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


a. (1.0002)⁵⁰

Common linear approximations at x = 0 Find the linearizations of the following functions at x = 0.

b. cos x

Linearization for Approximation

In Exercises 7–12, find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.

f(x) = ∛x, a = 8.5

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

a. f(x) = (1 − x)⁶

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)

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

What is the value of the local linearization of the function $f\left(x\right)=x^2$ at $x=2$, evaluated at $x=2.1$?

If $f\left(x\right)=x^3+1$, use the linearization $L\left(x\right)$ at $a=5$ to approximate $f\left(5.1\right)$.

If $f\left(x\right)=\sqrt{x}+12$, use the linearization $L\left(x\right)$ at $a=16$ to approximate $f\left(16.01\right)$.