What is the linear approximation of f?

How it works

The linear approximation calculator finds the linearization of f at x = a:

L(x) = f(a) + f′(a)(x − a)

This is the tangent line to y = f(x) at a, the Taylor polynomial of order 1. A computer algebra system (nerdamer, open source) finds it with exact numbers and writes it expanded; the algebra runs in the background after you start typing.

The page shows:

• L(x) = the linearization, expanded, exactly (for example x/6 + 3/2).
• Estimate L(x): L at the x you typed under "Estimate f at x", as a decimal to 10 significant figures: f(a) + f′(a)(x − a), worked out with the decimal values of f(a) and f′(a).
• Exact f(x): f at that x, to compare: the algebra's exact value of f at the typed x when it agrees with f(x) in decimals (to 10⁻¹² × (|f(x)| + s)), else f(x) in decimals. It is shown when it is a real number and holds its 10 figures: the algebra's text is 0 (ln(x) at 1), or rounding x to a computer number, or rounding inside f, cannot move its 10th figure: s × 2.2 × 10⁻¹⁶ under 5 × 10⁻¹¹ × |f(x)|, where s = |x f′(x)| is read from f at x(1 ± 10⁻⁷). For ln(x) at x = 1.0000000005 it is left out, as rounding x changes ln x in its 8th figure.

With the estimate field empty, only L(x) is shown.

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 just beside a (at a ± 10⁻⁶ × max(1, |a|): ln(x) or 1/x at 0), means f is not defined at x = a; ±∞ on both sides as well means f(a) is past the largest computer number, about 1.8 × 10³⁰⁸ (e^x at 710), and the page says f(a) is too large. The one-sided difference quotients (f(a + h) − f(a))/h and (f(a) − f(a − h))/h, with h = 10⁻⁶ × max(1, |a|), must agree to 1 part in 1,000 of max(1, |right quotient|), plus 10⁻¹⁵ × |f(a)|/h for rounding; if not, the page says the slopes of f just left and right of a differ (a corner, a jump, a vertical tangent, or a curve that bends too sharply for h). The derivative the algebra uses is compared with a numeric difference quotient at 20 points (to 1 part in a million), and f(a) and f′(a) with the checked derivative at a. The slope of L, read as (L(10⁸) − L(−10⁸))/(2 · 10⁸), or as (L(1) − L(−1))/2 when that is past the largest computer number, must agree with the average of the two quotients to 10⁻⁵ × max(1, |f′(a)|), plus 10⁻¹⁵ × |f(a)|/h for rounding. The estimate uses f(a) worked out exactly by the algebra when that agrees with f(a) in decimals to 10⁻¹² × (|f(a)| + |f′(a) a|). A slope below 2⁻¹⁰²² (about 2.2 × 10⁻³⁰⁸, the smallest normal computer number), or 0 while the line has an x term, gives no answer. If any check fails, the algebra finds no formula, or the work takes over 3 seconds, the page shows no line.

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 centre a and the x to estimate are numbers or constant expressions: 9, 9.1, pi/3, 31pi/90.

How answers are written

• L(x) is written expanded, in the syntax you type, with exact numbers; the terms can come in any order (sqrt(3)/2 − π/6 + x/2). A line that would hold 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 not shown.
• The estimate and f(x) are decimals to 10 significant figures, rounded half up.

What gets no answer

• f not defined at a, f(a) too large, or slopes just left and right of a that differ (a corner, jump or vertical tangent).
• A step that fails its check, finds no formula, or takes over 3 seconds.
• A value to estimate that is not a number gives a message next to that field.

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

√x at a = 9, estimating √9.1 (OpenStax Calculus Volume 1, section 4.2, Example 4.5). f(9) = 3 and f′(x) = 1/(2√x), so f′(9) = 1/6. L(x) = 3 + (x − 9)/6 = x/6 + 3/2. The estimate is L(9.1) = 3 + 0.1/6 = 3.016666667; √9.1 = 3.016620626.

sin(x) at a = π/3, estimating sin 62° (OpenStax Calculus Volume 1, section 4.2, Example 4.6). sin(π/3) = √3/2 and cos(π/3) = 1/2, so L(x) = √3/2 + (x − π/3)/2 = sqrt(3)/2 − π/6 + x/2. 62° is 31π/90 radians, so the estimate is L(31π/90) = √3/2 + π/180 = 0.8834786963; sin(31π/90) = 0.8829475929.