What is the curvature of y = f(x)?

How it works

For the graph y = f(x) the curvature is

κ(x) = |f″(x)| / (1 + f′(x)²)^(3/2)

and the radius of curvature is R = 1/κ. A computer algebra system (nerdamer, open source) finds f′ and f″ (f″ as the derivative of f′); the algebra runs in the background after you start typing.

The page shows:

• Curvature κ: κ(a) as a decimal to 10 significant figures, rounded half up: the value of the exact κ(a) when the two agree (to 10⁻⁹ of the exact value plus 10⁻¹⁴ × max(1, |a|)), otherwise the formula worked out at a in decimals. So sin(x) at π has κ = 0, not the 4.3 × 10⁻¹⁷ that π rounded to a computer number gives.
• Exact κ: κ(a) written exactly: the formula with x = a, simplified by the algebra (for example 6, 1/2, 1/sqrt(2), 2/37^(3/2)). It is left out when the simplified form fails its check, holds a rounded number, or takes over 3 seconds; the decimal still shows.
• Radius of curvature: 1/κ(a) as a decimal, left out when κ(a) = 0.
• κ(x) = the formula |f″(x)|/(1 + f′(x)²)^(3/2) with f′ and f″ as the algebra gives them.

Every answer is checked before it is shown. f must be a real number at a: f(a) that is not a number, or is ±∞ while f is a number at a ± 10⁻⁶ × max(1, |a|) (ln(x) at 0), means f is not defined there; ±∞ on both sides as well means f(a) is too large, past about 1.8 × 10³⁰⁸ (e^x at 1000). Each derivative is compared with a numeric difference quotient at 20 points (to 1 part in a million). κ(a) must be a finite number, or the page says f′(a) or f″(a) does not exist. The exact value is simplified by the algebra, which checks it against the formula at a. A result that holds a number the algebra could only give rounded (a whole number past 2⁵³; a fraction p/q, p and q being the numbers that multiply its top and its bottom, such as 288557167/(342919925e), when q after removing its factors 2 and 5 is over 1,000,000, or is over 1 while |p| times it is over 10¹²; or a decimal with more than 12 significant digits) is never shown as exact. Two more cases give "No verified answer": κ(a) below 2⁻¹⁰²² (about 2.2 × 10⁻³⁰⁸, the smallest normal computer number; 0 included) when the exact κ(a) is not the text 0 (e^x at −800, where κ is about 3.7 × 10⁻³⁴⁸, below the smallest computer number; the digits of a number that small are not all kept), and a sum or difference inside κ(a) (the exact form when there is one, else the formula at a) that cancels to under 10⁻⁶ of the size of its two parts, so digits are lost (cos(x) − 1 + x²/2 at 10⁻⁵, where 1 − cos x cancels). If a derivative fails its check, finds no formula, or the work takes over 3 seconds, the page says "No verified answer" too.

What you can type

• A number has at most 15 digits in a row. A computer number keeps only about 16 digits, so a longer one (9007199254740993) would stand for a nearby number (9007199254740992), and the page asks for fewer digits instead. Write very large or very small numbers with a power of ten (1e-20).
• The function f uses the variable x. Numbers can have decimals (2.5) and powers of ten (1e-3).
• Operations: + − * / and ^ for powers. Brackets group. A number or bracket next to a letter multiplies: 2x, 3(x + 1), x sin(x).
• Constants: pi (or π) and e.
• Functions: sqrt, cbrt, ln (and log, the same natural logarithm), log10, exp, abs (or |x|), sin, cos, tan, sec, csc, cot, asin, acos, atan (arcsin, arccos, arctan also work), sinh, cosh, tanh and their inverses. Angles are in radians. sin x without brackets means sin(x); sin x^2 means sin(x²).
• The point a is a number or a constant expression: 1, −2.5, pi/2, e.

The chart

The chart draws y = f(x) from x = a − 3 to a + 3, with the point (a, f(a)) marked.

Worked examples by hand

y = x³ − 3x + 1 at x = 1 (OpenStax Calculus Volume 3, section 3.3, Example 3.13). f′(x) = 3x² − 3, so f′(1) = 0; f″(x) = 6x, so f″(1) = 6. κ = |6|/(1 + 0²)^(3/2) = 6, and the radius of curvature is R = 1/6.

y = x − x²/4 at x = 2 (OpenStax Calculus Volume 3, section 3.3, Exercise 133). f′(x) = 1 − x/2, so f′(2) = 0; f″(x) = −1/2. κ = |−1/2|/(1 + 0)^(3/2) = 1/2, and R = 2.