MATH 250a
Fall Semester 2007
Section 2 (J. M. Cushing)
Tuesday, November 13
http://math.arizona.edu/~cushing/250a.html
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Notes on First Degree Polynomial Approximation | MATH 250A, Study notes of Differential Equations
Material Type: Notes; Professor: Bergevin; Class: Calculus and Differential Equations I; Subject: Mathematics Main; University: University of Arizona; Term: Fall 2007;
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MATH 250aFall Semester 2007
Section 2 (J. M. Cushing)^ Tuesday, November 13
http://math.arizona.edu/~cushing/250a.html
First Degree Polynomial Approximation Approximate
at^
by a first degree polynomial ( )^
y^ f^
x^
x^ a =^
x
y
a Geometrically, approximate graph of
f(x) by that of its tangent at
x = a
.
( ,^ ( ))a f a
(^ )
tangent line:( ) ( )^
( ) p^ x^
f a^
f^ a^ ′x^ a =^
+^
− p(x) passes through same point as
f^ (x).
and has same slope as
f^ (x).
First Degree Polynomial Approximation
(^
1
for
( )^
( )^
( )^
p^ x^
f a^
f^ a^
x^ a^
f^ x^
x^ a
=^
+^
−^ ≈
Examples ( )
at^
x f^ x^
e^
a =^
First Degree Polynomial Approximation
(^
1
for
( )^
( )^
( )^
p^ x^
f a^
f^ a^
x^ a^
f^ x^
x^ a
=^
+^
−^ ≈
Examples ( )
at^
x f^ x^
e^
a =^
(0)^1 ( )^
(0)^1
x f f^ x^
e^
f = ′
=^
⇒^
1
for
( )^
x
p^ x^
x^ e^
x
=^ +^
≈^
First Degree Polynomial Approximation
(^
1
for
( )^
( )^
( )^
p^ x^
f a^
f^ a^
x^ a^
f^ x^
x^ a
=^
+^
−^ ≈
Examples ( )
at^
x f^ x^
e^
a =^
(0)^1 ( )^
(0)^1
x f f^ x^
e^
f = ′
=^
⇒^
1
for
( )^
x
p^ x^
x^
x
e
=^ +^
≈^
First Degree Polynomial Approximation
(^
1
for
( )^
( )^
( )^
p^ x^
f a^
f^ a^
x^ a^
f^ x^
x^ a
=^
+^
−^ ≈
Examples ( )
at^
x f^ x^
e^
a =^
(0)^1 ( )^
(0)^1
x f f^ x^
e^
f = ′
=^
⇒^
1
for
( )^
x
p^ x^
x^
x
e
=^ +^
≈^
First Degree Polynomial Approximation
(^
1
for
( )^
( )^
( )^
p^ x^
f a^
f^ a^
x^ a^
f^ x^
x^ a
=^
+^
−^ ≈
Examples ( ) sin^
at^
f^ x^
x^
a =^
(0)^
( )^
cos^
(0)^1
f f x^
x^
f = ′
=^
⇒^
1
for
( )^
sin^
p^ x^
x^
x^
x
=^ ≈
First Degree Polynomial Approximation
(^
1
for
( )^
( )^
( )^
p^ x^
f a^
f^ a^
x^ a^
f^ x^
x^ a
=^
+^
−^ ≈
Examples ( ) sin^
at^
f^ x^
x^
a =^
(0)^
( )^
cos^
(0)^1
f f x^
x^
f = ′
=^
⇒^
1
for
( )^
sin^
p^ x^
x^
x^
x
=^ ≈
First Degree Polynomial Approximation
(^
1
for
( )^
( )^
( )^
p^ x^
f a^
f^ a^
x^ a^
f^ x^
x^ a
=^
+^
−^ ≈
Examples ( ) ln^
at^
f^ x^
x^
a =^
(1)^0 ( )^ 1/
(1)^1
f f x^
x^
f = ′
=^
⇒^
1
for
( )^
1 ln
p^ x^
x^
x^
x
=^ −^
≈^
First Degree Polynomial Approximation
(^
1
for
( )^
( )^
( )^
p^ x^
f a^
f^ a^
x^ a^
f^ x^
x^ a
=^
+^
−^ ≈
Examples ( ) ln^
at^
f^ x^
x^
a =^
1
for
( )^
1 ln
p^ x^
x^
x^
x
=^ −^
≈^
(1)^0 ( )^ 1/
(1)^1
f f x^
x^
f = ′
=^
⇒^
If the graph of
f^ (x) is significantly curved at
x = a
,
the tangent will be a poor approximation. A better approximation would try to approximate some of
f^ ’s curvature.
Graphs of
second
degree polynomials are parabolas. IDEA: match
f^ ,^ f^
'^ and the
second
derivative
f^ ″
with a quadratic polynomial.
Second Degree Polynomial Approximation Goal:^ find a quadratic polynomial
p(x) such that
( ),^ equal respectively
( ),^
( )
( ),^
( ),^
( )
p^ a^
p^ a^
p^ a
f a^
f^ a^
f^ a ′^
′′ ′^
′′
Second Degree Polynomial Approximation Goal:^ find a quadratic polynomial
p(x) such that
( ),^ equal respectively
( ),^
( )
( ),^
( ),^
( )
p^ a^
p^ a^
p^ a
f a^
f^ a^
f^ a ′^
′′ ′^
′′
2
0 1
2
1
(^22) ( )^
(^
)^
(^
)
( )^
2 (^
)
( )^
2 p^ x^
c^ c
x^
a^ c
x^
a
p^ x^
c^
c^ x^
a
p^ x^
=^ +^ c
−^ +
−
′^ =
+^
−
′′^ =
Second Degree Polynomial Approximation Goal:^ find a quadratic polynomial
p(x) such that
( ),^ equal respectively
( ),^
( )
( ),^
( ),^
( )
p^ a^
p^ a^
p^ a
f a^
f^ a^
f^ a ′^
′′ ′^
′′ 0 1 2 ( ) ( ) ( )^
2 p^ a^
c p^ a^
c p^ a^
= c ′^ = ′′^ =
2
0 1
2
( )^
(^
)^
(^
)
p^ x^
c^ c
x^
a^ c
x^
a
=^ +
−
+^
−