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

Question

The graph of a piecewise-linear function f , for \(−1 ≤ x  ≤4 \), is shown above. What is the value of \(\int_{-1}^{4}f(x)dx\)?

(A) 1                                         (B) 2.5                              (C) 4                               (D) 5.5                                        (E) 8

▶️Answer/Explanation

Ans:B

\(\int_{-1}^{4}f(x)dx=\int_{-1}^{2}f(x)dx+\int_{2}^{4}f(x)dx\)
= Area of trapezoid(1) – Area of trapezoid(2) = 4 -1.5= 2.5 

Question

\(\int \frac{e^{\sqrt{x}}}{\sqrt{x}}dx\)=

(A) \(2e^{\sqrt{x}}+C\)

(B)\(\frac{1}{2}e^{\sqrt{x}}+C\)

(C) \(e^{\sqrt{x}}+C\)

(D) \(2\sqrt{x}e^{\sqrt{x}}+C\)

(E) \(\frac{1}{2}\frac{e^{\sqrt{x}}}{\sqrt{x}}+C\)

▶️Answer/Explanation

Ans:A

Question

 For all  x >1, if \( f(x)=\int_{1}^{x}\frac{1}{t}dt\),then f'(x)=

(A) 1                                             (B) \(\frac{1}{x}\)                                                         (C) lnx −1                                            (D) ln x                                                  (E) \(e^{x}\)

▶️Answer/Explanation

Ans:B

Use the Fundamental Theorem of Calculus.\( f'(x) =\frac{1}{x}\)

Question

\(\frac{d}{dx}\int_{0}^{x}cos(2\pi u)du \) is
(A)0                         (B)\(\frac{1}{2\pi}sinx\)                                 (C)\(\frac{1}{2\pi}cos(2\pi x)\)                          (D)\(cos(2\pi x)\)                   (E)\(2\pi cos(2\pi x)\)

▶️Answer/Explanation

Ans:D

Answer follows from the Fundamental Theorem of Calculus.

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