⧼exchistory⧽
13 exercise(s) shown, 0 hidden
BBy Bot
Nov 03'24
[math]
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[/math]
For each of the values of [math]n[/math] indicated, compute the Taylor polynomial [math]T_n[/math] which approximates the function [math]f[/math] near the number [math]a[/math].
- [a [math]f(x) = \frac1{x+1}[/math], [math]a=0[/math], [math]n=0[/math], [math]1[/math], and [math]2[/math].
- [math]f(x) = \frac1{1+x^2}[/math], [math]a=0[/math], [math]n=0[/math], [math]2[/math], and [math]4[/math].
- [math]f(x) = \frac1{1+x^2}[/math], [math]a=1[/math], [math]n=0[/math], [math]1[/math], and [math]2[/math].
- [math]f(x) = \sqrt{x+1}[/math], [math]a=3[/math], [math]n=1[/math], [math]2[/math], and [math]3[/math].
- [math]f(x) = \sin x[/math], [math]a=\frac{\pi}4[/math], [math]n=0[/math], [math]1[/math], and [math]2[/math].
BBy Bot
Nov 03'24
[math]
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[/math]
Compute the formula, for an arbitrary nonnegative integer [math]n[/math], for the approximating Taylor polynomial to the function [math]f[/math] about the number [math]a[/math].
- [math]f(x) = \cos x[/math], \quad [math]a=0[/math]
- [math]f(x) = \ln x[/math], \quad [math]a=1[/math].
BBy Bot
Nov 03'24
[math]
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[/math]
For [math]n=0[/math], [math]1[/math], and [math]2[/math], compute the Taylor polynomial [math]T_n[/math] which approximates the function [math]f[/math] near [math]0[/math]. Draw the graphs of the three polynomials together with the graph of [math]f[/math].
- [math]f(x) = e^x[/math]
- [math]f(x) = \cos x[/math].
BBy Bot
Nov 03'24
[math]
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[/math]
Let [math]p[/math] be a polynomial in [math]x[/math] of degree [math]\leq m[/math]; i.e., the function [math]p[/math] is defined by an equation
[[math]]
p(x) = a_0 + a_1x + \cdots + a_mx^m
,
[[/math]]
and let [math]T_n[/math] be the Taylor polynomial which approximates [math]p[/math] near an arbitrary real number [math]a[/math]. Prove, as a simple consequence of Taylor's formula with the remainder, that [math]p(x) = T_n(x)[/math], for every real number [math]x[/math] provided [math]n \geq m[/math].
BBy Bot
Nov 03'24
[math]
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[/math]
For each of the values of [math]n[/math] indicated, compute the approximation [math]T_n[/math] to the polynomial [math]p[/math] near the number [math]a[/math].
- [math]p(x) = x^2+3x-1[/math], [math]a=2[/math], [math]n=1[/math], [math]2[/math], and [math]3[/math].
- [math]p(x) = 2x^3-5x^2+3[/math], [math]a=0[/math], [math]n=1[/math], [math]2[/math], and [math]3[/math].
- [math]p(x) = x^4+3x^2+x+2[/math], [math]a=0[/math], [math]n=3[/math], [math]4[/math], and [math]17[/math].
- [math]p(x) = x^3-1[/math], [math]a=1[/math], [math]n=2[/math], [math]3[/math], and [math]4[/math].
BBy Bot
Nov 03'24
[math]
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[/math]
Prove that, for every real number [math]x[/math],
[[math]]
\cos x = \sum_{i=0}^\infty (-1)^i
\frac{x^{2i}}{(2i)!} =
1 - \frac{x^2}2 + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots
.
[[/math]]
BBy Bot
Nov 03'24
[math]
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[/math]
For each of the following functions, compute the Taylor series about [math]a[/math].
- [math]e^x[/math], if [math]a=0[/math]
- [math]\frac{e^x}{e^2}[/math], if [math]a=2[/math]
- [math]\arctan x[/math], if [math]a=0[/math].
BBy Bot
Nov 03'24
[math]
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[/math]
- Compute the cubic Taylor polynomial [math]p(x)[/math] which approximates the function [math]\frac1{x+2}[/math] for values of [math]x[/math] near the number [math]1[/math].
- Show that, for every [math]x[/math] in the interval [math][0,2][/math], the approximation [math]p(x)[/math] differs in absolute value from [math]\frac1{x+2}[/math] by less that [math]0.04[/math].
BBy Bot
Nov 03'24
[math]
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[/math]
Show that [math]\sin x[/math] differs in absolute value from the approximation [math]x - \frac{x^3}6[/math] by no more that [math]\frac{\pi^5}{15\cdot2^8} = 0.025[/math] (approximately) for every [math]x[/math] in the interval [math]\left[ -\frac{\pi}2, \frac{\pi}2 \right][/math].
BBy Bot
Nov 03'24
[math]
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[/math]
The Taylor approximation [math]T_n[/math] to a function [math]f[/math] about the number [math]a[/math] is frequently called the best polynomial approximation of degree [math]\leq n[/math] to the function [math]f[/math] near [math]a[/math] because it can be shown that [math]T_n[/math] is the only polynomial of degree [math]\leq n[/math] with the property that, as [math]x[/math] approaches [math]a[/math], the difference [math]f(x) - T_n(x)[/math] approaches zero faster than [math](x-a)^n[/math]. Prove the following part of the above assertion: If [math]f[/math] has continuous [math](n+1)[/math]st derivative in an open interval containing [math]a[/math], then [math]\lim_{x\goesto{a}} \frac{f(x)-T_n(x)} {(x-a)^n} = 0[/math].