AP Calculus AB 6.5 Interpreting the Behavior of Accumulation Functions Involving Area - MCQs - Exam Style Questions
No-Calc Question
The graph of the function \(f\) is shown for \(-3 \le x \le 8\). The function \(g\) is defined by \( g(x)=\int_{0}^{x} f(t)\,dt. \) Which of the following statements is true?
(A) \(g\) is decreasing for \(2<x<8\).
(B) \(g\) is increasing for \(1<x<6\).
(C) \(g\) has a relative maximum at \(x=1\).
(D) \(g\) is not differentiable at \(x=2\).
(B) \(g\) is increasing for \(1<x<6\).
(C) \(g\) has a relative maximum at \(x=1\).
(D) \(g\) is not differentiable at \(x=2\).
▶️ Answer/Explanation
By the Fundamental Theorem of Calculus, \(g'(x)=f(x)\).
\(g\) increases where \(g'(x)>0\), i.e., where \(f(x)>0\).
From the graph, \(f(x)>0\) on the interval \(1<x<6\).
Therefore \(g\) is increasing on \(1<x<6\).
✅ Answer: (B)
\(g\) increases where \(g'(x)>0\), i.e., where \(f(x)>0\).
From the graph, \(f(x)>0\) on the interval \(1<x<6\).
Therefore \(g\) is increasing on \(1<x<6\).
✅ Answer: (B)
Calc-Ok Question
The graph of an increasing function \(f\) is shown. Let \(g(x)=\displaystyle\int_{1}^{x} f(t)\,dt\). Which table could represent values of \(g\)?
(A) shows \(g(0)=-16,\ g(0.5)=-8,\ g(1)=0,\ g(1.5)=8,\ g(2)=16\)
(B) shows \(g(0)=16,\ g(0.5)=8,\ g(1)=0,\ g(1.5)=8,\ g(2)=16\)
(C) shows \(g(0)=-16,\ g(0.5)=-1,\ g(1)=0,\ g(1.5)=1,\ g(2)=16\)
(D) shows \(g(0)=16,\ g(0.5)=1,\ g(1)=0,\ g(1.5)=1,\ g(2)=16\)
(B) shows \(g(0)=16,\ g(0.5)=8,\ g(1)=0,\ g(1.5)=8,\ g(2)=16\)
(C) shows \(g(0)=-16,\ g(0.5)=-1,\ g(1)=0,\ g(1.5)=1,\ g(2)=16\)
(D) shows \(g(0)=16,\ g(0.5)=1,\ g(1)=0,\ g(1.5)=1,\ g(2)=16\)
▶️ Answer/Explanation
On \((0,1)\), \(f(t)<0\) ⇒ \(\int_{0}^{1} f(t)\,dt<0\).
Thus \(g(0)=\int_{1}^{0}f(t)\,dt= -\!\int_{0}^{1} f(t)\,dt>0\).
Also \(g(1)=0\), and on \((1,2)\) the area is positive ⇒ \(g(2)>0\).
Only table **(D)** shows \(g(0)>0\), \(g(1)=0\), and small positive values near \(x=1\) on both sides with symmetry of magnitudes shown in the picture.
✅ Answer: (D)
Thus \(g(0)=\int_{1}^{0}f(t)\,dt= -\!\int_{0}^{1} f(t)\,dt>0\).
Also \(g(1)=0\), and on \((1,2)\) the area is positive ⇒ \(g(2)>0\).
Only table **(D)** shows \(g(0)>0\), \(g(1)=0\), and small positive values near \(x=1\) on both sides with symmetry of magnitudes shown in the picture.
✅ Answer: (D)