Find the value and derivative of an integral function at a point, and use the Mean Value Theorem to bound its change over an interval.

Vault Of Euclid

Evaluating an Integral Function and Bounding its Change

The definite integral core

Given \( H(x) = \int_{1}^{x} \frac{1}{1+t^{2}} \, dt \), find \( H(1) \) and \( H'(1) \), and show that \( H(4)-H(2) < \frac{2}{5} \).

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What you need first

Properties of definite integrals
The Fundamental Theorem of Calculus
The Mean Value Theorem
Solving basic inequalities

What this problem tests

Tests the ability to differentiate an integral with a variable upper limit and to apply the Mean Value Theorem to establish bounds.

This problem is typical of a first-year calculus course, testing the understanding of the Fundamental Theorem of Calculus and the Mean Value Theorem in tandem.

Evaluating an Integral Function and Bounding its Change

What you need first

What this problem tests

More on The definite integral