4. Applications of Derivatives

Use the linear approximation (1 + x)ᵏ ≈ 1 + kx to find an approximation for the function f(x) for values of x near zero.

c. f(x) = 1/√(1 + x)

Faster than a calculator Use the approximation (1 + x)ᵏ ≈ 1 + kx to estimate the following.

a. (1.0002)⁵⁰

Show that the linearization of f(x) = (1 + x)ᵏ at x = 0 is L(x) = 1 + kx.

Linear approximation Find the linear approximation to ƒ(x) = cosh x at a = ln 3 and then use it to approximate the value of cosh 1.

Shallow-water velocity equation

a. Confirm that the linear approximation to ƒ(x) = tanh x at a = 0 is L(x) = x.