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Aliun [14]
3 years ago
8

Write the Taylor Series for f(x) = sin(x)center

Mathematics
1 answer:
tangare [24]3 years ago
3 0

Answer:

Taylor series of sin(x) centered at x = 0.

Step-by-step explanation:

Taylor series expansion:

\sum_{n=0}^N f^n(a)\displaystyle\frac{(x-a)^n}{n!}

Here, f(x) = sin (x) and a = 0

f(x) = \sin x, f(0) = 0\\f'(x) = \cos x, f'(0) = 1\\f''(x) = -\sin x, f''(0) = 0\\f'''(x) = -\cos x, f'''(0) = -1\\f^4(x) = \sin x, f^4(0) = 0\\f^5(x) = \cos x, f^5(0) = 0

Putting all the values and expanding, we get,

f(x) = f(a) + \displaystyle\frac{f'(a)(x-a)}{1!} + \displaystyle\frac{f''(a)(x-a)^2}{2!} + \displaystyle\frac{f'''(a)(x-a)^3}{3!} + \displaystyle\frac{f^4(a)(x-a)^4}{4!} + \displaystyle\frac{f^5(a)(x-a)^5}{5!} + ...\\\\= \sin 0 + \displaystyle\frac{x}{1!} +  \displaystyle\frac{(0)x^2}{2!} + \displaystyle\frac{(-1)x^3}{3!} + \displaystyle\frac{(0)x^4}{4!} + \displaystyle\frac{(1)x^5}{5!} + ...

Solving, we get

\sin x = x - \displaystyle\frac{x^3}{3!} + \displaystyle\frac{x^5}{5!} - \displaystyle\frac{x^7}{7!} + ...\\\\\sin x = x - \displaystyle\frac{x^3}{6} + \displaystyle\frac{x^5}{120} - \displaystyle\frac{x^7}{5040} + ...

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Answer:

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Step-by-step explanation:

Pints

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The pint is a US customary unit of volume. Pints can be abbreviated as pt; for example, 1 pint can be written as 1 pt.

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The US fluid ounce is a unit of volume equal to 1/16 of a pint or 1/8 of a cup. The fluid ounce is sometimes referred to as an "ounce" but should not be confused with the unit of mass. One fluid ounce is equal to just under 29.6 milliliters, but in nutrition labeling, one fluid ounce is rounded to exactly 30 milliliters.[1]

The fluid ounce is a US customary unit of volume. Fluid ounces can be abbreviated as fl oz, and are also sometimes abbreviated as fl. oz. or oz. fl.. For example, 1 fluid ounce can be written as 1 fl oz, 1 fl. oz., or 1 oz. fl..

Pint to Fluid Ounce Conversion Table

Pint measurements converted to fluid ounces

Pints Fluid Ounces

1 pt 16 fl oz

2 pt 32 fl oz

3 pt 48 fl oz

4 pt 64 fl oz

5 pt 80 fl oz

6 pt 96 fl oz

7 pt 112 fl oz

8 pt 128 fl oz

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10 pt 160 fl oz

11 pt 176 fl oz

12 pt 192 fl oz

13 pt 208 fl oz

14 pt 224 fl oz

15 pt 240 fl oz

16 pt 256 fl oz

17 pt 272 fl oz

18 pt 288 fl oz

19 pt 304 fl oz

20 pt 320 fl oz

21 pt 336 fl oz

22 pt 352 fl oz

23 pt 368 fl oz

24 pt 384 fl oz

25 pt 400 fl oz

26 pt 416 fl oz

27 pt 432 fl oz

28 pt 448 fl oz

29 pt 464 fl oz

30 pt 480 fl oz

31 pt 496 fl oz

32 pt 512 fl oz

33 pt 528 fl oz

34 pt 544 fl oz

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