Linear Approximation

What is ? You can’t get it exactly by hand, but you know , and 26 is very close to 25. Linear approximation uses that: near a point you know, a smooth curve is almost the same as its tangent line, and a line is easy to evaluate. The same idea is behind the small-angle approximation in physics, Newton’s method for solving equations, and estimating how measurement errors spread through a calculation.

What Is Linear Approximation?

The linearization of at is the tangent line at that point, written as a function:

Linear approximation means using in place of for values of close to :

The graph shows why this works. At the line and the curve meet and have the same slope, so they leave that point heading in the same direction. Close to they’re nearly indistinguishable. Farther away the curve bends and the line doesn’t, so the error grows. The approximation is a local one: it’s only good near .

The point is called the center of the linearization. Choose it close to the value you want, at a place where and are easy to compute exactly.

How to Do a Linear Approximation

1. Choose : a nearby value where the function is easy to evaluate.
2. Compute and .
3. Write .
4. Evaluate at the value you want.

Example 1: Estimating a Square Root

Estimate .

Step 1 — choose : and , since exactly.

Step 2 — the value and slope at :

Step 3 — the linearization:

Step 4 — evaluate:

The true value is , so the estimate is off by less than 0.001.

Example 2: A Cube Root

Estimate .

Use at . Then and , so :

The true value is .

Example 3: Approximations Near Zero

Many standard approximations come from linearizing at . For , and , so :

That’s the small-angle approximation from physics: , and the true value is . The same steps give the whole table:

The last row is worth remembering. is why 1% growth for 10 years is roughly, but a bit more than, 10% in total.

Overestimate or Underestimate?

The second derivative tells you which side of the curve the tangent line is on:

• , concave up: the curve bends up away from the tangent line, so the line is below the curve and is an underestimate.
• , concave down: the line is above the curve and is an overestimate.

Check this against the examples. For , , so 5.1 should be too big, and it is: . For , , so 1.05 should be too small, and it is: .

How Accurate Is It?

The error depends on two things: how far is from , and how sharply the curve bends. Back to at , where :

Moving 10 times farther from (from 0.1 to 1) makes the error about 100 times bigger. That’s typical: the error grows roughly with the square of the distance from . Taylor polynomials, later in the course, fix this by adding a curved correction.

Differentials

The same idea, written in terms of changes, is called differentials. Let be a small change in . The matching change along the tangent line is

The true change in the function is , and for small , . On at with , while . That’s a large , chosen so the gap is visible. With , and .

Estimating Measurement Error

Differentials are how scientists estimate how an error in a measurement affects a result. Suppose a ball’s radius is measured as 10 cm, with a possible error of 0.1 cm. How far off could the computed volume be?

Step 1 — the volume and its differential:

Step 2 — substitute , :

Step 3 — compare to the volume. cm³, so the relative error is

A 1% error in the radius becomes about a 3% error in the volume. The exponent 3 in triples the relative error.

Common Mistakes

• Choosing far from . The approximation is only accurate near . For , works; gives 5.25, which is much worse.
• Using degrees in trig approximations. only holds in radians. is about 0.087, not 5.
• Evaluating at instead of . The slope in is the number .
• Mixing up and . is the tangent-line estimate; is the exact change. They’re close only when is small.

Practice Problems

Try each one before opening the answer. The practice set has 12 more with instant checking.

Problem 1. Find the linearization of at .

Problem 2. Use a linear approximation to estimate .

Problem 3. Estimate .

Problem 4. Use the linearization of at to estimate .

Problem 5. Is the estimate from Problem 2 too high or too low?

Problem 6. The side of a square is measured as 5 cm, with a possible error of 0.02 cm. Use differentials to estimate the possible error in its area.

The tangent line has now given you slopes, equations and estimates. Next, the derivative helps find where a function reaches its highest and lowest values, starting with critical points.