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Anonymous users2024-02-07Honestly, that's right, no problem, you can do it. But you'll find that the coefficients preceded by all odd terms of x are equal to zero. This is because when exp(-x 2 2) is the derivative of x, the derivation once is -xexp(-x 2 2), and substituting x=0 is 0;
The derivative quadratic is -exp(-x 2 2) +x 2exp(-x 2 2), and substituting x=0 is -1;
The third lead is -xexp(-x 2 2) +2xexp(-x 2 2) -x 3exp(-x 2 2), substituting x=0 or 0;
If you lead the odd number of times, there will always be an x before exp, so if you substitute x=0, it will always be 0, and the corresponding Taylor series term will not exist. The remaining even-power terms are exactly the answer.
The reason why substitution is possible is that when x tends to 0, the substitution variable t=-x 2 2 also tends to 0. This can only be done if x and the substitution variable are at the same limit at the point. If it's at x=1, you have to do the math honestly (you can make t = x-1, or t=(x-1) 2, etc., guarantee the same limit, and then use the substitution method of the case problem, but this is undoubtedly asking for trouble).
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Anonymous users2024-02-06Open the book of advanced mathematics and look at the range of the absolute value of fx in a and b definite integrals, I don't know much about this problem.
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Anonymous users2024-02-05Taylor can calculate functions. It comes from Taylor's theorem in calculus.
If the function is smooth enough, the Taylor formula is given given the derivatives of the function at a certain point.
These derivatives can be used as coefficients to construct a polynomial to approximate the value of the function in the neighborhood at this point.
The importance of Taylorism is reflected in the following five aspects:
1. The derivative and integration of power series can be done item by term, so the summation function is relatively easy.
2. An analytic function can be extended to a definition in the complex plane.
and make the method of complex analysis feasible.
3. Taylor series.
It can be used to approximate the value of the excavation function and estimate the error.
4. Prove inequality.
5. Find the limit of the pending formula.
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Anonymous users2024-02-04Taylor's is 1+x+x 2 2!+x^3/3!+.x^n/n!+rn(x) 。
Taylor's formula. is a formula that describes the value of a function near a point with information about it. If the function satisfies certain conditions, Taylor's formula can construct a polynomial using the coefficients of each derivative of the function at a certain point.
to approximate this function.
Taylor's formula takes its name from the English mathematician Brooke Taylor, who first described it in a letter in 1712. Taylor's formula is one of the approximations often used to study the properties of complex functions, and it is also an important application of functional differentiation.
Taylor's formula is mathematical analysis.
It is also important to study the limits of functions.
and estimation errors, an indispensable mathematical tool, the Taylor formula epitomizes calculus.
The essence of the approximation method has unique advantages in approximate calculation.
The Taylor formula can be used to reduce nonlinear problems to linear problems, and it has a high degree of accuracy, so it has important applications in all aspects of calculus. Taylor's formula can be used to find the limit, to determine the extreme value of a function, to find the value of a certain point in destroying a higher derivative, and to judge a generalized integral.
convergence, approximate calculation, inequality proof, etc.
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Anonymous users2024-02-03Common Taylor styles are as follows:
Taylor's formula: If a function is derivable of order n, then this function can be of order n by Taylor's formula, i.e., f(x)=f(x0)+f'(x0)(x-x0)+f''(x0)(x-x0) 2!+.
f＾(n)(x0)(x-x0)＾(n)/n!+0x。
f (n) (x0) denotes the nth derivative of f(x) at x0, and 0x denotes the infinitesimal of a higher order than mountain brigade (x-x0) (n). Expressed by the Lagrangian remainder, then 0x=f (n+1)( x- )n+1) n+1!, while McLaughlin's formula is a special case of Taylor's formula at point 0.
Taylor's formula can be used to get the coefficients of the power term of x in the formula f(x), or it can be used to find the limit problem by deducing the original function from the derivative of a known function. For example, find the limit of lim (e x-x-1) x when x approaches 0, f(x)=e x quadratic at x=0 =e (0)+e (0)*(x-0)+e (0)(x-0) 2!+0x=1+x+x/2。
Then lim (e x-x-1) x=lim (1+x+x 2-x-1) x=1 2 is understood by the derivative definition, f'(x)=lim [f(x)-f(x0)] x-x0) where x- u003ex0. Then there is when x- u003ex0 lim f(x)-f(x0)=f'(x)(x-x0), lim f(x) and its error in f(x) Lagrangian remainder term is f (2) ( x- )2) 2!is (x-x0) and the high touch is infinitesimal in order only.
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Anonymous users2024-02-02Common Taylor styles are:
1. e^x = 1+x+x^2/2!+x^3/3!+…x^n/n!+…
2.ln(1+x)=x-x 2 Yubu 2+x 3 3 3-......1)^(k-1)*(x^k)/k(|x|"Town Book Sui 1).
3. sin x = x-x^3/3!+x^5/5!-…1)^(k-1)*(x^(2k-1))/2k-1)!+x<∞)
4. cos x = 1-x^2/2!+x^4/4!-…1)k*(x^(2k))/2k)!+x<∞)
5. arcsin x = x + 1/2*x^3/3 + 1*3/(2*4)*x^5/5 + x|<1)
6. arctan x = x - x^3/3 + x^5/5 -…x≤1)
These Taylor formulas are commonly used mathematical formulas, which are widely used in mathematical analysis, physical calculations and other fields.
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Anonymous users2024-02-01Taylor's formula is defined as if the function f(x) has a derivative of the (n+1) order on some open interval (a,b) containing x0, then for any x(a,b), there is f(x)=f(x0) 0!+f'(x0)/1!*(x-x0)+f''(x0)/2!
x-x0))^2+f(n)(x0)/n!*(x-x0) n+rn(x), where rn(x) f(n+1)( n+1)!*x-x0) (n+1), where is a value between x0 and x.
Brief introduction. In mathematics, the Taylor series is a function expressed by an infinite term plus a series of additives, and these additive terms are obtained from the derivative of the function at a certain point.
The Taylor series is named after the English mathematician Sir Brook Taylor, who published Taylor's formula in 1715.
The Taylor series, also known as the McLaurin series, is derived from the derivative of the function at the zero point of the independent variable, named after the Scottish mathematician Colin McLaughlin. Taylor series plays an important role in approximate calculations.
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Anonymous users2024-01-31Taylor. The formula is defined as if the function f(x) has a derivative of (n+1) on some open interval (a,b) containing x0, then for any x(a,b), there is f(x)=f(x0) 0!+f'(x0)/1!
x-x0)+f''(x0)/2!*(x-x0))^2+f(n)(x0)/n!*(x-x0)^n+rn(x）。
Among them, Hu and your friends rn(x) f(n+1)( n+1)!*x-x0) (n+1), where is a value between x0 and x.
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Anonymous users2024-01-30Commonly used Taylor formulas are as follows:
1、e^x = 1+x+x^2/2!+x^3/3!+…x^n/n!+…Bend to the line.
2、ln(1+x)=x-x^2/2+x^3/3-……1)^(k-1)*(x^k)/k(|x|<1)
3、sin x = x-x^3/3!+x^5/5!-…1)^(k-1)*(x^(2k-1))/2k-1)!
4、cos x = 1-x^2/2!+x^4/4!-…1)k*(x^(2k))/2k)!
5、arcsin x = x + 1/2*x^3/3 + 1*3/(2*4)*x^5/5 + x|<1)
6、arccos x = x + 1/2*x^3/3 + 1*3/(2*4)*x^5/5 + x|<1)
7、sinh x = x+x^3/3!+x^5/5!+…1)^(k-1)*(x^2k-1)/(2k-1)!+
The Taylor formula of tanx is: tanx=x+x 3 3+(2 x 5) 15+(17 x 7) 315+(62x 9) 2835+o[x] 11(|x|<π2)。
Taylor's formula. It is a formula that describes the value of a function in the vicinity of a point with information about it, if the function is smooth enough, when the derivatives of the function at a point are known.
Taylor's formula can be used as coefficients to construct a polynomial using these buried derivatives.
to approximate the value of the function in the field at this point.
The importance of the Taylor style is reflected in the following five aspects:
1. Power series.
The derivative and integration of can be done item by term, so the summation function is relatively easy.
2. An analytic function.
Can be extended to a definition in the complex plane.
on an open piece.
and make the complex analysis method feasible.
3. Taylor series.
It can be used to approximate the value of the calculated function and estimate the error.
4. Prove inequality.
5. Find the limit of the pending formula.
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Anonymous users2024-01-29Yes, Taylor Zhan Chun Dong Kai is only in a certain point of the neighborhood, and the neighborhood can be large or small.
Moreover, Taylor originally got an approximate value, and the error can be large or small (as Gao Zan said, when x tends to positive infinity, x n is the high-order infinity).
The equivalent infinitesimal that is generally used is only made in McLaughlin's reed pants at zero point, and the error is 0.
Give an example to accompany the balance.
Damage ff is one of the characteristics of the Dragon Palace sect.,Only dragons can do it.,70's is generally worthless.,No one wants it.,At most, it's only 1w block.。 Generally, the more advanced the more expensive, depending on the number of Dragon Palaces in your district, there are generally a lot of people in Dragon Palaces, and the damage ff below 120 is generally very low, let's talk about it from 130, 130 is generally 5w, 140 is about 5-8w, 150 8w-10w, 160 of 10w-15w, the reason why there is no accurate price, there is another reason, that is, special time, damage ff will be very expensive. For example, during the martial arts competition, the ** of FF skyrocketed.
Hehe, I've seen too much of the initial D.
The so-called heel toe action is to use the right foot fuel replenishment technique to lower the gear to the appropriate gear when exiting the corner before cornering, while braking. The "heel" here refers to the heel, and the "toe" refers to the toe. >>>More
Sweet potato mashed taro pie making process:
Ingredients: 1 sweet potato, 50g glutinous rice flour >>>More
Yogurt primer is the yogurt you ferment with lactic acid bacteria powder or buy back the finished yogurt to leave less than half a cup, and the next time you use milk to ferment yogurt, the half cup of yogurt left is the yogurt primer.
The tasks in the Glory Battlefield of Xinyue members will be updated with 3-5 tasks every day, and you can get 1 point for each completed one, and the points can be exchanged for corresponding things, and 200 points can be exchanged for a title, and the attributes are very good. If you encounter a task that is more difficult to complete, you can use the no-do card to complete the task directly.