Maclaurin's theorem and related problems

Maclaurin's theorem and related problems

1. Maclaurin as Taylor at $(0,0)$

Set $(a,b)=(0,0)$ in Taylor:

is Maclaurin's theorem for $f(x,y)$. Remainder $R_{n+1}= 1/(n+1)! [x ∂/∂x + y ∂/∂y]^{n+1} f_{θx,θy}$ (Lagrange) with $0<θ<1$.

This is the workhorse for small $x,y$ expansions about the origin — most exam expansions are Maclaurin, not general Taylor.

2. When to Use Maclaurin Directly vs Product of Single-Variable Series

Both work; pick faster:

• If $f(x,y)= g(x)·h(y)$ or $g(x+y)$, expand $g, h$ singly and multiply — e.g., $e^x sin y$.
• If $f$ mixes $x,y$ inseparably ($e^{x y}$), use formula or substitute $t = x y$.

3. Worked Examples — Standard Maclaurin Problems

(a) f(x,y)= e^{x} sin y to third order about (0,0):
    e^x =1+x+x²/2+x³/6 ; sin y = y - y³/6 ; product keep total degree ≤3:
    f ≈ (1+x+x²/2)(y - y³/6) ≈ y + x y + x² y/2 - y³/6 - x y³/6  (last term degree 4? Actually x y³ degree 4 → drop for order 3) ⇒ y + x y + (x² y)/2 - y³/6 ✓
    Check via formula: f(0,0)=0, f_x= e^x sin y→0, f_y= e^x cos y→1, etc. matches.

(b) f(x,y)= log(1+ x + y) to second order:
    t=x+y small: log(1+t)= t - t²/2 + t³/3 ; ⇒ f≈ (x+y) - (x²+2xy+y²)/2 ✓
    Remainder O((x+y)³).

(c) f(x,y)= sin(x+y) + cos(x+y) mixture? Expand sin→(x+y)-(x+y)³/6, cos→1-(x+y)²/2 ⇒ sum ✓

(d) f(x,y)= (1+x+y)^{1/2} to second order: use binomial (1+t)^{1/2}=1+t/2 -t²/8 ; t=x+y ⇒ 1+(x+y)/2 -(x+y)²/8 ✓

(e) f(x,y)= e^{x y} to second order: t= x y ⇒ e^t=1+t +t²/2 ⇒ 1+ x y + x² y²/2 (≈1+xy for order 2) ✓

(f) Approximate value: f(0.1,0.05)= e^{0.1} cos 0.05 using Maclaurin second order from Taylor topic:
    ≈1+0.1+(0.01-0.0025)/2=1.10375 vs actual 1.10396 error 0.00021 (<R3 bound) ✓

4. Remainder Estimate for Maclaurin

Lagrange $R_3 = 1/6 [x ∂/∂x + y ∂/∂y]^3 f_{θx,θy}$. Bound by max third derivative in disc $√(x²+y²) ≤ ρ$. For $e^x cos y$, $|R_3| ≤ e^{ρ}(|x|+|y|)³/6$.

5. Convergence Domain

• $e^x, sin, cos, sinh, cosh$: all $x,y$.
• $log(1+x+y), (1+x+y)^α$: $|x+y|<1$.
• $1/(1 - x - y)$: $|x+y|<1$.
  Mention if question asks for region of validity.

6. Exam Points

• State Maclaurin's theorem for $f(x,y)$ with remainder — 3-5 marks, frequent.
• Expand $e^{x+y}, e^{x} sin y, log(1+x+y), √(1+x²+y²)$ to second/third order — 7 marks, very frequent.
• Use expansion to approximate $f(0.1,0.2)$ and bound error — 5 marks, frequent.
• Show product method matches formula — 5 marks.