Use linear approximation, i.e. the tangent line, to approximate 1/0.101 as follows:

Question

Let f(x)=1/x and find the equation of the tangent line to f(x) at a "nice" point near 0.101. Then use this to approximate 1/0.101

Answer 1

f(01) = 1/0.1 = 10; f'(x) = - 1/x²; slope of tangent line at point (0.1, 10) is f'(0.1) = - 1/(0.1)² = - 100.

Tangent line is y = 10 - 100(x - 0.1) = - 100x + 20;

Linear approximation L(x) = -100x + 20; f(0.101) ≈ L(0.101) = - 100·0.101 + 20 = 9.9