Question

In Exercises 43 and 44, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The slope of the graph of the inverse tangent function is positive for all \(x\).

Short answer

The statement 'The slope of the graph of the inverse tangent function is positive for all \(x\)' is true. The derivative of the inverse tangent function is \(\frac{1}{1+x^2}\), which is always positive for any value of \(x\).

Step-by-step solution

Understand the Statement

The statement under consideration is 'The slope of the graph of the inverse tangent function is positive for all \(x\)'. Here, the slope of the graph refers to the derivative of the function.

Derivation of the Inverse Tangent Function

The derivative of the inverse tangent function, \(arctan(x)\), can be derived using implicit differentiation. It's derivative is given by \[\frac{d}{dx} arctan(x) = \frac{1}{1+x^2}\]

Analyze the Derivative

From the above, it's clear that for any given \(x\), the denominator of the derivative, \(1 + x^2\) is always positive as \(x^2\) is always non-negative. Therefore, the derivative of \(arctan(x)\), \(\frac{1}{1+x^2}\), is always positive for all \(x\). Thus, according to the definition of slope, as per the derivative, the slope of the graph of the inverse tangent function is always positive.