Before the final

Getting the factorial in a Taylor term wrong

The habit: The coefficient of (x − a)^k is f^(k)(a)/k!, and the remainder after degree n divides by (n + 1)!.

15 practice questions are built to catch it, in 1 topic.

See it fail in a figure: Leave the factorials out, in the Taylor polynomials lesson

Practise against it: 6 questions

What it looks like

A question from Taylor polynomials, with the answer this slip gives beside the right one.

Introductory2 marks · 3 min

The third-order Maclaurin polynomial of a function $f$ is given below. Find $f'''(0)$ .

T_{3}(x) = 2 - x + 5x^{2} - \tfrac{4}{3}x^{3}

With the slip: $\displaystyle \approx -1.333$
Worked through: $\displaystyle f'''(0) = -8$

−4/3 is the coefficient of x³, which is f‴(0) divided by 3!. The question asks for f‴(0) itself.

See the worked solution

Where it comes up

Taylor polynomials15 questions

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

01
Find the coefficient of $x^{3}$ in the third-order Maclaurin polynomial of $f(x) = \log(1 + 2x)$ .
Taylor polynomialsIntroductory
02
The third-order Taylor polynomial of a function $f$ about $x = -1$ is given below. Find $f''(-1)$ . $\displaystyle T_{3}(x) = 6 + 3(x + 1) - \tfrac{9}{2}(x + 1)^{2} + \tfrac{1}{3}(x + 1)^{3}$
Taylor polynomialsIntroductory
03
Find the fourth-order Maclaurin polynomial $T_{4}(x)$ of $f(x) = \cos(3x)$ .
Taylor polynomialsIntroductory
04
Use the Maclaurin polynomial of $\dfrac{1}{1 - u}$ to find the fourth-order Maclaurin polynomial $T_{4}(x)$ of $f(x) = \dfrac{3}{1 + 2x}$ .
Taylor polynomialsIntroductory
05
A function $f$ has $f(1) = 3$ , $f'(1) = -2$ and $f''(1) = 6$ . Use the quadratic approximation of $f$ centred at $x = 1$ to estimate $f(1.1)$ .
Taylor polynomialsStandard
06
Find the coefficient of $x^{3}$ in the third-order Maclaurin polynomial of $f(x) = e^{2x}\sin x$ .
Taylor polynomialsStandard