<abstract xmlns="http://www.w3.org/1999/xhtml">If <italic>f</italic>, <italic>g</italic>: [<italic>a</italic>, <italic>b</italic>] → 𝕉 are two continuous functions, then there exists a point <italic>c</italic> ∈ (<italic>a</italic>, <italic>b</italic>) such that<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2020-0005_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msubsup><mo>∫</mo><mi>a</mi><mi>c</mi></msubsup><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow><mi>d</mi><mi>x</mi><mo>+</mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow><mo>)</mo></mrow><mi>g</mi><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow><mo>=</mo><mrow><msubsup><mo>∫</mo><mi>c</mi><mi>b</mi></msubsup><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow><mi>d</mi><mi>x</mi><mo>+</mo><mrow><mo>(</mo><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow><mo>.</mo></mrow></math><tex-math>\int_a^c {f\left(x \right)} dx + \left({c - a} \right)g\left(c \right) = \int_c^b {g\left(x \right)} dx + \left({b - c} \right)f\left(c \right).</tex-math></alternatives></disp-formula>In this paper, we study the approaching of the point <italic>c</italic> towards <italic>a,</italic> when <italic>b</italic> approaches <italic>a.</italic></abstract>

If f, g: [a, b] → 𝕉 are two continuous functions, then there exists a point c ∈ (a, b) such that∫acf(x)dx+(c-a)g(c)=∫cbg(x)dx+(b-c)f(c).\int_a^c {f\left(x \right)} dx + \left({c - a} \right)g\left(c \right) = \int_c^b {g\left(x \right)} dx + \left({b - c} \right)f\left(c \right).In this paper, we study the approaching of the point c towards a, when b approaches a.

Calculus for the intermediate point associated with a mean value theorem of the integral calculus

Faculty of Mathematics and Computer Science Department of Computer Science

Faculty of Mathematics and Computer Science Department of Mathematics

Faculty of Science Department of Mathematics and Computer Science

General Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

If f, g: [a, b] → 𝕉 are two continuous functions, then there exists a point c ∈ (a, b) such that∫acf(x)dx+(c-a)g(c)=∫cbg(x)dx+(b-c)f(c).\int_a^c {f\left(x...

Calculus for the intermediate point associated with a mean value theorem of the integral calculus

Published Online: Jul 31, 2020

Page range: 59 - 66

Received: Jan 08, 2020

Accepted: May 27, 2020

DOI: https://doi.org/10.2478/gm-2020-0005

Keywords
intermediate point, mean-value theorem

© 2020 Emilia-Loredana Pop et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Calculus for the intermediate point associated with a mean value theorem of the integral calculus

Published Online: Jul 31, 2020

Page range: 59 - 66

Received: Jan 08, 2020

Accepted: May 27, 2020

DOI: https://doi.org/10.2478/gm-2020-0005

Keywordsintermediate point, mean-value theorem

© 2020 Emilia-Loredana Pop et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
intermediate point, mean-value theorem