation of tangent lines
Equation of LinesEquation of Lines
Write the equation of a line that passes Write the equation of a line that passes through (-3, 1) with a slope of – ½ .through (-3, 1) with a slope of – ½ .
oror
oror( 3)
10.5
y
x
0.5(1 3)xy
( 3 10.5 )y x
Equation of LinesEquation of Lines
Write the equation of a line that passes Write the equation of a line that passes through (0, 1) with a slope of ½ .through (0, 1) with a slope of ½ .
oror
oror0
01
.5y
x
0 51 .y x
5 10.y x
Equation of LinesEquation of Lines
Write the equation of the line .Write the equation of the line .
oror
oror00
1.5
x
y
0.5 1y x
20
0.5
x
y
0.5( 2)y x
LinesLines
When writing the equation of a line that When writing the equation of a line that passes through (passes through (00, , 11) with a slope of ) with a slope of -3-3 . .
What is the missing What is the missing blueblue number? number?
Save your answer.Save your answer.
13
y
x
Passes through (Passes through (00, , 11) with ) with a slope of a slope of -3-3. What is the . What is the missing missing blueblue number? number?
0.00.0
0.10.1
13
y
x
( _1 _)3y x
Write the equation of a Write the equation of a greengreen line line that passes through (0, 1) with a that passes through (0, 1) with a slope of -3 .slope of -3 .What is the missing What is the missing greengreen number number mm??
-3.0-3.0
0.50.5
1
0
y
xm
1y mx
1y mx
Secant LinesSecant Lines
Write the equation of Write the equation of the secant line that the secant line that passes through passes through
and (and (200200, , 220220).).
(0,0)(95,70)( ,117 120)( ,143 170)( ,184 210)( ,163 195)
What is the slope of this secant What is the slope of this secant line (line (184184, , 210210) and () and (200200, , 220220)? )?
0.6250.625
0.20.2
Secant LinesSecant Lines
Write the equation Write the equation of the secant line of the secant line that passes that passes through through
and (and (200200, , 220220).).
( ,184 210)( ,191 215)
http://math.georgiasouthern.edu/~bmclean/java/p6.html Secant Lines Secant Lines
http://www.youtube.com/watch?v=P9dpTTpjymE Derive Derive
http://www.9news.com/video/player.aspx?aid=52138&bw= Kids Invest= Kids Invest
The slope of f(x) =xThe slope of f(x) =x2 2
when x=xwhen x=x00 is is
and when x = 1and when x = 1
0 0
00 0
( ) ( )
( )limh
f fx h x
x h x
0 0
0m
) (l
( )ih
x h
h
f f x
0
1( ) ( )lim
1h
h
h
f f
Find the slope of the Find the slope of the tangent line of f(x) = 2x tangent line of f(x) = 2x + 3 when x = 1.+ 3 when x = 1.
1. Calculate f(1+h) – f(1)1. Calculate f(1+h) – f(1)
f(1+h) = 2(1+h) + 3f(1+h) = 2(1+h) + 3
f(1) = 5 f(1) = 5
f(1+h) – f(1) = 2 + 2h + 3 – 5 f(1+h) – f(1) = 2 + 2h + 3 – 5 =2h=2h
2. Divide by h and get 22. Divide by h and get 2
3. Let h go to 0 and get 23. Let h go to 0 and get 2
0
1( ) ( )lim
1h
h
h
f f
Find the slope of the Find the slope of the tangent line of f(x) = xtangent line of f(x) = x22 when x = 1.when x = 1.
1. Calculate f(1+h) – f(1)1. Calculate f(1+h) – f(1)
f(1+h) = 1f(1+h) = 122 + 2h + h + 2h + h22
f(1) = 1f(1) = 122
f(1+h) – f(1) = 2h + hf(1+h) – f(1) = 2h + h2 2 ..
2. Divide by h and get 2 + 2. Divide by h and get 2 + hh
3. Let h go to 0 and get 23. Let h go to 0 and get 2
0limslop
(e
) ( )h
f fx h x
h
Find the slope of the Find the slope of the tangent line of f(x) = xtangent line of f(x) = x22 when x = x.when x = x.
1. Calculate f(x+h) – f(x)1. Calculate f(x+h) – f(x)
f(x+h) = xf(x+h) = x22 + 2xh + h + 2xh + h22
f(x) = xf(x) = x22
f(x+h) – f(x) = 2xh + hf(x+h) – f(x) = 2xh + h2 2 ..
2. Divide by h and get 2x + 2. Divide by h and get 2x + hh
3. Let h go to 0 and get 2x3. Let h go to 0 and get 2x
0limslop
(e
) ( )h
f fx h x
h
Find the slope of the Find the slope of the tangent line of f(x) = xtangent line of f(x) = x22. . f(x+h) - f(x) =f(x+h) - f(x) =
A.A. (x+h)(x+h)22 – x – x22
B.B. xx22 + h + h22 – x – x22
C.C. (x+h)(x – h)(x+h)(x – h)
(x+h)(x+h)22 – x – x22 = =
A.A. xx2 2 + 2xh + h+ 2xh + h22
B.B. hh22
C.C. 2xh2xh + h+ h22
= =
A.A. 2x2x
B.B. 2x + h2x + h22
C.C. 2xh2xh
0limslop
(e
) ( )h
f fx h x
h
22xh + h
h0limh
Average slopeAverage slope
Find the rate of change if it takes 3 hours to drive 210 miles.
What is your average speed or velocity?
( ) (3 0
3
)
0
f f
If it takes 3 hours to drive 210 miles If it takes 3 hours to drive 210 miles
then we averagethen we average
A.A. 1 mile per minute1 mile per minute
B.B. 2 miles per minute2 miles per minute
C.C. 70 miles per hour70 miles per hour
D.D. 55 miles per hour55 miles per hour
Instantaneous slopeInstantaneous slope
What if h went to What if h went to zero?zero?
0'( ) l
( (im
) )h
f x h x
hf x
f
DerivativeDerivative
if the limit exists as one real if the limit exists as one real number. number.
0'( ) l
( (im
) )h
f x h x
hf x
f
..DefinitionDefinitionIf f : D -> K is a function then the derivative of f If f : D -> K is a function then the derivative of f
is a new function, is a new function, f ' : D' -> K' as defined above if the limit f ' : D' -> K' as defined above if the limit
exists. exists. Here the limit exists every where except at x = 1Here the limit exists every where except at x = 1
0'( ) l
( (im
) )h
f x h x
hf x
f
Guess at Guess at
0
( ) ( )1lim
1'( )
hf x
f h
h
f
..Guess at Guess at
0
( ) ( )1lim
1'( )
hf x
f h
h
f
ThusThus
d.n.e.d.n.e.
0'( ) li
1 1m
( ) ( )h
f fhf
hx
..Guess at Guess at
f’(0) – slope of f when x = 0f’(0) – slope of f when x = 0
0'( ) l
( (im
) )h
f x h x
hf x
f
Guess at f ’(3)Guess at f ’(3)
-1.0-1.0
0.490.49
Guess at f ’(-2)Guess at f ’(-2)
-3.0-3.0
1.991.99
Note that the rule is Note that the rule is f '(x) is the slope at the point ( x, f(x) ), f '(x) is the slope at the point ( x, f(x) ), D' is a subset of D, butD' is a subset of D, butK’ has nothing to do with KK’ has nothing to do with K
0'( ) l
( (im
) )h
f x h x
hf x
f
K is the set of distances from homeK is the set of distances from homeK' is the set of speeds K' is the set of speeds K is the set of temperaturesK is the set of temperaturesK' is the set of how fast they rise K' is the set of how fast they rise K is the set of today's profits , K is the set of today's profits , K' tells you how fast they changeK' tells you how fast they changeK is the set of your averages K is the set of your averages K' tells you how fast it is changing. K' tells you how fast it is changing.
0'( ) l
( (im
) )h
f x h x
hf x
f
Theorem If f(x) = c where c Theorem If f(x) = c where c is a real number, then f ' (x) is a real number, then f ' (x) = 0.= 0.
Proof : Lim [f(x+h)-f(x)]/h = Proof : Lim [f(x+h)-f(x)]/h =
Lim (c - c)/h = 0.Lim (c - c)/h = 0.
Examples Examples
If f(x) = 34.25 , then f ’ (x) = 0If f(x) = 34.25 , then f ’ (x) = 0
If f(x) = If f(x) = , then f ’ (x) = 0, then f ’ (x) = 0
If f(x) = 1.3 , find f’(x)If f(x) = 1.3 , find f’(x)
0.00.0
0.10.1
Theorem Theorem If f(x) = x, then f ' (x) = 1. If f(x) = x, then f ' (x) = 1.
Proof : Lim [f(x+h)-f(x)]/h = Proof : Lim [f(x+h)-f(x)]/h =
Lim (x + h - x)/h = Lim h/h = 1Lim (x + h - x)/h = Lim h/h = 1
What is the derivative of x What is the derivative of x grandson?grandson?
One grandpa, one.One grandpa, one.
Theorem If c is a constant,Theorem If c is a constant,(c g) ' (x) = c g ' (x) (c g) ' (x) = c g ' (x)
Proof : Lim [c g(x+h)-c g(x)]/h =Proof : Lim [c g(x+h)-c g(x)]/h =
c Lim [g(x+h) - g(x)]/h = c g ' (x) c Lim [g(x+h) - g(x)]/h = c g ' (x)
Theorem If c is a constant,Theorem If c is a constant,(cf) ' (x) = cf ' (x) (cf) ' (x) = cf ' (x)
( 3 x)’ = 3 (x)’ = 3 or( 3 x)’ = 3 (x)’ = 3 or
If f(x) = 3 x then If f(x) = 3 x then
f ’(x) = 3 times the derivative of xf ’(x) = 3 times the derivative of x
And the derivative of x is . . And the derivative of x is . .
One grandpa, one !!One grandpa, one !!
If f(x) = -2 x then f ’(x) If f(x) = -2 x then f ’(x) = =
-2.0-2.0
0.10.1
TheoremsTheorems
1. (f + g) ' (x) = f ' (x) + g ' (x), and 1. (f + g) ' (x) = f ' (x) + g ' (x), and
2. (f - g) ' (x) = f ' (x) - g ' (x) 2. (f - g) ' (x) = f ' (x) - g ' (x)
1. (f + g) ' (x) = f ' (x) + g ' 1. (f + g) ' (x) = f ' (x) + g ' (x) (x) 2. (f - g) ' (x) = f ' (x) - g ' 2. (f - g) ' (x) = f ' (x) - g ' (x) (x)
If f(x) = 3If f(x) = 322 x + 7, find f ’ x + 7, find f ’ (x)(x)
f ’ (x) = 9 + 0 = 9f ’ (x) = 9 + 0 = 9
If f(x) = x - 7, find f ’ (x)If f(x) = x - 7, find f ’ (x)
f ’ (x) = - 0 = f ’ (x) = - 0 =
55 5
If f(x) = -2 x + 7, find f ’ If f(x) = -2 x + 7, find f ’ (x)(x)
-2.0-2.0
0.10.1
If f(x) = xIf f(x) = xnn then f ' (x) = n x then f ' (x) = n x (n-(n-
1)1)
If f(x) = xIf f(x) = x44 then f ' (x) = 4 xthen f ' (x) = 4 x33
If If 2
3( )g x
x 23x
2 2 3'( ) (3 ) ' 3( ) ' 3( 2 )g x x x x 3
3
66x
x
If f(x) = xIf f(x) = xnn then f ' (x) = n x then f ' (x) = n xn-1 n-1
If f(x) = xIf f(x) = x44 + 3 x+ 3 x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4
f ' (x) = 4 xf ' (x) = 4 x3 3 + . . . .+ . . . .
f ' (x) = 4xf ' (x) = 4x33 + 9 x+ 9 x22 - 4 x – 3 + 0 - 4 x – 3 + 0
f(1) = 1 + 3 – 2 – 3 + 4 = 3f(1) = 1 + 3 – 2 – 3 + 4 = 3
f ’ (1) = 4 + 9 – 4 – 3 = 6f ’ (1) = 4 + 9 – 4 – 3 = 6
3y
If f(x) = xIf f(x) = xnn then f ' (x) = n x then f ' (x) = n x (n-(n-
1)1)If f(x) = If f(x) = xx44 then f ' (x) = 4then f ' (x) = 4 x x33
If f(x) = If f(x) = 44 then f ' (x) = 0then f ' (x) = 0 If If ( ) 3g x x
1
23x1 1 1
2 2 21
'( ) (3 ) ' 3( ) ' 3( )2
g x x x x
1
23 3
2 2x
x
If f(x) = then f ‘(x) =If f(x) = then f ‘(x) =x
1 1
2 21
'( ) ( ) '2
f x x x
1
2 x
Find the equation of the line Find the equation of the line tangent to g when x = 1. tangent to g when x = 1.
If g(x) = xIf g(x) = x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4
g ' (x) = 3 xg ' (x) = 3 x22 - 4 x – 3 + 0 - 4 x – 3 + 0
g (1) =g (1) =
g ' (1) =g ' (1) =
If g(x) = xIf g(x) = x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4find g (1)find g (1)
0.00.0
0.10.1
If g(x) = xIf g(x) = x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4find g’ (1)find g’ (1)
-4.0-4.0
0.10.1
Find the equation of the Find the equation of the line tangent to f when x line tangent to f when x = 1. = 1.
g(1) = 0g(1) = 0
g ' (1) = – 4g ' (1) = – 4
14
0
x
y
4(0 1)xy
( 1)4y x
Find the equation of the line Find the equation of the line tangent to f when x = 1. tangent to f when x = 1.
If f(x) = xIf f(x) = x44 + 3 x+ 3 x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4
f ' (x) = 4xf ' (x) = 4x33 + 9 x+ 9 x22 - 4 x – 3 + 0 - 4 x – 3 + 0
f (1) = 1 + 3 – 2 – 3 + 4 = 3f (1) = 1 + 3 – 2 – 3 + 4 = 3
f ' (1) = 4 + 9 – 4 – 3 = 6 f ' (1) = 4 + 9 – 4 – 3 = 6
Find the equation of the Find the equation of the line tangent to f when x line tangent to f when x = 1. = 1.
f(1) = 1 + 3 – 2 – 3 + 4 = 3f(1) = 1 + 3 – 2 – 3 + 4 = 3
f ' (1) = 4 + 9 – 4 – 3 = 6 f ' (1) = 4 + 9 – 4 – 3 = 6
61
3Y
X
Write the equation of the Write the equation of the tangent line to f when x = 0. tangent line to f when x = 0.
If f(x) = xIf f(x) = x44 + 3 x+ 3 x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4
f ' (x) = 4xf ' (x) = 4x33 + 9 x+ 9 x22 - 4 x – 3 + 0 - 4 x – 3 + 0
f (0) = write downf (0) = write down
f '(0) = for last questionf '(0) = for last question
Write the equation of the Write the equation of the line tangent to f(x) when x line tangent to f(x) when x = 0.= 0.A.A. y - 4 = -3xy - 4 = -3x
B.B. y - 4 = 3xy - 4 = 3x
C.C. y - 3 = -4xy - 3 = -4x
D.D. y - 4 = -3x + 2y - 4 = -3x + 2
http://www.youtube.com/watch?v=P9dpTTpjymE Derive Derive
http://www.9news.com/video/player.aspx?aid=52138&bw= Kids= Kids
http://math.georgiasouthern.edu/~bmclean/java/p6.html Secant Lines Secant Lines
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