Home / AP Calculus BC : 6.4 The Fundamental Theorem of Calculus and Accumulation Functions- Exam Style questions with Answer- MCQ

AP Calculus BC : 6.4 The Fundamental Theorem of Calculus and Accumulation Functions- Exam Style questions with Answer- MCQ

Question

If \(f(x)=\int _{0}^{x^{3}}\cos (t^{2})dt\) , then \(f'(\sqrt{\pi })=\)
A \(3π sin(\pi ^{3})\)
B \(cos(\pi ^{3})\)
C 3π cosπ
D 3π \(cos(\pi ^{3})\)

Answer/Explanation

 

Question

\(\int_{-1}^{2}\frac{|x|}{x}dx\) is

(A) –3                 (B) 1                            (C) 2                                      (D) 3                                                 (E) nonexistent

Answer/Explanation

Ans:C

 

Question

 Let \(f(t)=t^{4}+2t^{2}-3\) and consider the function \(F(x)=\int_{-3}^{x}f(t)dt\) What are the x-value(s) for which F(x) has a local minimum?
(A) 1
(B) -1
(C) \(\pm \sqrt{3}\)
(D) ±1

Answer/Explanation

Ans:(A)

Question

 If \(g(x)=\int_{0}^{x}\left ( \log _{2} (t-1)+\log _{2}(t+1)\right )dt\)  find all x-values at which the line tangent to the graph of y = g(x) is horizontal.
(A) \(\sqrt{2}\)
(B) \(\sqrt{3}\)
(C) 2
(D) 3

Answer/Explanation

Ans:(A)

 You are looking for solutions to g′(x) = 0, which by the Fundamental Theorem of Calculus is equivalent to solving f(x)=0. To begin, use properties of logarithms to simplify the equation:
\(\log _{2}(x-1)+\log _{2}(x+1)=0\Leftrightarrow \log _{2}\left [ (x-1)(x+1) \right ]=0\)
Exponentiating both sides of the last equation base 2, this becomes
\((x-1)(x+1)=1\Leftrightarrow x^{2}-2=0\)

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