Question

For approximately what values of x can you replace sin x by x − x^3/6 with an error of magnitude no greater than 6×10^−4?

Answer 1

To replace sin x by x − x³/6 with an error of magnitude no greater than 6×10⁻⁴, the values of x should be approximately less than or equal to 0.4.

Step-by-step explanation:

To replace sin x by x − x3/6 with an error of magnitude no greater than 6×10−4, we need to determine the values of x for which the difference between sin x and x − x3/6 is less than or equal to 6×10−4. Let's consider the Taylor series expansion of sin x: sin x = x − x3/6 + x5/120 + ...

By comparing the terms, we can see that the error term x5/120 is smaller than 6×10−4 when x is around 0.4 or smaller. Therefore, we can replace sin x by x − x3/6 with an error of magnitude no greater than 6×10−4 for values of x approximately less than or equal to 0.4.