AP Calculus AB 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative - MCQs - Exam Style Questions
Calc-Ok Question
The graph of \(y=f(x)\) on the interval \(0<x<5\) is shown above. Which of the following could be the graph of \(y=f'(x)\)?
▶️ Answer/Explanation
Where \(f\) increases, \(f'(x)>0\); where \(f\) decreases, \(f'(x)<0\).
Where the slope of \(f\) becomes steeper upward (concave up), \(f’\) should be increasing; where concave down, \(f’\) should be decreasing.
Matching these features to the choices gives the correct derivative graph.
✅ Answer: (A)
Where the slope of \(f\) becomes steeper upward (concave up), \(f’\) should be increasing; where concave down, \(f’\) should be decreasing.
Matching these features to the choices gives the correct derivative graph.
✅ Answer: (A)
Calc-Ok Question
Let \(g\) be a twice-differentiable function with \(g'(x)>0\) and \(g”(x)>0\) for all real numbers \(x\), such that \(g(3)=12\) and \(g(5)=18\). Which of \(20,\;21,\) and \(22\) are possible values for \(g(6)\) ?
(A) 21 only
(B) 22 only
(C) 20 and 21 only
(D) 21 and 22 only
▶️ Answer/Explanation
Reasoning (Mean Value + convexity)
Average rate on \([3,5]\): \(\dfrac{g(5)-g(3)}{5-3}=\dfrac{18-12}{2}=3\).
Given \(g'(x)>0\) and \(g”(x)>0\), the derivative is positive and increasing (function is increasing and convex).
Hence for \(x>5\), the slope \(g'(x)\) is ≥ the average slope on \([3,5]\), so \(g'(x)\ge 3\) for \(x\ge5\).
Then \(g(6) \ge g(5) + 3\cdot(6-5) = 18+3 = 21\), and strict convexity gives \(g(6) > 21\).
Thus \(g(6)\) cannot be \(20\) or \(21\); among \(20,21,22\) only \(22\) is possible.
✅ Answer: (B) 22 only