Linear Approximation and Concavity: How the Second Derivative Sharpens Your Estimates

Tangent lines are one of the quickest ways to estimate a function near a known point. But a tangent line is only as good as the curve is straight. This article shows how concavity tells you whether your estimate is too high or too low, and how the second derivative gives you a better approximation.

Linear approximation in one formula

Near a point x = a, a smooth function behaves a lot like its tangent line:

f(x) ≈ f(a) + f'(a)(x − a)

You only need the value and the slope at a. That is what makes it so handy for estimating things like roots, logs, and trig values by hand.

What concavity means

Concavity describes how a curve bends, and it is measured by the second derivative, f”.

• f” > 0 (concave up): the curve bends upward like a cup and sits above its tangent line.
• f” < 0 (concave down): the curve bends downward like a frown and sits below its tangent line.

That gives a simple rule for the error in a linear approximation:

• Concave up: the tangent line underestimates the true value.
• Concave down: the tangent line overestimates the true value.

Worked example: estimating √4.1

Let f(x) = √x and a = 4.

• f(4) = 2
• f'(x) = 1 / (2√x), so f'(4) = 1 / 4

The linear approximation is:

√4.1 ≈ 2 + (1 / 4)(0.1) = 2.025

Now check the concavity. Since f”(x) = -(1 / 4)x^-3/2, which is negative, the function is concave down. So 2.025 is an overestimate. The actual value is about 2.02485.

Three ways concavity improves the approximation

1. It tells you the direction of the error. Knowing the estimate is too high lets you state a bound. For √4.1, the true value is slightly below 2.025.
2. It lets you add a correction term. This is the quadratic approximation, explained below.
3. It tells you how much to trust the estimate. The larger |f”| is, or the farther x is from a, the worse the linear estimate gets.

Where the quadratic approximation comes from

The idea is to match the function’s value, slope, and bend at x = a. Start with a quadratic:

P(x) = c₀ + c₁(x − a) + c₂(x − a)²

and require that P and f agree in value, first derivative, and second derivative at a.

• P(a) = c₀, so c₀ = f(a)
• P'(x) = c₁ + 2c₂(x − a), so P'(a) = c₁, giving c₁ = f'(a)
• P”(x) = 2c₂, so P”(a) = 2c₂, giving c₂ = f”(a) / 2

Putting it together:

f(x) ≈ f(a) + f'(a)(x − a) + (1 / 2)f”(a)(x − a)²

The 1 / 2 appears because differentiating (x − a)² twice produces a factor of 2. Dividing by 2 cancels it so the second derivatives match exactly.

What the new term does

The first two terms are the tangent line. The extra term (1 / 2)f”(a)(x − a)² bends that line to follow the curve.

• If f”(a) > 0, it adds a positive amount, fixing the underestimate.
• If f”(a) < 0, it subtracts, fixing the overestimate.

Back to √4.1

Here f”(4) = -1 / 32, so the correction is:

(1 / 2)(-1 / 32)(0.1)² ≈ -0.00016

The improved estimate is 2.025 − 0.00016 = 2.02484, which is almost exactly the true value of 2.02485.

Key takeaways

• The tangent line gives a quick estimate using f(a) and f'(a).
• The sign of f” tells you whether that estimate is too low (concave up) or too high (concave down).
• Adding (1 / 2)f”(a)(x − a)² gives the quadratic approximation, which is usually far more accurate.
• The farther you move from a, the more the error grows, so stay close to your base point.