Taylor's Theorem
Approximating a function by a polynomial, with a precise formula for the error. This is the rigorous heart of "good enough" approximation, and the bridge from the Mean Value Theorem to the infinite series of the next two units.
Polynomials are easy to compute, differentiate, and integrate; most functions are not. Taylor's theorem says how well a polynomial approximates a function near a point, and — crucially — puts an exact, provable bound on how big the error can possibly be.
It is how calculators and computers evaluate $\sin$, $\cos$, $e^x$ to many digits; how physics linearizes hard problems (small-angle $\sin\theta\approx\theta$ in a pendulum); and how Newton's method for root-finding is derived from the linear (degree-1) Taylor approximation.
Brook Taylor published the series in 1715; Madhava in 14th-century India had found the key cases (sine, cosine, arctangent) three centuries earlier. A function can fail to equal its own Taylor series everywhere it converges — Cauchy's 1823 example $e^{-1/x^2}$ (at $x=0$) has every derivative zero at the origin, so its Taylor series is identically $0$, yet the function itself is not.
- Write the Taylor polynomial of a function at a point.
- State Taylor's theorem with the Lagrange remainder, and sketch why it follows from the Mean Value Theorem.
- Bound approximation errors using the remainder formula.
- Explain why a convergent Taylor series need not equal the function it came from.
Taylor polynomials and the remainder theorem
$\displaystyle T_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k$ — the unique degree-$n$ polynomial matching $f$ and its first $n$ derivatives at $a$.
If $f$ is $n+1$ times differentiable on an interval containing $a$ and $x$, then $f(x)=T_n(x)+R_n(x)$ where $R_n(x)=\dfrac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$ for some $c$ strictly between $a$ and $x$.
For $e^x$ about $a=0$: $e^x=1+x+\tfrac{x^2}{2}+\cdots+\tfrac{x^n}{n!}+R_n$, with $R_n(x)=\tfrac{e^c}{(n+1)!}x^{n+1}$ for some $c$ between $0$ and $x$.
Error bounds in practice
View $1-\tfrac{x^2}{2}$ as the degree-3 Taylor polynomial of cosine (the cubic coefficient is zero, so $T_3=T_2$). Then $|f^{(4)}|=|\cos|\le1$ everywhere, so the Lagrange error at $x=0.1$ is at most $\dfrac{(0.1)^4}{4!}\approx4.2\times10^{-6}$.
Estimate $e$ using $T_n(1)$ for $e^x$ at $a=0$, accurate to $10^{-4}$.
• The remainder point $c$ is unknown and generally different for each
$x$ — bound $|f^{(n+1)}|$ over the whole interval, don't try to pin down
$c$.
• A Taylor series can converge yet not equal $f$ — only $R_n\to0$ as
$n\to\infty$ guarantees equality (Cauchy's $e^{-1/x^2}$ example above is the
standard warning).
• Centre matters: expanding about a nearby $a$ shrinks $(x-a)^{n+1}$ and usually
improves accuracy dramatically.
• $T_n$ has degree $n$ but may have fewer than $n$ nonzero terms (odd/even
symmetry can make coefficients vanish, as with $\cos x$ above).
Practice
Write the degree-3 Taylor polynomial of $\sin x$ about $0$.
Answer
Bound the error of approximating $e^{0.5}$ by $1+0.5+\tfrac{0.25}{2}$.
Solution
Use a Taylor polynomial to estimate $\sqrt{1.1}$.
Answer
How large must $n$ be so that $T_n(1)$ for $\sin x$ about $0$ is accurate to within $10^{-6}$?
Solution
Bound the error in $\sin\theta\approx\theta$ for $|\theta|\le0.1$ radians.
Answer
Why does expanding $\ln x$ about $a=1$ (rather than $a=0$, where it's undefined) make sense for estimating $\ln(1.05)$?
Answer
Explain why "the Taylor series of $f$ at $a$ converges for all $x$" does not by itself prove $f(x)=T_\infty(x)$.
Answer
Show that Newton's method update $x_{n+1}=x_n-f(x_n)/f'(x_n)$ is exactly solving $T_1(x)=0$ for the degree-1 Taylor polynomial of $f$ centred at $x_n$.
Answer
Expanded topic checklist
Course coverage. This unit is part of the complete MAT159H1 topic map. It should cover the following items before you move on:
- Taylor polynomials. Compute Taylor polynomials around a center with derivative data.
- Remainders. Use Lagrange, integral, or alternating remainders to justify approximation.
- Local approximation. Know what order of contact means and how error scales.
- Applications. Use Taylor theorem for limits, inequalities, and numerical estimates.
Use the current UofT MAT159H1 calendar topics as the coverage spine, and pair lecture notes with a Spivak-style proof-calculus text for extra reading. The exact assigned textbook should still be checked against the instructor syllabus.
The Taylor spine is: polynomial matching derivative data, theorem with remainder, and explicit error control on a stated interval.
Deep practice extensions
Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.
Find the degree-4 Taylor polynomial for $\cos x$ at $0$.
Guide
Use Taylor expansion to compute $\lim_{x\to0}(\sin x-x)/x^3$.
Guide
Bound the error of $\sin xpprox x$ for $|x|\le0.1$.
Guide
Why is a Taylor polynomial about $a=1$ different from one about $a=0$?