In arithmetic, the mean cost theorem is used to evaluate the conduct of a feature. The suggest price theorem asserts that if the f is a non-stop function at the closed c language [a, b], and differentiable at the open c programming language (a, b), then there is at least one point c at the open c language (a, b), then the mean fee theorem formulation is:
$$f’ (c) = [f(b) – f (a)] / b – a$$
The mean price theorem for vital states that the slope of a line consolidates at unique points on a curve (easy) may be the very equal because the slope of the tangent line to the curve at a selected point among the 2 individual points. let f be the function on [a, b]. Then the common f (c) of c is
$$1/ b – a∫_a^b f(x) d(x) = f (c)$$
Example:
Find the value of \( f(x) = 5x^2 - 4x + 7 \) on the interval \([2, 6]\).
Solution:
In the given equation, \( f(x) \) is continuous on \([2, 6]\).
$$ F(c) = \frac{1}{b - a} \int_a^b f(x) \, dx = \frac{1}{6 - 2} \int_2^6 (5x^2 - 4x + 7) \, dx $$
Evaluate the integral:
$$ = \frac{1}{4} \left[ \frac{5x^3}{3} - \frac{4x^2}{2} + 7x \right]_2^6 $$
Substitute the limits:
$$ = \frac{1}{4} \left[ \left( \frac{5(6^3)}{3} - \frac{4(6^2)}{2} + 7(6) \right) - \left( \frac{5(2^3)}{3} - \frac{4(2^2)}{2} + 7(2) \right) \right] $$
$$ = \frac{1}{4} \left[ \left( \frac{5(216)}{3} - \frac{4(36)}{2} + 42 \right) - \left( \frac{5(8)}{3} - \frac{4(4)}{2} + 14 \right) \right] $$
$$ = \frac{1}{4} \left[ \left( 360 - 72 + 42 \right) - \left( \frac{40}{3} - 8 + 14 \right) \right] $$
Simplify:
$$ = \frac{1}{4} \left[ 330 - \frac{46}{3} \right] $$
$$ = \frac{1}{4} \left[ \frac{990 - 46}{3} \right] $$
$$ = \frac{1}{4} \cdot \frac{944}{3} $$
$$ = \frac{944}{12} = 78.67 $$
Final Answer: The average value of the function \( f(x) = 5x^2 - 4x + 7 \) on the interval \([2, 6]\) is approximately \( 78.67 \).
here you could also confirm the outcomes using an internet suggest fee theorem calculator for short computation.
This free Rolle’s Theorem calculator may be used to compute the price of trade of a feature with a theorem by means of upcoming steps:
A restrained form of the suggest price theorem became proved by M Rolle in the yr 1691; the final results turned into what's now known as Rolle's theorem, and was proved for polynomials, with out the strategies of calculus. The imply value theorem in its modern shape which became proved via Augustin Cauchy inside the year of 1823.
f(b)−f(a) = f′(c)(b−a). This theorem is also known as the first mean fee Theorem that allows showing the increment of a given characteristic (f) on a selected c language via the cost of a spinoff at an intermediate point.
