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.

145. The linearization of eˣ at x = 0

a. Derive the linear approximation eˣ ≈ 1 + x at x = 0.

153. The linearization of 2ˣ

a. Find the linearization of f(x) = 2ˣ at x = 0. Then round its coefficients to two decimal places.

154. The linearization of log₃x

a. Find the linearization of

f(x) = log₃xatx = 3.

Then round its coefficients to two decimal places.

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function f(x) at x = a is called the quadratic approximation of f at x = a. In Exercises 41–46, find the (a) linearization (Taylor polynomial of order 1)

f(x) = ln(cos x)

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function f(x) at x = a is called the quadratic approximation of f at x = a. In Exercises 41–46, find the (a) linearization (Taylor polynomial of order 1)

f(x) = 1 / √(1 − x²)