3.9 Differentials
Key Concept:
Sometimes we can approximate complicated functions with simpler ones that give the accuracy we want for specific applications without being so hard to work with!
the tangent line approximation; "linearization"
Idea:
for a brief interval to either side of x = c,
the y­values along the tangent line the y­values on the curve.
1
y = f(x)
P(c, f (c))
slope = f '(c)
tangent
line
c
.
at (c, f(c)), using y ­ y1 = m (x ­ x
1)
y ­ f (c) = f '(c) . (x ­ c)
T(x) : y = f (c) + f '(c). (x ­ c)
example:
Find the tangent line approximation of f (x) = 1 + x at x = 0.
2
Differentials
Purpose:
used to estimate change
Concept:
f(x) suppose we know the value of a differentiable function at a particular point c and want to predict how much this value will change if we move nearby to a point c + x or since x is small anyway, say x dx (read as "the differential of x"),
= ...
c + dx .
nearby point is at 3
y = f(x)
(c, f (c))
T(x) : y = f (c) + f '(c). (x ­ c)
c
4
y = f(x)
(c + dx, f (c + dx))
(c, f (c))
dx
T(x) : y = f (c) + f '(c).(x ­ c)
c
c + dx
5
y = f(x)
(c + dx, f (c + dx))
}
y = curve
f (c + dx) ­ f (c)
(c, f (c))
dx
T(x) : y = f (c) + f '(c).(x ­ c)
c
c + dx
6
y = f(x)
(c + dx, f (c + dx))
} T(x) = dy = ?!!
(c, f (c))
dx
T(x) : y = f (c) + f '(c).(x ­ c)
c
c + dx
7
y = f(x)
(c + dx, f (c + dx))
(c + dx, T(c + dx))
} T(x) = dy = ?!!
(c, f (c))
dx
T(x) : y = f (c) + f '(c).(x ­ c)
c
c + dx
8
Exact Change:
y = f (c + dx) ­ f (c)
T(x) = dy = ?!!
Estimated Change:
T(x) = T(c + dx) ­ T(c)
using:
y = T(x) = f (c) + f '(c) .(x ­ c)
T(x) = T(c + dx) ­ T(c)
= T(c + dx) ­ T(c)
same!
= f (c) + f '(c) .((c+dx) ­ c) ­ f (c)
= f (c) + f '(c) .( c + dx ­ c) ­ f (c)
.
= f (c) + f '(c) ( c + dx ­ c) ­ f (c)
dy = T(x) = f '(c) dx
.
Def.
The differential of y , is
.
dy = f '(x) dx
9
y = f(x)
(c + dx, f (c + dx))
} T(x) = dy = f '(c) dx
(c, f (c))
.
dx
T(x) : y = f (c) + f '(c).(x ­ c)
c
c + dx
10
Geometric Interpretation:
y = f(x)
}
y
}dy
(c, f (c))
x = dx
.
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NOTE:
The change in the tangent line, T(x) or dy is just a constant ( f '(c)) times dx !!
Notation Alert!
If y = f(x)
dy
dx
= f '(x)
dy = f '(x). dx
dy
. . . we may regard the derivative dx
as a quotient of differentials.
12
example:
Comparing y and dy.
2
Let y = x
+ 2x , x = 0 , and x = dx = 0.1 .
Evaluate and compare y and dy.
13
3.9 Differentials
[Day 2]
Calculating Differentials
Procedure: If y = f(x), to find dy :
1. Find dy dx
.
2. "Multiply" both sides of the equation by the dx. differential form
Key: A differential on one side of an equation calls for a differential on the other side!
14
differential form of the Product Rule
15
Differential Formulas p231
example:
Find the differential, dy.
y2 = 6x4
Prep Students
for
Application
Example 1
﴾blue Handout!﴿
Find the differential, dV.
V
=
sphere
4
3
r3
16
Differentials Applied To Approximating Function Values
y = f(x)
(x + x, f (x + x))
} dy
(x, f (x))
x = dx
} f (x)
f (x + x)
.
x
f (x + x)
x + x f (x) + dy
or
y + y
y + dy
17
example:
Key:
implies
that you
know!
4
625
4
Approximate 624
[p234 #47]
choose a value for x that makes the calculations easier!
=5
18
example:
[p233 #22]
. . . use differentials and the graph of f to approximate (a) f(1.9) and (b) f(2.04).
19
example:
[p233 #28]
. . . use differentials and the graph of g' to approximate (a) g(2.93) and (b) g(3.1) given that g(3) = 8.
20
Error Propagation ­ tolerance
One way this occurs in practice is in the estimation of errors propagated by physical measuring devices.
Measurement
error
}
}
Propagated
error
f (x + x) ­ f (x) = y
Exact
value
{
{
Measured
value
DVD #4 Lesson ﴾~ 17:00﴿
* All Cell Phones Off!! *
21
Error Propagation ­ Summary
True
(Exact)
change
relative
change
percentage
change
Estimate
(Approximation)
f
df
f
df
f
f
f
f
. 100
df
f
. 100
22
Assignment:
p233­234 #1, 2, 4, 5
#7­10
#11­20
#45, 46, 48
#21, 23
#25, 27
#30, 31, 33, 35
Slide 2
Slide 13
Slide 16
Slide 18
Slide 19
Slide 20
Slide 22
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