AP Calculus BC 2.9 The Quotient Rule - MCQs - Exam Style Questions
No-CalcQuestion
\(f(x)=\dfrac{\sin x}{x}\). What is \(f'(x)\) at \(x=\dfrac{\pi}{2}\)?
A. \(-\dfrac{4}{\pi^2}\)
B. \(0\)
C. \(\dfrac{2}{\pi}\)
D. \(\dfrac{4}{\pi^2}\)
B. \(0\)
C. \(\dfrac{2}{\pi}\)
D. \(\dfrac{4}{\pi^2}\)
▶️ Answer/Explanation
Detailed solution
Quotient rule: \(f'(x)=\dfrac{x\cos x-\sin x}{x^2}\).
At \(x=\pi/2\): \(\cos(\pi/2)=0,\ \sin(\pi/2)=1\).
\(f'(\pi/2)=\dfrac{-1}{(\pi/2)^2}=-\dfrac{4}{\pi^2}\).
✅ Correct: A
No-Calc Question
If \(y=\dfrac{x^{3}}{x+2}\), then \(\dfrac{dy}{dx}=\)
(A) \(3x^{2}\)
(B) \(\dfrac{2x^{3}-6x^{2}}{(x+2)^{2}}\)
(C) \(\dfrac{2x^{3}+6x^{2}}{(x+2)^{2}}\)
(D) \(\dfrac{4x^{3}-6x^{2}}{(x+2)^{2}}\)
(B) \(\dfrac{2x^{3}-6x^{2}}{(x+2)^{2}}\)
(C) \(\dfrac{2x^{3}+6x^{2}}{(x+2)^{2}}\)
(D) \(\dfrac{4x^{3}-6x^{2}}{(x+2)^{2}}\)
▶️ Answer/Explanation
Detailed solution
Quotient rule: \(\displaystyle y’=\frac{(3x^{2})(x+2)-x^{3}(1)}{(x+2)^{2}}=\frac{3x^{3}+6x^{2}-x^{3}}{(x+2)^{2}}=\frac{2x^{3}+6x^{2}}{(x+2)^{2}}.\)
✅ Answer: (C)