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.