Show that (x)/(1+x^2)<arctanx<x for all x ∈(0,1] and 1-(1)/(2 x)<arctanx<2-(1)/(x) for al

step 1 Okay, so here we consider a function f of x, which is equal to cosine of x minus the quantity 1

Show that (x)/(1+x^2)<arctanx<x for all x ∈(0,1] and...

Key Concepts

Arctan Function

The arctan function, which is the inverse of the tangent function on its principal branch, is defined for all real numbers and maps to the interval (??/2, ?/2). Its properties, such as continuity and smoothness, make it a frequent subject of analysis in calculus and trigonometry, especially when establishing inequalities or approximations in analysis.

Derivative Analysis

The derivative of the arctan function is 1/(1+x^2), which plays a crucial role in comparing its growth rate relative to linear or other functions. By analyzing this derivative, one can understand how the function increases and establish bounds by integrating or applying comparison tests, particularly over specific intervals.

Monotonicity

Monotonicity is a fundamental concept when working with functions like arctan. Since arctan is strictly increasing, one can assert that if a function is compared pointwise with arctan, the order is maintained. This property is key to formulating and proving inequalities such as evaluating its lower and upper bounds on different intervals.

Inequality Bounding Techniques

Bounding techniques in calculus involve using properties like the derivative, monotonicity, and sometimes convexity or mean value theorem to enclose a function within specified limits. In the context of the arctan function, employing integration of its derivative and comparing it with simpler functions helps to establish explicit inequalities over distinct subdomains.

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