AP Calculus BC 6.1 Exploring Accumulations of Change - MCQs - Exam Style Questions
Question
Let \(f\) be a vector-valued function with \(f(0)=\langle -2,3\rangle\). If the instantaneous rate of change of \(f\) is given by \(\langle 3t^{2}+4t+1,\ 2t^{3}+t-2\rangle\), what is \(f(1)\)?
(A) \(\langle 2,2\rangle\)
(B) \(\langle 4,-1\rangle\)
(C) \(\langle 8,10\rangle\)
(D) \(\langle 10,7\rangle\)
(B) \(\langle 4,-1\rangle\)
(C) \(\langle 8,10\rangle\)
(D) \(\langle 10,7\rangle\)
▶️ Answer/Explanation
Correct answer: (A) \(\langle 2,2\rangle\)
Integrate each component:
\(x(1)=-2+\int_{0}^{1}(3t^{2}+4t+1)\,dt=-2+[t^{3}+2t^{2}+t]_{0}^{1}=2.\)
\(y(1)=3+\int_{0}^{1}(2t^{3}+t-2)\,dt=3+[{\tfrac{1}{2}t^{4}+\tfrac{1}{2}t^{2}-2t}]_{0}^{1}=2.\)