Math  /  Calculus

Question(1 point)
Use the third-order Taylor polynonial for exsin(3x)e^{x} \sin (3 x) at x=0x=0 to approximate e1tsin(3/8)e^{\frac{1}{t}} \sin (3 / 8) by a rational number. e11sin(3/8)27/256\mathrm{e}^{\frac{1}{1}} \sin (3 / 8) \approx 27 / 256
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Studdy Solution
Simplify the expression:
3×38=98 3 \times \frac{3}{8} = \frac{9}{8}
92×(964)=81128 \frac{9}{2} \times \left(\frac{9}{64}\right) = \frac{81}{128}
4×(27512)=108512=27128 -4 \times \left(\frac{27}{512}\right) = -\frac{108}{512} = -\frac{27}{128}
Combine these terms:
98+8112827128 \frac{9}{8} + \frac{81}{128} - \frac{27}{128}
Convert 98\frac{9}{8} to a fraction with denominator 128:
98=144128 \frac{9}{8} = \frac{144}{128}
Add the fractions:
144128+8112827128=19812827128=171128 \frac{144}{128} + \frac{81}{128} - \frac{27}{128} = \frac{198}{128} - \frac{27}{128} = \frac{171}{128}
The rational approximation of e1tsin(38) e^{\frac{1}{t}} \sin \left(\frac{3}{8}\right) is:
171128 \boxed{\frac{171}{128}}

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