AP Calculus AB 2.9 The Quotient Rule - MCQs - Exam Style Questions
Calc-Ok Question
The functions \(f\) and \(g\) are defined for \(-4\le x\le 4\). The graph of \(y=g(x)\) is a line segment and the graph of \(y=f(x)\) is a semicircle, as shown in the figure. The function \(h\) is defined by \(h(x)=\dfrac{f(x)}{g(x)}\). What is the instantaneous rate of change of \(h\) with respect to \(x\) at \(x=0\)?
(A) \(-\tfrac{1}{2}\)
(B) \(0\)
(C) \(\tfrac{1}{2}\)
(D) \(2\)
▶️ Answer/Explanation
From the figure:
\(f(0)=4\) (top of the semicircle).
\(f'(0)=0\).
\(g(0)=2\) (y-intercept of the line segment).
\(g'(0)=\dfrac{0-2}{4-0}=-\tfrac{1}{2}\).
Quotient rule:
\(h'(x)=\dfrac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}\).
\(h'(0)=\dfrac{0\cdot 2-4(-\tfrac12)}{2^2}=\dfrac{2}{4}=\tfrac12.\)
✅ Answer: (C)
No-Calc Question
The graph of a function \(f\) is shown above. If \(g\) is the function defined by \(g(x)=\dfrac{x^{2}+1}{f(x)}\), what is the value of \(g'(2)\) ?
(A) \(-\tfrac{8}{9}\)
(B) \(\tfrac{1}{9}\)
(C) \(1\)
(D) \(\tfrac{32}{9}\)
(B) \(\tfrac{1}{9}\)
(C) \(1\)
(D) \(\tfrac{32}{9}\)
▶️ Answer/Explanation
From the line segment \(x=0\to3\): slope \(=\dfrac{7-(-5)}{3-0}=4\Rightarrow f'(2)=4\).
Value at \(x=2\): \(f(2)=-5+4(2)=3\).
Quotient rule: \(g'(x)=\dfrac{2x\,f(x)-(x^{2}+1)f'(x)}{[f(x)]^{2}}\).
Plug \(x=2\): \(g'(2)=\dfrac{4\cdot3-5\cdot4}{3^{2}}=\dfrac{12-20}{9}=-\tfrac{8}{9}\).
✅ Answer: (A)