Home / AP Calculus BC 10.14 Finding Taylor or Maclaurin Series for a Function – Exam Style Questions – MCQs

AP Calculus BC 10.14 Finding Taylor or Maclaurin Series for a Function - Exam Style Questions - MCQs - New Syllabus

AP Calculus BC-MCQs and Answers - All Topics

No-CalcQuestion

What is the sum of the series \[ 1-\frac{(\pi/3)^{2}}{2!}+\frac{(\pi/3)^{4}}{4!}-\frac{(\pi/3)^{6}}{6!}+\cdots+(-1)^n\frac{(\pi/3)^{2n}}{(2n)!}+\cdots \, ? \]

(A) \( \dfrac{1}{2} \)
(B) \( \dfrac{\sqrt{3}}{2} \)
(C) \( e^{-(\pi/3)^{2}} \)
(D) \( \dfrac{1}{\,1+(\pi/3)^{2}\,} \)

▶️ Answer/Explanation

Detailed solution

Recognize the Maclaurin series for \( \cos x \): \( \cos x=1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\cdots \).
Substitute \( x=\dfrac{\pi}{3} \): \( \cos\!\left(\dfrac{\pi}{3}\right)=\dfrac{1}{2} \).

✅ Correct Answer: (A) \( \dfrac{1}{2} \)

No-CalcQuestion

Let \( f(x)=\sin(x^{2}) \). What are the first three nonzero terms of the Maclaurin series for \( f'(x) \)?
(A) \( -2x^{3}+\dfrac{7}{3}x^{7}-\dfrac{x^{11}}{60} \)
(B) \( 1-\dfrac{x^{4}}{2}+\dfrac{x^{8}}{24} \)
(C) \( 2x-x^{5}+\dfrac{x^{9}}{12} \)
(D) \( 2x+2x^{3}+x^{5} \)
▶️ Answer/Explanation
Detailed solution

\(\sin u = u-\dfrac{u^{3}}{3!}+\dfrac{u^{5}}{5!}-\cdots\). Substitute \(u=x^{2}\): \[ f(x)=x^{2}-\frac{x^{6}}{6}+\frac{x^{10}}{120}-\cdots. \]
Differentiate term-by-term: \[ f'(x)=2x-x^{5}+\frac{x^{9}}{12}-\cdots. \]

✅ Answer: (C)

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