How to use the mean value theorem of definite integral to find the limit? Just posted a paper of others in Baidu Library. You can download it. It is very detailed.
The search title is the limit in the mean value theorem of integral (Yang Yonghong 05). document
How to use the differential mean value theorem to find the limit? Write the expression of the mean value theorem first, and then use L'H?pital's law to solve it.
Mean value theorem of definite integral differential? (x) = x2 f (x), and 2 (1) 22 () 22 ()110f = ∫ x2f x dx =? c f c？ , where ]2c∈[0, 1, that is, 12 f (1) = c2 f (c), so there is? (c) =？ (1), according to Rolle's theorem, there is ξ ∈(c, 1)? (0, 1), which makes? ′(ξ ) = 0。 And then what? (x '(x) = x2 f '(x)+2xf (x x), so ξ 2 f '(ξ)+2ξf (ξ) = 0, so ξ f' (ξ)+2 f is noticed.
The limit mean value theorem is the basic theorem of differential calculus, which means that there must be a point on a continuous smooth curve whose slope is the same as the average slope of the whole curve.
content
If the function f(x) satisfies
Continuous on the closed interval [a, b];
Derivable in the open interval (a, b),
Then there is at least one point ξ (a
f(b)-f(a)= f′(ξ)(b-a)& lt； /horse
Established.
Mean value theorem is divided into differential mean value theorem and integral mean value theorem;
The integral of f(x) on a to b is equal to f(a)-f(b)ξ, which is twice that of a-b.
What is the integral mean value theorem?
If the function f(x) is continuous in the closed interval [a, b], then there is at least one point ξ in the product division slot [a, b], so the following formula holds ∫ lower limit A, and upper limit BF (x) dx = f (ξ) (b-a) (A ≤ξ≤/.
What is the mean value theorem of definite integral? Write a general form, often using the first integral mean value theorem:
If the function f(x) is continuous in the closed interval [a, b] and the function g(x) is integrable and has the same sign, then there is at least one point ξ on the product division slot [a, b] such that ∫ (a, b) f (x) * g (x) dx = f (\) ξ b)
How to use integral mean value theorem and Rolle theorem to find definite integral? Unless the integrand function is constant, it is impossible to find a definite integral by using the integral mean value theorem and Rolle theorem.
But these two things can be used to prove some useful inequalities.
Note that there are two integral mean value theorems, the first is the inference of the mean value theorem, and the second is obtained by partial integration.
The theorem content of integral mean value theorem is divided into integral first mean value theorem and integral second mean value theorem, and each theorem contains two formulas. Its degenerate state means that there is a moment in the process of ξ change that makes the areas of two graphs equal.
The mean value theorem of integral is divided into the first mean value theorem of integral and the second mean value theorem of integral, and each mean value theorem contains two formulas. Its degenerate state means that there is a moment in the process of ξ change that makes the areas of two graphs equal.
Integral Mean Value Theorem Integral Mean Value Theorem: If f(x) is continuous on [a, b], then (a, b) has at least a little ε, which satisfies.
b
∫f(x)dx=f(ε)(b-a)
a