AP Calculus AB 6.4 The Fundamental Theorem of Calculus and Accumulation Functions - MCQs - Exam Style Questions
Calc-Ok Question
Let \(g\) be the function with first derivative \(g'(x)=\sqrt{x^{3}+x}\) for \(x>0\). If \(g(2)=-7\), what is the value of \(g(5)\) ?
(A) \(4.402\)
(B) \(11.402\)
(C) \(13.899\)
(D) \(20.899\)
▶️ Answer/Explanation
Use the FTC: \(g(5)=g(2)+\displaystyle\int_{2}^{5} g'(x)\,dx\).
So \(g(5)=-7+\displaystyle\int_{2}^{5}\sqrt{x^{3}+x}\,dx\).
Numerically, \(\displaystyle \int_{2}^{5}\sqrt{x^{3}+x}\,dx\approx 20.899\).
Therefore \(g(5)\approx -7+20.899=13.899\).
✅ Answer: (C)
So \(g(5)=-7+\displaystyle\int_{2}^{5}\sqrt{x^{3}+x}\,dx\).
Numerically, \(\displaystyle \int_{2}^{5}\sqrt{x^{3}+x}\,dx\approx 20.899\).
Therefore \(g(5)\approx -7+20.899=13.899\).
✅ Answer: (C)