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Beginning CalculusApplications of Differentiation
- Approximations and Differentials -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,FSMT - UPSI
(LECTURE SLIDES SERIES)
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Linear Approximation Quadratic Approximation Differentials
Outlines
Linear Approximation
Quadratic Approximation
Use differentials to estimate values.
Compare linear approximations and differentials.
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Linear Approximation Quadratic Approximation Differentials
Linear Approximation
Definition 1
Let y = f (x) be a curve, and (x0, f (x0)) be a point on the curve.Thelinear approximation of f near x = x0 (x ≈ 0) is
f (x) ≈ f (x0) + f ′ (x0) (x − x0) (1)
where f (x0) + f ′ (x0) (x − x0) is the equation of the tangent line nearx = x0.
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Linear Approximation Quadratic Approximation Differentials
Example
The linear approximation of f (x) =√x + 3 near x = 1:
f (x) =√x + 3, f ′ (x) =
1
2√x + 3
f (1) = 2, f ′ (1) =14
f (x) ≈ f (x0) + f′ (x0) (x − x0)
= f (1) + f ′ (1) (x − 1)
= 2+14(x − 1)
=7+ x4
⇒√x + 3 ≈ 7+ x
4
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Linear Approximation Quadratic Approximation Differentials
Example - continue
4 2 0 2 4
1
2
3
x
y
√x + 3 ≈ 7+ x
4only near x = 1.
√3.98 = 1. 995 0
√3+ 0.98 ≈ 7+ 0.98
4= 1.995
√4.05 = 2. 012 5
√3+ 1.05 ≈ 7+ 1.05
4= 2.0125
√8 = 2. 828 4
√3+ 5 ≈ 7+ 5
4= 3
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Linear Approximation Quadratic Approximation Differentials
Example
The linear approximation of f (x) = ln x near 1:
f (x) = ln x , f ′ (x) =1x
f (1) = 0, f ′ (1) = 1
f (x) ≈ f (1) + f ′ (1) (x − 1)= 0+ (1) (x − 1)= x − 1
⇒ ln x ≈ x − 1
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Linear Approximation Quadratic Approximation Differentials
Example - continue
ln x ≈ x − 1
1 1 2
1
1
2
x
y
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Linear Approximation Quadratic Approximation Differentials
Remark
f ′ (x0) = lim∆x→0
∆y∆x
= lim∆x→0
f (x0 + ∆x)− f (x0)∆x
lim∆x→0
∆y∆x
= f ′ (x0)
∆y∆x≈ f ′ (x0) (2)
Equation (1) is equivalence to Equation (2).
f (x) ≈ f (x0) + f ′ (x0) (x − x0)⇔∆y∆x≈ f ′ (x0) (3)
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Linear Approximation Quadratic Approximation Differentials
Proof of Remark
Proof:
∆y∆x
≈ f ′ (x0)
∆y ≈ f ′ (x0)∆xf (x)− f (x0) ≈ f ′ (x0) (x − x0)
f (x) ≈ f (x0) + f′ (x0) (x − x0)
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Linear Approximation Quadratic Approximation Differentials
Linear Approximations Near 0
f (x) ≈ f (0) + f ′ (0) x (4)
sin x :cos x :ex :
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Linear Approximation Quadratic Approximation Differentials
Geometric Representation of Linear Approximation Near 0
sin x ≈ x
4 2 2 4
4
2
2
4
x
y
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Linear Approximation Quadratic Approximation Differentials
Geometric Representation of Linear Approximation Near 0
cos x ≈ 1
4 2 2 4
1.0
0.5
0.5
1.0
x
y
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Linear Approximation Quadratic Approximation Differentials
Geometric Representation of Linear Approximation Near 0
ex ≈ 1+ x
4 2 0 2 4
2
4
x
y
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Linear Approximation Quadratic Approximation Differentials
More Linear Approximation Near 0
f (x) ≈ f (0) + f ′ (0) x
ln (1+ x)
(1+ x)r
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Linear Approximation Quadratic Approximation Differentials
Approximate The Values
ln (1.5) = 0.405 47
ln (1.3) = 0.262 36
ln (1.1) = 0.095 31
The approximations get more accurate as x takes the values closerand closer to 0.
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Linear Approximation Quadratic Approximation Differentials
Example - Linear Approximation Near 0
e−3x√1+ x
=(e−3x
)(1+ x)−1/2
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Linear Approximation Quadratic Approximation Differentials
Quadratic Approximation
Quadratic approximation is used when linear approximation is not enough.
f (x) ≈ f (x0) + f ′ (x0) (x − x0) +f ′′ (x0)2
(x − x0)2 (5)
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Linear Approximation Quadratic Approximation Differentials
Discussion on Quadratic Approximation near 0
f (x) ≈ f (x0) + f ′ (x0) (x − x0) +f ′′ (x0)2
(x − x0)2 (6)
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Linear Approximation Quadratic Approximation Differentials
Quadratic Approximation Near 0
f (x) ≈ f (0) + f ′ (0) x + f ′′ (0)2
x2 (7)
sin x :cos x :ex :
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Linear Approximation Quadratic Approximation Differentials
Geometric Representation of Quadratic ApproximationNear 0
sin x ≈ x
4 2 2 4
4
2
2
4
x
y
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Linear Approximation Quadratic Approximation Differentials
Geometric Representation of Quadratic ApproximationNear 0
cos x ≈ 1− 12x2
4 2 2 4
2
1
1
2
x
y
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Linear Approximation Quadratic Approximation Differentials
Geometric Representation of Quadratic ApproximationNear 0
ex ≈ 1+ x + 12x2
4 2 0 2 4
2
4
x
y
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Linear Approximation Quadratic Approximation Differentials
More on Quadratic Approximation Near 0
f (x) ≈ f (0) + f ′ (0) x + f ′′ (0)2
x2
ln (1+ x)
(1+ x)r
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Linear Approximation Quadratic Approximation Differentials
Example
Linear approximation of ln (1+ x) near x = 0 :Quadratic approximation of ln (1+ x) near x = 0 :Quadratic approximation gives much more accuracy than linearapproximation (near x = 0 ).
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Linear Approximation Quadratic Approximation Differentials
Example - Quadratic Approximation Near 0
e−3x√1+ x
=(e−3x
)(1+ x)−1/2
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Linear Approximation Quadratic Approximation Differentials
Linear Approximation of e Near 0
ak =(1+
1k
)k→ e as k → ∞
Take ln:
ln ak = ln(1+
1k
)k= k ln
(1+
1k
)≈ k
(1k
)= 1
with x =1k. (Note: as k → ∞, x → 0 )
ln ak → 1 as k → ∞ near x = 0.
The rate of convergence (how fast ln ak → 1)
ln ak − 1→ 0
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Linear Approximation Quadratic Approximation Differentials
Quadratic Approximation of e Near 0
ln ak = ln(1+
1k
)k= k ln
(1+
1k
)≈ k
(1k− 12k2
)= 1− 1
2k
ln ak → 1 as k → ∞ near x = 0.
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Linear Approximation Quadratic Approximation Differentials
Differentials
Definition 2
Let y = f (x) . The differential of y (or differential of f )is denoted by
dy = f ′ (x) dx
⇔ dydx
= f ′ (x)
Leibniz interpretation of derivative as a ratio of "infinitesimals".
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Linear Approximation Quadratic Approximation Differentials
Use in Linear Approximations
dx replaces ∆xdy replaces ∆y
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Linear Approximation Quadratic Approximation Differentials
Example
Estimate: (64.1)1/3
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