Notes 3.9 Differentials
Read summary of Differentials from Princeton Review.
Summarized below are the types of problems you are expected to know for the AP Exam.
Use differentials to approximate the value of a function:
Use the formula: f ( x + Δx) ≈ f ( x) + f ' ( x)Δx
Example 1: Approximate
3
126 and compare to the calculator answer.
Let x = 125, Δx = 1, f ( x) = 3 x` = ( x )
1
3
Apply the given values to the formula.
⎛ 1 −2 ⎞
f ( x + Δx) ≈ 3 x + ⎜ x 3 ⎟(Δx )
⎝3
⎠
1
(1)
≈ 3 125 +
2
33 125
1
≈ 5+
≈ 5.013333...
75
The calculator value is 5.013297935… Not bad!
Determine the differential dy given y = f(x):
Use the formula: dy = f ' ( x)dx
Example 2: Determine the differential, dy, given y =
sin 2 x
x3
dy 2 sin x cos x ⋅ x 3 − 3x 2 sin 2 x
=
dx
x6
x 2 sin x(2 x cos x − 3 sin x )
dy =
dx
x6
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Use differentials to determine propagated error.
Use the formula: dy = f ' ( x)dx
Example 3: Determine the possible error in calculating the area of a square with side measurement = 15 cm
with a possible error of 0.05 cm. Calculate the percent error.
A = s2
dA
= 2s
ds
dA = 2 s ⋅ ds
300
dA = 2(15)(.05)
dA = 1.5 cm
(15,225)
200
2
A=s
2
100
A = 15 2 = 225
-20
percent error =
-10
10
1.5
= .00666666... ≈ .67%
15 2
-100
(15.05, 226.5025)
(15.05, 226.5)
A=s
dA
Actual
area is (15,225)
225
2
20
Due to
measurement
error 0.05,
Area is
approximated
as 226.5
ΔA
ds = Δs
15
Actual side = 15
15 +0.05
Measured side =
15 + 0.05
dA is the approximation of Δ A, the change in area
for a given change in the measurement of the side.
As ds gets smaller, dA becomes more accurate.
Animation of Differential Graph
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