Question.
\(\lim_{n\rightarrow \infty }\frac{1}{n}\left [ \sqrt{\frac{1}{2}}+\frac{2}{n} +….+\frac{n}{n}\right ]\)=
(A)\(\frac{1}{2}\int_{0}^{1}\frac{1}{\sqrt{x}}dx\) (B)\(\int_{0}^{1}\sqrt{x}dx\) (C)\(\int_{0}^{1}xdx\) (D) \(\int_{1}^{2}xdx\) (E)\(2\int_{1}^{2}x\sqrt{x}dx\)
Answer/Explanation
Ans:B
Question
For \(-1<x<1\) if \(f(x)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^{2n-1}}{2n-1}\) , then \({f}'(x)\)
(A) \(\sum_{n=1}^{\infty}(-1)^{n+1}x^{2n-2}\)
(B) \(\sum_{n=1}^{\infty}(-1)^{n}x^{2n-2}\)
(C) \(\sum_{n=1}^{\infty}(-1)^{2n}x^{2n}\)
(D) \(\sum_{n=1}^{\infty}(-1)^{n}x^{2n}\)
(E) \(\sum_{n=1}^{\infty}(-1)^{n+1}x^{2n}\)
Answer/Explanation
Ans:A
Question
If n is a positive integer, then \(\left [ \left ( \frac{1}{n} \right )^{2} +\left ( \frac{2}{n} \right )^{2}+…+\left ( \frac{3n}{n} \right )^{n}\right ] \)can be expressed as
(A)\(\int_{0}^{1}\frac{1}{x^{2}}dx \) (B)\(3 \int_{0}^{1}\left ( \frac{1}{x} \right )^{2}dx\) (C)\(\int_{0}^{3}\left ( \frac{1}{x} \right )^{2}dx\) (D)\(3 \int_{0}^{3}x^{2}dx\) (E)\(3 \int_{0}^{3}x^{2}dx\)
Answer/Explanation
Ans:D
The expression is a Riemann sum with \(\Delta x=\frac{1}{n}\) and \(f(x)=x^2\)
The evaluation points are:\( \frac{1}{n},\frac{2}{n},\frac{3}{n},…,\frac{3n}{n}\)
Question
The closed interval [a b, ] is partitioned into n equal subintervals, each of width ∆x , by the numbers where \(x_{0},x_{1}…,x_{n} where a=x_{0}<x_{1}<x_{2}<…..x_{n-1}<x_{n}=b\) . What is\(\lim_{n\rightarrow \infty }\sum_{i=1}^{n}\sqrt{x_{i}}\Delta x?\)
(A)\(\frac{2}{3}\left (b^{\frac{3}{2}} -a^{\frac{3}{2}} \right )\)
(B)\(b^{\frac{3}{2}}-a^{\frac{3}{2}}\)
(C)\(\frac{3}{2}\left (b^{\frac{3}{2}} -a^{\frac{3}{2}} \right )\)
(D)\(b^{\frac{1}{2}}-a^{\frac{1}{2}}\)
(E)\(2\left ( b^{\frac{1}{2}}-a^{\frac{1}{2}} \right )\)
Answer/Explanation
Ans:A
