Final answer:
The Alternating Series Estimation Theorem can be used to estimate the range of values for x that make an approximation accurate within a certain error. In this case, the range is x < 0.39.
Step-by-step explanation:
The Alternating Series Estimation Theorem provides a way to estimate the range of values of x for which the given approximation is accurate within a certain error. In this case, we have the approximation sinx ≈ x - x³/6, and we want the error to be less than 0.01.
The theorem states that the error in approximating a convergent alternating series is less than or equal to the absolute value of the next term in the series. So we can set up the inequality |x³/6| < 0.01 and solve for x to find the range of values.
To solve this inequality, we can multiply both sides by 6 to get |x³| < 0.06. Since |x³| is always positive, we can remove the absolute value and write the inequality as x³ < 0.06.
Taking the cube root of both sides gives us x < ∛0.06. Evaluating this expression gives x < 0.39. So the range of values for x that satisfy the given approximation with an error less than 0.01 is x < 0.39.