Use Taylor series to find infinitely many parabolas, third-degree polynomials,

Walter Clyburn 2021-12-16 Answered
Use Taylor series to find infinitely many parabolas, third-degree polynomials, etc that all have f(x) = 3x + 5 as one of their tangent lines.
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Expert Answer

eninsala06
Answered 2021-12-17 Author has 37 answers
Step 1
Given that, f(x)=3x+5.
Formula used:
General form of Taylor expansion centered at a f(x)=n=0fn(a)n!(xa)n.
Where fn is the n th derivative of f.
Third degree Taylor polynomial consisting of first 4 terms of the Taylor expansion.
Polynomials:
f(a)+f(a)(xa)+f(a)(xa)22+f(a)(xa)36
Step 2
f(x)=3x+5
f'(x)=3
f''(x)=0
f'''(x)=0
Third degree Taylor polynomials is
3a+5+3(x-a)+0+0=3a+5+3x-3a
=3x+5
Thus, the polynomial is 3x+5.

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Jeffery Autrey
Answered 2021-12-18 Author has 35 answers
f(x)=3x+5
general form of Taylor expansion centored at a
f(x)=n=0f(n)(a)n!(xa)n. Here f(n) is the nth derivative of
f(a)+f(a)(xa)+f(a)2(xa)2+f(a)6(xa)3
f(x)=3x+5
f'(x)=3
f''(x)=0
f'''(x)=0
3a+5+3(x-a)+0+0
=3a+5+3x-3a
=3x+5
Polynomial = 3x+5

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