Math 161

Sample First Midterm

Problem 8

Dr. Wilson

8. Find the best linear approximation for ln x which is tangent to the graph of the function when x = 1. Use this to approximate ln 1.05. Check your answer with a calculator.

The formula for the best linear approximation to y = f(x) at a point where x = a is

f(x) = f(a) + f'(a)(x - a)

If f(x) = ln x and a = 1, we get

ln(x) = ln 1 + x - 1

= 0 + x - 1

= x - 1

To use this to approximate ln 1.05,

1.05 - 1 = 0.05

According to a calculator

ln 1.05 = 0.048790164