Second-Order Taylor Polynomial in 2D – Quick
Reference Sheet
General Formula (centered at (x■, y■)):
P■■■,■■■(x,y) = f(x■, y■) + f■(x■, y■)(x - x■) + f■(x■, y■)(y - y■)
+ ½ f■■(x■, y■)(x - x■)² + f■■(x■, y■)(x - x■)(y - y■) + ½ f■■(x■, y■)(y - y■)²
Step 1
Evaluate f(x■, y■).
Step 2
Find f■ and f■, then evaluate them at (x■, y■).
Step 3
Find f■■, f■■, and f■■, then evaluate them at (x■, y■).
Step 4
Plug into the general formula.
Tip
The quadratic terms come from the Hessian matrix: H = [[f■■, f■■], [f■■, f■■]]. The quadratic part is ½ [
Example: f(x,y) = e■ cos(y), center = (0,0)
Step 1: f(0,0) = 1
Step 2: f■ = e■ cos y → f■(0,0) = 1
f■ = -e■ sin y → f■(0,0) = 0
Step 3: f■■ = e■ cos y → f■■(0,0) = 1
f■■ = -e■ sin y → f■■(0,0) = 0
f■■ = -e■ cos y → f■■(0,0) = -1
Step 4: P(x,y) = 1 + 1·x + 0·y + ½(1)x² + 0·xy + ½(-1)y²
Final: P(x,y) = 1 + x + ½ x² - ½ y²