NOTES
Section 2.6 Differentials
The derivative of f at x is the limit of the difference
quotient:
f ′( x) = lim
h→0
f ( x + h) − f ( x )
h
Example: Concept of Increment
For y = f (x) = x3, a change in x from 2 to 2.1 corresponds
to a change in y from y = f (2) = 8 to y = f (2.1) = 9.261.
Increment Notation
Change in x (the increment in x) is denoted by ∆x.
Change in y (the increment in y) is denoted by ∆y.
In the example, ∆x = 2.1 – 2 = 0.1
∆y = f (2.1) – f (2) = 9.261 – 8 = 1.261.
The difference quotient can be written as
f ( x2 ) − f ( x1 ) f ( x + h) − f ( x) f ( x1 + ∆x) − f ( x1 )
=
=
where h = ∆x
∆x
x2 − x1
h
For y = f (x), ∆x = x2 – x1, so x2 = x1 + ∆x, and
∆y = y2 – y1 = f (x2) – f (x1) = f (x1 + ∆x) – f (x1)
∆x can be either positive or negative.
∆y represents the change in y corresponding to a ∆x change
in x.
As shown in the figure,
∆y f ( x + ∆x) − f ( x)
=
∆x
∆x
gives the slope of the secant
line connecting the points
with first coordinates x1 and
x2= x1 + ∆x
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Section 2.6 Differentials
2
NOTES
Example from PPT
Given the function
=
y f=
( x)
(A) Find ∆x, ∆y, and
x2
,
2
∆y
for
=
x1 1 and
=
x2 2.
∆x
(B) Find
f ( x1 + ∆x) − f ( x1 )
for=
x1 1 and ∆
=
x 2.
∆x
(B) Find
f ( x1 + ∆x) − f ( x1 )
for x1 = 1 and x2 = 2.
∆x
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Section 2.6 Differentials
NOTES
Differentials
∆y
exists.
∆x→0 ∆x
Suppose that the limit f ′( x) = lim
For small ∆x, f ′( x) ≈
∆y
∆x
and ∆y ≈ ∆x ⋅ f ′( x)
The differential=
dy
f ′( x) ⋅ ∆x or =
df
f ′( x) ⋅ ∆x
Definition: Differentials
If y = f ( x) defines a differentiable function, then the
differential dy, or df , is defined as the product of
f ′( x) and dx, where dx = ∆x.
Symbolically,=
dy f ′( x) ⋅ dx, or
df =
f ′( x) ⋅ dx where dx =
∆x
The tangent line has slope f´(x) with horizontal change dx.
The vertical change is given by dy = f´(x) dx.
This is illustrated in the figure
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Section 2.6 Differentials
2
Interpretation of Differentials
∆x and dx both represent change in x.
The increment ∆y stands for the actual change in y
corresponding to the change in x.
The differential dy stands for the approximate change in y,
estimated by using derivatives.
∆y ≈ dy =f ′( x) dx
In applications, we use dy to estimate ∆y.
Example: Differentials
Find dy for f (x) = x2 + 3x
Evaluate dy for x = 2 and dx = 0.1.
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125
NOTES
Section 2.6 Differentials
Example : Comparing Increments and Differentials
Let =
y
f ( x=
) 6 x − x2.
(A) Find ∆y and dy when x = 2.
(B) Find ∆y and dy from part (A) for Δx = 0.1, 0.2, and 0.3
Example: Cost-Revenue
A company manufactures and sells x transistor radios per
week. If the weekly cost and revenue equations are
C ( x)= 5, 000 + 2 x
x2
R( x)= 10 x −
1, 000
0 ≤ x ≤ 8, 000
use differentials to approximate changes in revenue and
profit if production is increased from 2,000 to 2,010
units/week.
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NOTES