The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval [0, 6]. Express solutions to four decimal places.
(arctan x)³ ― x + 2 = 0

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the values of \( x \) in the interval \([0, 6]\) that satisfy the equation \((\arctan x)^3 - x + 2 = 0\).
2. Step 2: Recognize that this equation involves the inverse trigonometric function \( \arctan x \), which is not easily solvable by algebraic methods. Therefore, we will use a graphing calculator to find the solutions.
3. Step 3: Enter the function \( f(x) = (\arctan x)^3 - x + 2 \) into the graphing calculator.
4. Step 4: Use the graphing calculator to plot the function \( f(x) \) over the interval \([0, 6]\). Look for the points where the graph intersects the x-axis, as these are the solutions to the equation.
5. Step 5: Use the calculator's root-finding feature to determine the x-values of these intersection points, ensuring the solutions are expressed to four decimal places.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, are used to find angles when given a ratio of sides in a right triangle. The function arctan(x) returns the angle whose tangent is x. Understanding how to manipulate and interpret these functions is crucial for solving equations involving them, especially when they appear in non-linear forms.

Graphing Calculators

Graphing calculators are powerful tools that allow users to visualize functions and their intersections. They can plot equations and help identify solutions graphically, which is particularly useful for equations that cannot be solved algebraically. Familiarity with using a graphing calculator to find roots and analyze graphs is essential for solving complex equations.

Numerical Solutions

Numerical solutions refer to methods for approximating the solutions of equations when exact solutions are difficult or impossible to obtain. Techniques such as graphing or using numerical algorithms (like Newton's method) can provide approximate values for roots. In this context, expressing solutions to a specified decimal place indicates the precision required in the answer.