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Lecture 22

MAT187H1 Lecture 22: 10.1 Approximating Functions with PolynomialsPremium


Department
Mathematics
Course Code
MAT136H1
Professor
Anthony Lam
Lecture
22

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MAT187H1Lecture2210.1ApproximatingFunctionswithPolynomials
Approximatingfunctions
Givenknownfunctionf(x),wherevalueweknowatx=a
estimatevalueatx≠a(ief(c)=?)
Firstapproximateuses:slopeoff(x)atx=a,ief’(a)
f(c)≈f(a)+ywherey=f’(a)x=f’(a)(ca)
f(c)≈f(a)+f’(a)(ca)
Ingeneral,f(x)≈f(a)+f’(a)(xa)
Withoutproof
f(x)≈f(a)+f’(a)(xa)+f’’(a)(xa)
2
/2
Ingeneral,wedefine:
N
th
orderTaylorPolynomialforf(x)centredatx=a
f(x)≈P
n
(x)=f(a)+f’(a)(xa)+f’’(a)(xa)
2
/2!+f’’’(x)(xa)
3
/3!+...+f
(n)
(a)(xa)
n
/n!
= (a)
n
k=0
f(k)
k!
(xa)k
Basedonproperty:P
n
(a)=f(a) 
P’
n
(a)=f’(a)
P’’
n
(a)=f’’(a)
…
P
(n)
(a)=f
(n)
(a)
Forspecialcase,whena=0:
N
th
orderMaclaurinPolynomial:P
n
(x)=f(0)+f’(0)x+f’’(0)x
2
/2!+f’’’(0)x
3
/3!+...+f
(n)
(0)x
n
/n!
ExFindP
n
(x)forf(x)=e
x
atx=0
Sincef(x)=f’(x)=f’’(x)=...=f
(n)
(x)=e
x
f
(n)
(0)=e
0
=1forallk≤n
f(x)=e
x
≈P
n
(x)
P
n
(x)=f(0)+f’(0)x+...+f
(n)
(0)x
n
/n!
=e
0
+e
0
x+e
0
x
2
/2!+...+e
0
x
n
/n!
e
x
≈1+x+x
2
/2!+...+x
n
/n!
e
1
≈1+1+1/2!+...+1/n! (x=1)
e
1
≈11+1/2!1/3!+...1/n! (x=1)
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