Approximations

First-Order Approximation or Linear Approximation

For a differentiable function \( f \), let the input be \( x \) and the output be \( y \), so that

\[ y = f(x) \]

When the input is perturbed by a small amount \( \Delta x \), the new input is \( x + \Delta x \), and the corresponding output is

\[ y + \Delta y = f(x + \Delta x) \]

Thus, the exact change in output is

\[ \Delta y = f(x+\Delta x) - f(x). \]

To approximate \( \Delta y \), we use the definition of the derivative:

\[ f'(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}. \]

For small \( \Delta x \), we approximate:

\[ f'(x) \approx \frac{\Delta y}{\Delta x} \]

which implies

\[ \Delta y \approx f'(x) \Delta x. \]

Since the derivative is often written as \( \frac{dy}{dx} \), this can also be expressed as

\[ \Delta y \approx \frac{dy}{dx} \Delta x. \]

This is the fundamental idea behind linear approximation in calculus, where small changes in the output are approximated using the derivative.

We want to approximate \( 28^{1/3} \) using the first-order approximation.

Let

\[ y = x^{1/3} \]

For \( x = 27 \), we know

\[ y = 27^{1/3} = 3. \]

Now, consider a small change in input:

\[ \Delta x = 28 - 27 = 1. \]

We approximate the corresponding change in output using the derivative. The derivative of \( y = x^{1/3} \) is:

\[ \frac{dy}{dx} = \frac{1}{3} x^{-2/3}. \]

Evaluating at \( x = 27 \):

\[ \left. \frac{dy}{dx} \right|_{x=27} = \frac{1}{3} \cdot 27^{-2/3} = \frac{1}{27}. \]

Thus, the small change in \( y \) is:

\[ \Delta y \approx f'(x) \Delta x = \frac{1}{27} \times 1. \]

So, the estimated value of \( 28^{1/3} \) is:

\[ 28^{1/3} \approx y + \Delta y = 3 + \frac{1}{27}. \]

The actual value of \(28^{1/3}\) is 3.03658897 and the linear approximation is 3.03703704.

The alternative formula for quickly answering approximations is

\[ f(x+\Delta x) \approx f(x) + f'(x) \Delta x. \]

For \( f(x) = x^{1/3} \),

\[ f(27) = 27^{1/3} = 3, \quad f'(x) = \frac{1}{3} x^{-2/3}. \]

Evaluating at \( x = 27 \),

\[ f'(27) = \frac{1}{3} \cdot 27^{-2/3} = \frac{1}{27}. \]

Applying the first-order approximation,

\[ 28^{1/3} \approx f(27) + f'(27) \cdot (28 - 27). \]