Use the tangent line approximation to approximate the value of ln(1.004)
To use the tangent line approximation, you need to follow these steps:
1. Choose a convenient point near the value you want to approximate. In this case, we can choose x = 1 as our point since it is easier to calculate the natural logarithm at x = 1.
2. Calculate the derivative of the function you want to approximate at that point. In this case, the derivative of ln(x) is 1/x.
3. Use the point-slope form of a line to find the equation of the tangent line. The point-slope form is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point. Plugging in x1 = 1 and m = 1/1 = 1, we have y - ln(1) = 1*(x - 1), which simplifies to y = x - 1.
4. Plug in the value you want to approximate into the equation of the tangent line. In this case, we want to approximate ln(1.004), so we substitute x = 1.004 into the equation y = x - 1:
ln(1.004) ≈ 1.004 - 1
≈ 0.004
Therefore, using the tangent line approximation, ln(1.004) is approximately equal to 0.004.