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first derivative
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Expression 12: "w" Subscript, "i" "d" "t" "h" , Baseline equals 10 Superscript, negative "s" Subscript, "m" "a" "l" "l" "n" "e" "s" "s" , Baseline , Baseline
w
i
d
t
h
=
1
0
−
s
m
a
l
l
n
e
s
s
equals
=
0.0 1
0
.
0
1
12
Click the display button to show the coordinates of P₂ (you'll have to scroll very close to see the triangle!)
Click the display button to show the coordinates of P₂ (you'll have to scroll very close to see the triangle!)
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13
...and the slope formula, remember slope=(y₂-y₁)/(x₂-x₁)
...and the slope formula, remember slope=(y₂-y₁)/(x₂-x₁)
16
Expression 17: "s" Subscript, "l" "o" "p" "e" , Baseline equals StartFraction, "P" Subscript, 2 , Baseline . "y" minus "P" Subscript, 1 , Baseline . "y" Over "P" Subscript, 2 , Baseline . "x" minus "P" Subscript, 1 , Baseline . "x" , EndFraction
s
l
o
p
e
=
P
2
.
y
−
P
1
.
y
P
2
.
x
−
P
1
.
x
equals
=
1.1 2 4
1
.
1
2
4
17
If you get the width as small as possible, the slope gets closer and closer to a specific value (usually different for each x-coordinate)
If you get the width as small as possible, the slope gets closer and closer to a specific value (usually different for each x-coordinate)
18
Expression 19: "f" prime left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis
f
′
x
0
equals
=
1.1 1 4
1
.
1
1
4
19
If you do this for every point on the graph, you get a tangent slope for each of them
If you do this for every point on the graph, you get a tangent slope for each of them
20
So we shorten this process by doing the slope calculation *without* choosing a specific x₀ coordinate
So we shorten this process by doing the slope calculation *without* choosing a specific x₀ coordinate
21
And this results in something we can use to calculate the tangent slope anywhere
And this results in something we can use to calculate the tangent slope anywhere
22
Click the display button on the left!
Click the display button on the left!
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23
Other visuals
Other visuals
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27
29
powered by
powered by
left parenthesis, 0.5 5 7 , 0.3 1 0 2 5 , right parenthesis
0
.
5
5
7
,
0
.
3
1
0
2
5
(x,y)
"x"
x
"y"
y
"a" squared
a
2
"a" Superscript, "b" , Baseline
a
b
7
7
8
8
9
9
over
÷
functions
(
(
)
)
less than
<
greater than
>
4
4
5
5
6
6
times
×
| "a" |
|
a
|
,
,
less than or equal to
≤
greater than or equal to
≥
1
1
2
2
3
3
negative
−
A B C
StartRoot, , EndRoot
pi
π
0
0
.
.
equals
=
positive
+
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