Taylor polynomials

Warm-up

Three quick questions from earlier lessons.

Discover

The violet curve is $y = e^x$ . The blue curve is $1 + x + \tfrac{x^2}{2} + a\,\tfrac{x^3}{6}$ . Switch $a$ from 0 to 1 to add the cubic term.

The curve y = eˣ and its Maclaurin polynomial of degree 2 (a = 0) or 3 (a = 1). The cubic term keeps the polynomial close to eˣ over a wider range.

What happens when you add the cubic term?

The polynomial hugs $e^x$ over a wider range
Nothing changes near the curve

Key idea

Match the value and derivatives at a point.

The Taylor polynomial of degree

n

of

f

centered at

a

is

P_n(x) = \displaystyle\sum_{k=0}^{n} \dfrac{f^{(k)}(a)}{k!}(x - a)^k

. Centered at 0 it is a Maclaurin polynomial.

Worked example

$P_3$ of $\sin x$ at 0.

f(0) = 0

,

f'(0) = 1

,

f''(0) = 0

,

f^{(3)}(0) = -1

Derivatives at 0: sin, cos, −sin, −cos.

P_3(x) = x - \dfrac{x^3}{6}

Divide by k!.

P_3(0.5) = 0.4792

sin(0.5) = 0.4794…

Your turn, guided

Your turn: $P_2$ of $\cos x$ at 0.

P_2(x) = 1 - \dfrac{x^2}{2}

1, 0, −1 divided by 0!, 1!, 2!.

Practice ★

$f(0) = -6$ , $f'(0) = -6$ and $f''(0) = 4$ . Write the degree-2 Maclaurin polynomial of $f$ .
$f(0) = 4$ , $f'(0) = 4$ and $f''(0) = 1$ . Write the degree-2 Maclaurin polynomial of $f$ .

Find the degree-2 Maclaurin polynomial of $f(x) = e^{ 8x}$ .
Find the degree-2 Maclaurin polynomial of $f(x) = e^{ 6x}$ .

Watch out!

Practice ★★

Find the degree-2 Taylor polynomial of $f(x) = 8\ln x$ centered at $x = 1$ .
Find the degree-2 Taylor polynomial of $f(x) = 6\ln x$ centered at $x = 1$ .

Boss problem. $f(x) = e^{ 2x}$ , approximated near 0 by its degree-3 Maclaurin polynomial $P_3$ .

Exit ticket

Three questions to check you have got it.

Find the degree-2 Maclaurin polynomial of $f(x) = e^{ 8x}$ .
Find the degree-2 Maclaurin polynomial of $f(x) = e^{ 6x}$ .
Find the degree-2 Maclaurin polynomial of $f(x) = e^{ 7x}$ .

Summary card

Taylor polynomials

$P_n(x) = \displaystyle\sum_{k=0}^{n} \dfrac{f^{(k)}(a)}{k!}(x - a)^k$ .

Watch out: divide by $k!$ ; use powers of $(x - a)$ .

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