Taylor's theorem with various forms of remainders

Taylor's theorem with various forms of remainders

1. Taylor for $f(x,y)$ About $(a,b)$

If $f$ has continuous partials up to order $n+1$ near $(a,b)$, then for $h = x-a, k = y-b$:

where $[h ∂/∂x + k ∂/∂y]^r$ expands binomially, e.g. $r=2$ gives $h² f_{xx}+2hk f_{xy}+k² f_{yy}$. This is the two-variable analogue of $f(a+h)=f(a)+h f'+h²/2! f''+...$.

For $(a,b)=(0,0)$, this becomes Maclaurin.

2. Remainder $R_{n+1}$ — Three Usual Forms

Let $(a+θh, b+θk)$ be some intermediate point $0<θ<1$.

• Lagrange form:

— useful for bounds (estimate max derivative on segment).

• Cauchy form:

— sometimes sharper.

• Peano / o-notation:

meaning remainder smaller than $ρ^n$ where $ρ=√(h²+k²)$ — shows approximation order.

For exams, Lagrange suffices; mention others for completeness.

3. Linear and Quadratic Approximation (most used)

• First-order (tangent plane):

with error $O(h²+k²)$.

• Second-order: add $1/2[h² f_{xx}+2hk f_{xy}+k² f_{yy}]$ — needed for maxima theory and error $O(ρ³)$.

4. Worked Examples

(a) Expand f(x,y)= e^x cos y about (0,0) to second order:
    f(0,0)=1, f_x= e^x cos y →1, f_y= -e^x sin y →0,
    f_{xx}= e^x cos y→1, f_{xy}= -e^x sin y→0, f_{yy}= -e^x cos y→-1
    ⇒ f(h,k) ≈ 1 + h + (h² - k²)/2 + R3 ; i.e. e^x cos y ≈1 + x + (x² - y²)/2 ✓
    Check: product of series e^x=1+x+x²/2, cos y=1-y²/2 confirms.

(b) Expand f(x,y)= sin(x+y) about (0,0) to third order: use single variable t=x+y
    sin t = t - t³/6 +... ⇒ f ≈ (x+y) - (x+y)³/6 ✓

(c) Estimate error for √(1+x+y) about (0,0) linear: √(1+h+k)≈1+(h+k)/2 ; Lagrange remainder
    = -(h+k)²/[8(1+θ(h+k))^{3/2}] ≤ (h+k)²/8 bounds ✓

(d) f(x,y)= log(1+x+y) : expansion log(1+t)=t -t²/2 +t³/3 ; t=x+y ⇒ log≈(x+y)-(x+y)²/2 ✓

5. Convergence and Use

Taylor's infinite series ($R_n→0$) converges where $f$ analytic (polynomials, $e^{x+y}$, $sin, cos$ everywhere; $log(1+x+y)$ for $|x+y|<1$). Remainder estimate tells you how many terms keep error < tolerance.

6. Exam Points

• State Taylor for $f(x,y)$ about $(a,b)$ with Lagrange remainder — 5 marks, very frequent.
• Expand $e^{x+y}, sin(x+y), log(1+x+y), √(1+x+y)$ to second/third order about $(0,0)$ — 5-7 marks, very frequent.
• Estimate remainder / error bound — 3-5 marks.
• Second-order Taylor for maxima derivation (next topics) — 5 marks, frequent.