Use the MVT to show that 10.049<√(101)<10.05 and 10.24<√(105)< 10.25. Also, find better estimate

step 1 Okay, so we let F of X here be equal to the square root of X, and therefore we have that F prime

Use the MVT to show that 10.049<√(101)<10.05 and...

Key Concepts

Mean Value Theorem (MVT)

The Mean Value Theorem is a fundamental theorem in calculus which states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there is a point c in (a, b) such that the derivative at c is equal to the average rate of change over [a, b]. This theorem is crucial for deriving inequalities and error estimates by linking the instantaneous rate of change to the overall change of the function between two points.

Taylor's Theorem (Linear Approximation)

Taylor's Theorem provides a way to approximate a function near a specific point using its derivatives. When the expansion is truncated after the first derivative (n=1), the result is known as the linear approximation. This approximation gives a first-order estimate of the function’s value and is useful for obtaining more accurate bounds by considering both the value of the function and its rate of change, along with an error term to quantify the approximation’s accuracy.

Error Estimation Using Remainder Terms

Error estimation in the context of Taylor's Theorem involves analyzing the remainder term that represents the difference between the actual function value and its finite Taylor series approximation. By bounding this remainder term, one can assess the accuracy of the linear approximation and refine the estimates within a desired range. This approach is fundamental in numerical analysis, ensuring that approximations like those of square roots stay within specified limits.

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