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What does it mean to say that a line is tangent to a curve at a point? . P For a circle, the tangent line at a point P is the line that is perpendicular to the radial line at point P. For a general curve, however, the problem is more difficult.
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Lesson 2.1The Derivative and the Tangent Line Problem
Quiz
What does it mean to say that a line is tangent to a curve at a point?
.. P For a circle, the tangent line at a point P is the line that is perpendicular to the radial line at point P.
For a general curve, however, the problem is more difficult.
Essentially, the problem of finding the tangent line at a point P boils down to the problem of finding the slope of the tangent line at point P. You can approximate this slope using a secant line through two points on the curve.
(c+x, f(c+ x).
.(c, f(c))-------------
---------------x
y
0 0 0
Find the slope of 2 3 at 2,1 .
2 2 2 2 3 1 2lim l i 2im l mx x x
f x x
f x f x xx x x
2.1 The Derivative and the Tangent Line Problem
0
limx
f c x f cm
x
2.1 The Derivative and the Tangent Line Problem
0
limx
f c x f cm
x
2
2 2 2 2 2
0 0
2
0 0
Find the derivative of +1.
1 1 2lim lim
2lim lim 2 2
x x
x x
f x x
x x x x x x x xx x
x x x x x xx
0
' limx
f x x f xf x
x
The slope of a function is its
derivative.
2.1 The Derivative and the Tangent Line Problem
2
2 2
0
Find the slope of +1 at 0,1 & 1,2 .
1 1lim
' 0 2 0
' 1
2
0
2 1 2
x
f x x
x xx
x
xf
f
2.1 The Derivative and the Tangent Line Problem
'f x
dydx
'y d f x
dx
xD y
2.1 The Derivative and the Tangent Line Problem
3 2 ' ?f x x x f x
0' lim
x
f x x f xf x
x
3 3 2 2 3
0 0
3 2 2 2 2 3 3
0
2 2 32 2 2
0 0
2 2 2 2 2 2lim lim
2 2 2 2 2lim
3 3 2lim lim 3 3 2 3 2
x x
x
x x
x x x x x x x x x x x x x x x x
x xx x x x x x x x x x x x x x
xx x x x x x x x x x x
x
2' 3 2f x x
3 2 22a b a b a ab b
Complete onWhiteboard
2.1 The Derivative and the Tangent Line Problem
' ?f x x f x
0' lim
x
f x x f xf x
x
0limx
x x xx
1'2
f xx
0
0
0
lim
lim
1lim 12
x
x
x
x x x x x x
x
x x x xx x x
x x x x
x x x
Complete onWhiteboard
2.1 The Derivative and the Tangent Line Problem
Find the slope of at 1,1 and 4,2 . Discuss the
behavior of at the origin.
f x x
f x
1'2
f xx
1' 1
2121
f
1' 42
144
f 0
1 1lim02x x
has a vertical tangent @ 0,0 .f0x
2.1 The Derivative and Tangent Line Problem
is differentiable on a,b
is a continuous, smooth curve on , and
does not have a vertical tangent.
f
fa b
f
AP EXAM
2If , find '.y yt
0
0
0
0
20
20 2
' lim
2 / 2 /lim
22
lim
2 21lim
1 2lim
2li 2m
t
t
t
t
t
t
f t t f ty
tt t t
tt tt
t t t t t tt
t t tt t
t
t t
tt t t t
t t t
Differentiability and continuityThe following alternative limit form of the derivative
is useful in investigating the relationship between differentiability and continuity. The derivative of f at c is
' lim
x c
f x f cf c
x c
}
2.1 The Derivative and the Tangent Line Problem
2 2
2 2
2 02lim lim 1
2 22 02
lim lim 12 2
x x
x x
xf x fx x
xf x fx x
' 2f DNE
2 Find ' at (2,0).f x x f x ' lim
x c
f x f cf c
x c
2.1 The Derivative and the Tangent Line Problem
1/ 3 Find ' at 0.f x x f x x
0
1/3
0
2/30
0lim
00lim
01lim tangent is vertic l ' 0a
x
x
x
f x fx
x
xNE
x
f D
Vertical Tangent LineIf a function is continuous at a point c and , then x = c
is a vertical tangent line for the function.
limx c
f x f cx c
2.1 The Derivative and the Tangent Line Problem
THM 2.1Differentiability Continuity
HW 2.1/3,4,5-15odd,16,21,24,25,27-32,33,35,37,41,45,47,62
x 0 x 0
x
2
0 x 0
x 0
113. ' ?1
1 11 1' lim lim
1 11 1lim lim1 1 1 1
1lim1 1
11
f x f xx
f x x f x x x xf xx x
x x x xx x x x x x x x
x x x x
x 0 x 0
x 0
x 0
x 0
x 0
116. ' ?
1 1
' lim li
1
m
1lim
1lim
1lim
1l m2 2
1i
f x f xx
f x x f x x x xf xx x
x x xx x x x
x x x x x xx x x x x x x
x x xx x x x x x x
x x x x x x xx x x
x 0 x 0
x 0 0
0 0
(
121. 1, 2
Find the tangent to @ .1 1
' lim lim
1 1 1 1lim lim
1lim l
)
im
1,2
x
x x
f x xxf
x x xf x x f x x x xf xx x
x x x x x x xx
x x x x x x x x
x x x x x x x xx x x x
1 1
2
2
1
0
2 0 1 2
1
0' 11
x x x
f
y
x
x
y
y m x
x y
x
3 / 2
3 / 23 / 2 3 / 2
124.
Find the tangent to that is to 2 6 0.1' (Earlier Problem)
21 1 1 1 1 1
2211 1 We need the line through 1,1 with 2
11 12
f xx
f x y
f xx
x xx
y x
x
f m
2 2
2 21 1 1 1
21
1 1 11 0 1lim lim lim lim1 1 1 1 1 1
1 2lim vertical tangent01
The limit from the right DNE
since is undefined for 1.is not different ab i le
x x x x
x
f x f x xx xx x x x x x
x
x
ff x
at 1.x
2
Find the derivative from the left and the right at 1.
Is the function differentiable at 11?
x
x f x x
2 Use the definition of derivative to find ' if 2 1.f x f x x x
0
' limx
f x x f xf x
x
2 2
0 0
2 2 2
0
2 2 2
0
2
0 0
2 1 2 1' lim lim
2 2 1 2 1lim
2 4 2 1 2 1lim
4 2lim lim 1 14 2 4
x x
x
x
x x
x x x x x xf x x f xf x
x xx x x x x x x x
xx x x x x x x x
xx x x x x x
xx
Yea! You finished the lesson!
Now get to work!
(c+x, f(c+ x)..(c, f(c)) ----------
------------
x
y
.(c+x, f(c+ x).
(c, f(c))-------------
---------------
x
y
.(c, f(c))
If f is defined on an open interval containing c, and if the limit
exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at point (c, f(c)).
lim
x→0
f(c + x) – f(c)
x= m
:4 1 4 3
4. (a) 4 1 4 3
4 1 (b) ' 1
4 1116.
2x x1 324. 2 2
28. (e)30. (a)32. (d)62. '
Even Answersf f f f
f ff
y x
f x DNE