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archaic
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Hello guys, is it possible to "see" the mean value theorem when one is only thinking of numerical values without visualizing a graph? Perhaps through a real world problem?
What do you want to know? A real world problem which can be solved due to the main value theorem? What should numerically mean? Numerically we always only get an approximation, as the likely mean value isn't rational.archaic said:Hello guys, is it possible to "see" the mean value theorem when one is only thinking of numerical values without visualizing a graph? Perhaps through a real world problem?
By numerically I meant reasoning without resorting to geometry. In other words, how can one foster an intuition for this without seeing a graph?fresh_42 said:What do you want to know? A real world problem which can be solved due to the main value theorem? What should numerically mean? Numerically we always only get an approximation, as the likely mean value isn't rational.
The mean value theorem without geometry is a mathematical theorem that states that for a continuous function on a closed interval, there exists a point within the interval where the slope of the tangent line is equal to the average rate of change of the function over the interval.
The mean value theorem without geometry is significant because it provides a way to relate the average rate of change of a function to the instantaneous rate of change at a specific point. This allows us to make more accurate predictions and calculations in various fields such as physics, economics, and engineering.
The mean value theorem without geometry is used in calculus to prove other important theorems, such as the fundamental theorem of calculus. It is also used to find the maximum and minimum values of a function, and to determine the concavity and inflection points of a curve.
No, the mean value theorem without geometry can only be applied to continuous functions on a closed interval. If a function is not continuous or the interval is not closed, the theorem does not hold.
The mean value theorem without geometry is a special case of the intermediate value theorem. It can be thought of as a "mean value" version of the intermediate value theorem, where instead of finding a specific value, we are finding a specific slope.