Home / AP Calculus AB : 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation- Exam Style questions with Answer- MCQ

AP Calculus AB : 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation- Exam Style questions with Answer- MCQ

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

Take the interval [0,1] and divide it into n pieces of equal length and form the right Riemann Sum for the function f(x)=√x. The limit of this sum is what is given and its value is given by \(\int_{0}^{1}\sqrt{x}dx\)

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

Take the derivative of the general term with respect to x:\( \sum_{n=1}^{\infty }(-1)^{n+1}x^{2n-2}\)

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:\( \frac{1}{n},\frac{2}{n},\frac{3}{n},…,\frac{3n}{n}\)

Thus the right Riemann sum is for x = 0 to x= 3  . The limit is equal to \(\int_{0}^{3}x^2dx\).

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

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