Taylor Series For Two Variables Examples

Taylor Series for Two Variables Examples: Seeing the Math Behind Surfaces

You know how a tangent line gives you a decent approximation of a curve near a point? Because of that, the Taylor series for two variables does something similar, but for surfaces. Here's the thing — instead of approximating a function of one variable with a polynomial, we're building a polynomial that hugs a surface near a specific point in the xy-plane. It sounds abstract, but once you see a few concrete examples, it clicks.

Let me walk you through what this actually looks like with real functions, real points, and real calculations. No fluff — just the kind of examples that make you go "oh, that's* how it works."

What Is the Taylor Series for Two Variables?

At its core, the Taylor series for a function of two variables f(x, y) is a way to approximate that function near a point (a, b) using a polynomial built from the function's partial derivatives evaluated at that point.

The general idea: if you know the value of the function and all its partial derivatives at a single point, you can construct a polynomial that matches the function's behavior in the neighborhood of that point. The more terms you include, the better the approximation.

The second-order Taylor expansion (which is what most examples focus on) looks like this:

f(x, y) ≈ f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b) + (1/2!)[f_xx(a, b)(x - a)² + 2f_xy(a, b)(x - a)(y - b) + f_yy(a, b)(y - b)²]

Where the subscripts denote partial derivatives. f_x means the partial derivative with respect to x, f_xx means the second partial derivative with respect to x twice, and so on.

Breaking Down the Formula

Each term has a clear geometric meaning:

• The constant term f(a, b) is just the height of the surface at the point.
• The linear terms f_x(a, b)(x - a) and f_y(a, b)(y - b) capture the slope in the x and y directions.
• The quadratic terms capture how the surface curves — whether it's bowl-shaped, saddle-shaped, or flat in different directions.

It's the multivariable cousin of the familiar single-variable Taylor series you saw in calculus. The complexity comes from having to track partial derivatives in multiple directions, but the principle is identical.

Why It Matters: Real Applications

So why should you care? Here's where this actually shows up:

Optimization problems: When you're trying to find maxima or minima of functions of two variables, the Taylor expansion tells you what the surface looks like near a critical point. Is it a peak? A valley? A saddle? The second-order terms answer that.

Physics and engineering: Many physical quantities depend on two or more variables. Temperature across a metal plate, pressure in a fluid as a function of position, potential energy in a mechanical system — the Taylor expansion gives you a local model you can actually work with.

Numerical methods: When you're solving differential equations or doing computational modeling, you often need to approximate a function locally. The Taylor series gives you a systematic way to do that with controlled error.

Economics and finance: Utility functions, production functions, option pricing models — economists use multivariable Taylor expansions to approximate complex relationships near equilibrium points.

The bottom line: if you're modeling anything that depends on two variables and you need to understand its local behavior, this tool is indispensable.

How It Works: Step-by-Step Examples

Let's get concrete. I'll work through several examples of increasing complexity.

Example 1: A Simple Quadratic Function

Let's start with f(x, y) = x² + xy + y², expanded around the point (0, 0).

First, we need the partial derivatives:

f_x = 2x + y
f_y = x + 2y
f_xx = 2
f_xy = 1
f_yy = 2

Evaluating at (0, 0):

f(0, 0) = 0
f_x(0, 0) = 0
f_y(0, 0) = 0
f_xx(0, 0) = 2
f_xy(0, 0) = 1
f_yy(0, 0) = 2

Plugging into the formula:

f(x, y) ≈ 0 + 0·x + 0·y + (1/2)[2x² + 2·1·xy + 2y²]
= x² + xy + y²

In this case, the Taylor series gives us back the original function exactly, because it's already a second-degree polynomial. That's a good sanity check — if the function is polynomial of degree n, the Taylor expansion of order n or higher reproduces it exactly.

Example 2: A Square Root Function

Now let's try something that's not a polynomial. Take f(x, y) = √(x² + y²), expanded around (1, 0).

This function represents the distance from the origin, and we want to approximate it near the point (1, 0). Simple as that.

Computing the partial derivatives:

f_x = x / √(x² + y²)
f_y = y / √(x² + y²)
f_xx = y² / (x² + y²)^(3/2)
f_xy = -xy / (x² + y²)^(3/2)
f_yy = x² / (x² + y²)^(3/2)

At (1, 0):

f(1, 0) = 1
f_x(1, 0) = 1
f_y(1, 0) = 0
f_xx(1, 0) = 0
f_xy(1, 0) = 0
f_yy(1, 0) = 1

So the second-order Taylor expansion is:

f(x, y) ≈ 1 + 1·(x - 1) + 0·(y - 0) + (1/2)[0·(x - 1)² + 2·0·(x - 1)(y - 0) + 1·y²]
= 1 + (x - 1) + y²/2

Let's test this. At the point (1.1, 0.1):

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Actual value: √(1.Consider this: 01) = √1. 22 ≈ 1.Because of that, 1 + 0. Also, 1045
Taylor approximation: 1 + 0. 21 + 0.01/2 = 1.

Pretty close. The approximation gets worse as you move further from (1, 0), which is exactly what you'd expect.

Example 3: An Exponential Function

Let's try f(x, y) = e^(x+y), expanded around (0, 0).

The partial derivatives are straightforward since the exponential function is its own derivative:

f_x = e^(x+y)
f_y = e^(x+y)
f_xx = e^(x+y)
f_xy = e^(x+y)
f_yy = e^(x+y)

At (0, 0), all of these equal 1.

So the second-order expansion is:

f(x, y) ≈ 1 + x + y + (1/2)[x² + 2xy + y²]
= 1 + x + y + (1/2)(x + y)²

This is actually the beginning of the familiar series for e^u where u = x + y. You can see how the multivariable version naturally reduces to the single-variable case when the function depends on a linear combination of the variables.

Example 4: A Trigonometric Function

Consider f(x, y) = sin(xy), expanded around (0, 0).

Partial derivatives:

f_x = y cos(xy)
f_y = x cos(xy)
f_xx = -y² sin(xy)
f_xy = cos(xy) - xy sin(xy

Continuing the Derivation

To finish the fourth example we need the remaining second‑order partials at the expansion point ((0,0)).

[ \begin{aligned} f_{yy}(x,y) &= -x^{2}\sin(xy) \quad\Longrightarrow\quad f_{yy}(0,0)=0,\[4pt] f_{xy}(x,y) &= \cos(xy)-xy\sin(xy) \quad\Longrightarrow\quad f_{xy}(0,0)=1. \end{aligned} ]

All first‑order derivatives also vanish at the origin:

[ f_x(0,0)=0,\qquad f_y(0,0)=0. ]

Plugging everything into the multivariable Taylor formula gives

[ \begin{aligned} \sin(xy) &\approx 0 ;+; 0\cdot x ;+; 0\cdot y \ &\quad +\frac12\Big[,0\cdot x^{2} +2\cdot 1\cdot xy +0\cdot y^{2}\Big] \ &= xy . \end{aligned} ]

Thus, near ((0,0)) the sine of a product behaves linearly: (\sin(xy)\approx xy).
If we kept the third‑order term, the next contribution would be (-\frac{(xy)^{3}}{6}), and so on.

General Observations

1. Order of Accuracy – The error of a second‑order Taylor polynomial is proportional to the cube of the distance from the expansion point. In symbols, if (|(x-a,y-b)|) is small, the remainder (R_2) satisfies
   [ |R_2| \le \frac{M}{6},|(x-a,y-b)|^{3}, ]
   where (M) bounds the third‑order partial derivatives in a neighbourhood of ((a,b)).

2. Choice of Expansion Point – The approximation is only reliable in a region where the higher‑order derivatives do not grow too large. For functions with singularities (e.g., (\sqrt{x^{2}+y^{2}}) near the origin) the safe domain may be limited.

3. Degenerate Cases – When the function is already a polynomial of degree (\le 2), the Taylor expansion reproduces it exactly, as we saw with (x^{2}+xy+y^{2}). This property is often used to verify symbolic implementations of the formula.

4. Connection to One‑Variable Series – If a multivariable function depends on a single linear combination, say (u = \alpha x + \beta y), then its Taylor expansion reduces to the familiar one‑variable series in (u). This explains why the exponential example collapsed to (1+u+\frac{u^{2}}{2}).

Practical Tips for Using Multivariable Taylor Expansions

Conclusion

The multivariable Taylor expansion is a powerful, systematic way to approximate smooth functions of several variables near a point of interest. By matching function value, gradient, and Hessian at the expansion point, we obtain a quadratic polynomial that captures the local geometry of the surface. The examples explored—quadratic polynomials, square‑root distance, exponentials, and sines—illustrate both the simplicity of low‑degree cases and the richness of more detailed functions.

When applied judiciously—choosing an expansion point where the function is well‑behaved and keeping enough terms to meet the desired accuracy—the Taylor polynomial serves as a reliable local model. It underpins many analytical techniques, from asymptotic analysis to perturbation methods, and it provides a bridge between exact calculus and practical numerical computation.

In short, mastering the multivariable Taylor expansion equips you with a versatile analytical lens: it lets you “zoom in” on a function’s behavior, linearize complex dependencies, and build approximations that are both mathematically sound and computationally tractable. Use it wisely, respect its domain of validity, and you’ll find it indispensable in fields ranging from physics and engineering to machine learning and economics.