AP Calculus AB 5.1 Using the Mean Value Theorem - MCQs - Exam Style Questions
No-Calc Question
The function \(f\) is defined by \(f(x)=\dfrac{2}{x}\) for \(x>0\). For what value of \(x\), if any, is the average rate of change of \(f\) over the interval \(4\le x\le 16\) equal to the instantaneous rate of change of \(f\) at \(x\)?
(A) \(\dfrac{32}{3}\)
(B) \(8\)
(C) \(10\)
(D) There is no such value of \(x\).
▶️ Answer/Explanation
Average rate on \([4,16]\):
\(\displaystyle \frac{f(16)-f(4)}{16-4}=\frac{\frac{2}{16}-\frac{2}{4}}{12}=\frac{\frac{1}{8}-\frac{1}{2}}{12}=-\frac{1}{32}.\)
Instantaneous rate: \(f'(x)=-\dfrac{2}{x^{2}}\).
Solve \(-\dfrac{2}{x^{2}}=-\dfrac{1}{32}\Rightarrow x^{2}=64\Rightarrow x=8\) (since \(x>0\)).
✅ Answer: (B) \(x=8\)
Calc-Ok Question
The graph of \(f\) is shown. On which interval \((a,b)\) is there guaranteed to exist \(c\) with \(\displaystyle f'(c)=\frac{f(b)-f(a)}{b-a}\)?
(A) \((2,4)\)
(B) \((4,6)\)
(C) \((6,8)\)
(D) \((8,10)\)
(B) \((4,6)\)
(C) \((6,8)\)
(D) \((8,10)\)
▶️ Answer/Explanation
Mean Value Theorem requires continuity on \([a,b]\) and differentiability on \((a,b)\).
From the figure, \([4,6]\) is smooth and continuous; other choices include a corner/jump or endpoint discontinuity near \(x=6\) or \(x=8\).
Thus MVT applies on \((4,6)\).
✅ Answer: (B)
From the figure, \([4,6]\) is smooth and continuous; other choices include a corner/jump or endpoint discontinuity near \(x=6\) or \(x=8\).
Thus MVT applies on \((4,6)\).
✅ Answer: (B)