Course Code: BMTC-133

Assignment Code: BMTC-133/TMA/2023

Maximum Marks: 100

1. Which of the following statements are true or false? Give reasons for your answers.

a) The singleton set \(\{x\}\) for any \(x \in \boldsymbol{R}\) is an open set.

(b) The series is \(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots\) is a convergent series.

c) The function \(f(x)=\left\{\begin{array}{cc}e^{-x}+e^{x}, & \text { when } x \neq 0 \\ 1, & \text { when } x=0\end{array}\right.\) is continuous on \([0,1]\).

d) The function \(f\) defined by \(f(x)=|x-\sqrt{2}| \forall x \in \boldsymbol{R}\) has a critical point at \(x=\sqrt{2}\).

e) If a function has finitely many points of discontinuities, then the function is not integrable.

2. a) Prove that the sequence \(\left\{a_{n}\right\}\) where \(a_{n}=\frac{2^{2}}{n^{2}+3^{2}}\), converges to 0 .

b) Find the following limit, if it exists:

\[
\lim _{x \rightarrow 0} \frac{x^{3} \sin x^{3}}{1-\cos x^{3}}
\]

c) Test the convergence of the following series.

i) \(\frac{1.2}{3^{2} \cdot 4^{2}}+\frac{3.4}{5^{2} \cdot 6^{2}}+\frac{5 \cdot 6}{7^{2} \cdot 8^{2}}+\cdots\)

ii) \(\sum \frac{\sqrt{n^{4+1}}-\sqrt{n^{4-1}}}{n}\)

3. a) Explain the order completeness property of \(\boldsymbol{R}\), and use it to show that the set \(S=\left\{\frac{n}{n+1} \mid n \in \boldsymbol{N}\right\}\) has a supremum as well as infimum in \(\boldsymbol{R}\).

b) Let \(f\) be the function defined by

\[
f(x)= \begin{cases}2 x-1, & \text { if } x \in] \infty, 1[ \\ \frac{3 x^{2}-2}{x}, & \text { if } x \in[1,2[ \\ (1+2 x)^{2}, & \text { if } x \in[2, \infty[\end{cases}
\]

Discuss the continuity of \(f\) on \(] \infty, \infty[\).

c) Check whether the following sets are open, closed or neither:

i) \(] 1,5[\cup[3,6]\)

ii) \([0,1] \cup\left\{\frac{5}{9}, \frac{3}{4}, \frac{10}{7}\right\}\)

iii) \(\{5 n: n \in N\}\) 4. a) Using the principle of mathematical induction, prove that 7 is a factor of \(3^{2 n-1}+2^{n+1}, \forall n \in \boldsymbol{N}\)

b) Show that the equation \(x^{3}-2 x^{2}+5 x-12=0\) has a root which is a positive real number.

c) Prove that the set \(\left\{\frac{3}{6}, \frac{3}{7}, \frac{3}{8}, \ldots\right\}\) is a countable set.

5. a) Show that the local maximum value of \(\left(\frac{1}{x}\right)^{x}\) is \(e^{1 / e}\).

b) Verify Cauchy Mean Value Theorem for the functions

\[
f(x)=x, g(x)=\frac{1}{x}, x \in[1,4]
\]

c) Show that \(1+x \leq e^{x}, \forall x \in[0, \infty[\). Does the inequality hold for \(x<0\) ? Justify your answer.

6. a) By showing that the remainder after \(n\)-terms tends to zero, find Maclaurin’s series expansion of \(\sin 2 x\).

b) Find the greatest value of the function \(f(x)=x^{4}-2 x^{3}-3 x^{2}+4 x+7\) over the interval \([0,1]\).

7. a) Consider the function \(f(x)=2 \cos x\) in the interval \(\left[0, \frac{\pi}{2}\right]\). Show that \(L\left(P_{1}, f\right) \leq L\left(P_{2}, f\right)\) and \(U\left(P_{2}, f\right) \leq U\left(P_{1}, f\right)\) where \(P_{1}=\left\{0, \frac{\pi}{3}, \frac{\pi}{2}\right\}\) and \(P_{2}=\left\{0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}\right\}\)

b) Show that the derivative \(f^{\prime}\) of the following function \(f\) given by

\[
f(x)=\left\{\begin{array}{cl}
x^{2} \sin \frac{1}{x} & \text { if } x \neq 0 \\
0 & \text { if } x=0
\end{array}\right.
\]

exists at \(x=0\) but \(f^{\prime}\) is not continuous at 0 .

8. a) Check whether the following function has a mean value in the interval \([2,5]\)

\[
f(x)=\left\{\begin{array}{lll}
1 & \text { if } & 2 \leq x<3 \\
3 & \text { if } & 3 \leq x \leq 5
\end{array}\right.
\]

Does this contradict the mean value theorem? Justify.

b) Find the limit as \(n \rightarrow \infty\), of the sum

\[
\frac{n}{3 n^{2}+1^{2}}+\frac{n}{3 n^{2}+2^{2}}+\frac{n}{3 n^{2}+3^{2}}+\cdots+\frac{1}{4 n} \text {. }
\]

c) Apply Weierstrass \(M\)-test to show that the series \(\sum \frac{10}{n^{4}+x^{4}}\) converges uniformly for all \(x \in \boldsymbol{R}\).

9. a) Using Riemann integration show that \(\int_{1}^{2}(3 x+1) d x=\frac{11}{2}\).

b) Show that the function \(f(x)=\frac{1}{x}\) is continuous on \(\left.] 0,1\right]\) but not uniformly continuous.

10. a) Give one example for the following. Justify your choice of examples.

i) A bounded set having no limit point.

ii) A bounded set having infinite number of limit points.

iii) A infinite compact set which is not an interval.

b) Prove that the function \(f\) defined by

\[
f(x)=\left\{\begin{array}{cc}
4, & \text { if } x \text { is rational } \\
-4, & \text { if } x \text { is irrational }
\end{array}\right.
\]

is discontinuous at each real number, using the sequential definition of continuity. (4)