Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.
f. tan⁻¹(sinh x) |₋₃³

Verified step by step guidance

1. Recall that the function given is the inverse tangent of the hyperbolic sine function, written as \(\tan^{-1}(\sinh x)\). This means we first need to evaluate \(\sinh x\) at the given bounds and then apply the inverse tangent function to those results.
2. Write down the definition of the hyperbolic sine function: \(\sinh x = \frac{e^{x} - e^{-x}}{2}\). This will help you calculate \(\sinh x\) at \(x = -3\) and \(x = 3\).
3. Calculate \(\sinh(-3)\) and \(\sinh(3)\) using the formula from step 2. Remember that \(\sinh\) is an odd function, so \(\sinh(-3) = -\sinh(3)\), which can simplify your calculations.
4. Next, apply the inverse tangent function \(\tan^{-1}\) to the values of \(\sinh(-3)\) and \(\sinh(3)\) you found. This means you will find \(\tan^{-1}(\sinh(-3))\) and \(\tan^{-1}(\sinh(3))\).
5. Finally, evaluate the definite expression \(\tan^{-1}(\sinh x) \big|_{-3}^{3}\) by subtracting the value at \(x = -3\) from the value at \(x = 3\): \(\tan^{-1}(\sinh 3) - \tan^{-1}(\sinh (-3))\). Use a calculator to find these values accurate to four decimal places.

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as sinh(x), cosh(x), and tanh(x), are analogs of trigonometric functions but based on hyperbolas. Sinh(x) is defined as (e^x - e^(-x))/2 and can take any real value. Understanding their properties is essential for evaluating expressions involving these functions.

Inverse Trigonometric Functions

The inverse tangent function, tan⁻¹(x), returns the angle whose tangent is x, with a range of (-π/2, π/2). It is important to know how to apply inverse trig functions to real numbers and interpret their outputs, especially when composed with other functions like sinh(x).

Definite Evaluation and Calculator Use

Evaluating an expression like tan⁻¹(sinh x) |₋₃³ requires substituting the limits into the function and computing values accurately, often using a calculator. Reporting answers to four decimal places demands careful rounding and understanding of function domains to ensure the values exist and are valid.