Linear Approximation: Estimating Both Function Values and Input Values

Introduction

Linear approximation is generally introduced as a method of estimating the value of a function near a known point. However, the same mathematical relationship can also be used in the opposite direction: estimating the input required to produce a particular function value.

These are two different applications of the same underlying idea: using a tangent line to approximate a curve locally.

Understanding this connection also helps explain why linear approximation is closely related to Newton’s method for finding roots.

The Basic Idea of Linear Approximation

Consider a differentiable function:

$y=f(x)$

Suppose the function is known at a particular point x₀.

At that point, we know three things:

The input value: x₀
The corresponding output: f(x₀)
The slope of the curve: f'(x₀)

Using these three pieces of information, we can construct a tangent line.

The equation of the tangent line is:

$y=f(x_0)+f'(x_0)(x-x_0)$

For points sufficiently close to x₀, the tangent line can provide a useful approximation of the actual curve.

This relationship allows two different types of estimation.

Forward Approximation: Finding the Function Value

The first application is the familiar one.

Suppose the input changes from x₀ to a nearby value x.

The change in the input is:

$dx=x-x_0$

The tangent line estimates the corresponding change in the function value:

$dy\approx f'(x_0)dx$

Therefore:

$f(x)\approx f(x_0)+f'(x_0)(x-x_0)$

In other words:

Estimated new output = Starting output + Estimated change in output.

Example

Consider:

$f(x)=x^2$

Suppose the objective is to estimate the square of 5.1 using the known value of 5².

The starting point is:

$x_0=5$

The starting function value is:

$f(5)=25$

The derivative is:

$f'(x)=2x$

Therefore:

$f'(5)=10$

The input changes by:

$dx=5.1-5=0.1$

The estimated change in the function value is:

$dy\approx10(0.1)=1$

Consequently:

$f(5.1)\approx25+1=26$

The actual value is 26.01.

The tangent line provides a close approximation.

Inverse Approximation: Finding the Input Value

Now consider a different problem.

Instead of asking what function value corresponds to a particular input, suppose the desired function value is already known.

The question becomes:

What input value will produce this desired output?

We can use exactly the same tangent-line equation.

$y\approx f(x_0)+f'(x_0)(x-x_0)$

This time, we know y and want to estimate x.

Rearranging:

$y-f(x_0)\approx f'(x_0)(x-x_0)$

Dividing by the slope:

$x-x_0\approx(y-f(x_0))/f'(x_0)$

Therefore:

$x\approx x_0+(y-f(x_0))/f'(x_0)$

This is the inverse application of linear approximation.

Example: Estimating a Square Root

Suppose we want to estimate the square root of 26.

This can be expressed as finding x such that:

$x^2=26$

We already know that:

$5^2=25$

Therefore, choose:

$f(x)=x^2,\quad x_0=5$

We also know:

$f(5)=25$

And:

$f'(5)=10$

The desired function value is 26.

Using the inverse approximation:

$x\approx5+(26-25)/10$ $x\approx5.1$

Therefore:

$26^{1/2}\approx5.1$

The actual value is approximately 5.09902.

Notice something interesting: this is the same tangent-line calculation used in the previous example, but the direction of the question has changed.

Previously, we knew x = 5.1 and estimated its square.

Now, we know the desired square is 26 and estimate the required x.

Why Must the Slope Be Nonzero?

There is an important mathematical condition in inverse approximation.

Consider the formula:

$x\approx x_0+(y-f(x_0))/f'(x_0)$

The denominator is the slope of the tangent line.

If the slope is zero, division by zero becomes impossible.

But there is also a deeper geometric reason.

Imagine a tangent line that is horizontal.

Moving horizontally along this line does not produce any change in its height.

Therefore, the tangent line cannot tell us how much the input must change to produce a different output.

For example, consider:

$f(x)=x^2$

At x = 0:

$f(0)=0$

And:

$f'(0)=0$

The tangent line is horizontal.

If we want to estimate the input corresponding to a small positive output, the inverse linear approximation fails because the tangent line does not provide a useful estimate of the required horizontal movement.

This is why a nonzero slope is important.

Why Does Local Invertibility Matter?

A nonzero slope has another important implication.

If the derivative is nonzero at a point and the function is continuously differentiable nearby, the function has a locally defined inverse around that point.

In simpler terms, nearby output values correspond to unique nearby input values.

Consider a steadily increasing function.

As the input increases, the output also increases. Therefore, if we know a nearby output value, we can identify the corresponding input.

A steadily decreasing function works similarly, although the slope is negative.

However, consider a function such as:

$f(x)=x^2$

Across its entire domain, the function is not one-to-one because:

$f(2)=4$

And:

$f(-2)=4$

The same output corresponds to two different inputs.

Nevertheless, if we restrict the function to positive x-values, it becomes one-to-one and can be inverted.

This distinction is important: inverse approximation is most useful when the function has a well-defined local inverse.

Connection with Newton’s Method

The inverse approximation formula has an especially interesting application.

Suppose we want to find a root of a function:

$f(x)=0$

We can treat zero as the desired output.

Using inverse linear approximation:

$x\approx x_0+(0-f(x_0))/f'(x_0)$

Therefore:

$x_1=x_0-f(x_0)/f'(x_0)$

This is precisely the iteration formula used in Newton’s method.

The process is repeated:

$x_{n+1}=x_n-f(x_n)/f'(x_n)$

Each iteration uses the tangent line at the current estimate to predict where the function will reach zero.

Thus, Newton’s method is not a completely separate mathematical idea. It is an application of inverse linear approximation.

The Bigger Picture: One Tangent Line, Two Directions

The same tangent line provides two different kinds of information.

Forward approximation:

Input is known.
Estimate the output.
Move horizontally and estimate the vertical change.
Uses the derivative.

Inverse approximation:

Desired output is known.
Estimate the input.
Move vertically and estimate the horizontal change.
Uses the reciprocal of the derivative.

The underlying geometric relationship is:

$dy\approx f'(x_0)dx$

Rearranging:

$dx\approx dy/f'(x_0)$

The first relationship estimates the vertical movement from a horizontal movement.

The second estimates the horizontal movement from a vertical movement.

Conclusion

Linear approximation is more versatile than simply estimating function values.

A tangent line provides a local relationship between inputs and outputs. Depending on which quantity is known, the same relationship can be used to estimate either one.

This leads to an important mathematical connection:

Ordinary linear approximation estimates function values.

Inverse linear approximation estimates input values.

Newton’s method uses inverse linear approximation to find roots.

The central insight is that a tangent line does not merely tell us how much the output changes when the input changes. It can also tell us approximately how much the input must change to achieve a desired change in the output.

Linear Approximation: Estimating Both Function Values and Input Values

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Calculus 1A: Differentiation

Calculus 1B: Integration

Calculus 1C: Coordinate Systems & Infinite Series