How accurate is your estimate Differential Notation The Linear Approximation to y f x is often written using the differentials dx and dy In this notation dx is used instead of ID: 616072
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Slide1Slide2
Use the Linear Approximation to estimate
How accurate is your estimate?Slide3
Differential Notation
The Linear Approximation to
y
=
f
(x) is often written using the “differentials”
dx
and
dy
. In this notation,
dx
is used instead of
Δ
x to represent the change in x, and dy is the corresponding vertical change in the tangent line:
This is simply another way of writing
Δ
f
≈
f
(a)Δ
x. Slide4
Differential Notation
How
much larger isSlide5
Thermal Expansion
A
thin metal cable has length
L
= 12 cm when the temperature is
T
= 21°C. Estimate the change in length when
T
rises to 24°C, assuming that
where
k
= 1.7 × 10
−5°C (k
is called the coefficient of thermal expansion).Slide6
Suppose that we measure the
diameter D
of a circle and use this result to compute the
area
of the circle. If our measurement of
D
is inexact, the area computation will also be inexact. What is the effect of the measurement error on the resulting area computation? This can be estimated using the Linear Approximation, as in the next example.Slide7
Effect
of an Inexact Measurement
The
Bonzo
Pizza Company claims that its pizzas are circular with diameter 50 cm
(a)
What is the area of the pizza?
(b)
Estimate the quantity of pizza lost or gained if the diameter is off by at most 1.2 cm.Slide8
Approximating
f(x)
by Linearization
If
f
is differentiable at
x = a
and
x
is closed to
a
, thenSlide9
Compute the linearization of at
a
= 1.Slide10
Estimate and compute the percentage error
.Slide11
The Size of the Error
The examples in this section may have convinced you that the Linear Approximation yields a good approximation to
Δ
f
when Δx is small, but if we want to rely on the Linear Approximation, we need to know more about the size of the error:
Remember that the error E is simply the vertical gap between the graph and the tangent
line
.
where
K
is the maximum value
of |
f
(
x
)| on the interval from
a
to
a
+
Δ
x
. Slide12
The Size of the Error
The Error Bound tells us two important things. First, it says that the error is small when the second
derivative
(and hence
K
) is small. This makes sense, because
f
(
x
) measures how quickly the tangent lines change direction. When |
f
(
x
)| is smaller, the graph is flatter and the Linear Approximation is more accurate over a larger interval around
x
=
a
(compare the graphs). Slide13
The Size of the Error
Second, the Error Bound tells us that the error is of
order two
in
Δ
x
, meaning that
E
is no larger than a constant times (
Δ
x
)
2
. So if Δx is small, say Δx = 10−n, then E has substantially smaller order of magnitude (Δx)2 = 10−2
n. In particular, E/Δx tends to zero (because E/Δx <
KΔx), so the Error Bound tells us that the graph becomes nearly indistinguishable from its tangent line as we zoom in on the graph around x = a. This is a precise version of the “local linearity” property discussed in Section 3.2Slide14
4.1
SUMMARY
Let
Δ
f
=
f (a
+
Δ
x
) −
f (a)
. The
Linear Approximation
is the estimate
Differential notation:
dx
is the change in
x
,
dy
=
f
(
a
)dx
,
Δy
= f (a
+ dx
) − f
(a
). In this notation, the Linear Approximation reads
The
linearization
of
f
(
x
) at
x
=
a
is the function
The Linear Approximation is equivalent to the approximation
The error in the Linear Approximation is the quantity
In
many cases, the percentage error is more important than the error itself:Slide15