Home / AP Calculus BC 6.5 Interpreting the Behavior of Accumulation Functions Involving Area – MCQs

AP Calculus BC 6.5 Interpreting the Behavior of Accumulation Functions Involving Area - MCQs - Exam Style Questions

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Question


For \( -4 \le x \le 6 \), the graph of the function \(f\) is piecewise linear (as shown in the figure on the prompt). If \( g(x)=\displaystyle\int_{0}^{x} f(t)\,dt \), what values of \(x\) in the closed interval \( [-4,6] \) satisfy \( g(x)=0 \)?
(A) \(0\) only
(B) \(0,\,2,\text{ and }6\) only
(C) \(-2,\,1,\text{ and }4\)
(D) \(-4,\,-2,\,0,\,2,\text{ and }6\)

▶️ Answer/Explanation
Detailed solution

By the definition of an accumulation function, \(g(x)\) is the signed area from \(0\) to \(x\) under \(f\). Zeros of \(g\) occur when the net signed area from \(0\) to \(x\) is \(0\).

From the piecewise-linear graph:
• At \(x=0\): \(\displaystyle g(0)=\int_{0}^{0}f(t)\,dt=0\).
• From \(0\) to \(2\): the positive and negative triangular regions cancel, so \( \displaystyle \int_{0}^{2} f(t)\,dt = 0 \Rightarrow g(2)=0\).
• From \(0\) to \(6\): the additional area from \(2\) to \(6\) returns the running total to \(0\), so \(g(6)=0\).
• Other listed points do not yield zero net area.

✅ Answer: (B) \(0,\,2,\text{ and }6\) only

No-Calc Question

Let \(f\) be defined by \(f(x)=-2x+\displaystyle\int_{2}^{x}\sqrt{1+t^{2}}\,dt\). Which statement is true on the interval \(0<x<1\)?
(A) \(f\) is decreasing, and the graph of \(f\) is concave up.
(B) \(f\) is decreasing, and the graph of \(f\) is concave down.
(C) \(f\) is increasing, and the graph of \(f\) is concave up.
(D) \(f\) is increasing, and the graph of \(f\) is concave down.
▶️ Answer/Explanation
Detailed solution

By FTC: \(f'(x)=-2+\sqrt{1+x^{2}}\).

For \(0<x<1\): \(\sqrt{1+x^{2}}<\sqrt{2}<2\) ⇒ \(f'(x)<0\) ⇒ decreasing.

Also \(f”(x)=\dfrac{x}{\sqrt{1+x^{2}}}>0\) on \(0<x<1\) ⇒ concave up.

✅ Answer: (A)

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