tangent line approximation

Question

Use the tangent line approximation of  at  to approximate .

Answer 1

We are approximating the actual y-value of the function f(x) = √(1 - x) at x = .1 by using instead the y-value of the tangent line to the curve at the point (0 , 1) when x = .1 .

The easiest way to find the tangent line's y-value there is to use the formula L(x) = f(x₀) + f^'(x₀)·(x - x₀) where x₀ = 0 , and x = .1:

f(0) = 1

f^'(x) = - 1 / (2√(1 - x)) (by power rule and chain rule) so f^'(0) = - 1/2 .

Finally, L(.1) = 1 - 1/2·(.1) = .95

Note that because f(x) is concave down at x = 0, this linear approximation is an overestimate of the actual value.