Taylor series and limits

Taylor series and limits Quiz

Spec 2.1–2 · Edexcel A-Level Further Maths: Quiz

Studying this course? Keep your progress in one place.

Pick an answer for each. Quizzes never use energy.

5 questions, one answer each

Each answer is checked as you go, with why.

See all 5 questions

  1. 1. What is the coefficient of (x−a)3(x-a)^3 in the Taylor series of f(x)f(x)?

    • A. f′′′(a)f'''(a)
    • B. f′′′(a)3\frac{f'''(a)}{3}
    • C. f′′′(a)6\frac{f'''(a)}{6}
    • D. f′′′(x)6\frac{f'''(x)}{6}

  2. 2. The Taylor series of cos⁡x\cos x about π3\frac{\pi}{3} starts 12−32(x−π3)+…\frac12-\frac{\sqrt3}{2}\left(x-\frac{\pi}{3}\right)+\dots. What is the coefficient of (x−π3)2\left(x-\frac{\pi}{3}\right)^2?

    • A. −14-\frac14
    • B. −12-\frac12
    • C. 14\frac14
    • D. 12\frac12

  3. 3. lim⁡x→0x−arctan⁡xx3\displaystyle\lim_{x\to0}\frac{x-\arctan x}{x^3} equals

    • A. 00
    • B. 11
    • C. 13\frac13
    • D. 15\frac15

  4. 4. lim⁡x→0e2x2−1x2\displaystyle\lim_{x\to0}\frac{\mathrm{e}^{2x^2}-1}{x^2} equals

    • A. 11
    • B. 22
    • C. 44
    • D. ∞\infty

  5. 5. Why is a truncated Taylor series about x=ax=a more accurate near aa?

    • A. The first term is larger.
    • B. The series is exact at every point.
    • C. Terms in (x−a)r(x-a)^r become very small when x−ax-a is small.
    • D. The factorials are smaller.

Quiz