MTH621 Quiz 4

Q1. Investigate\,\,the\,\,\lim it\,\,of\,\,\,\mathop {\lim }\limits_{x \to 0} \,\,x\sin \frac{{1\,\,}}{x} = - - - - .

1. none\,\,of\,\,these
2. 1
3. 0
4. x

Correct answer: 0

Q2. Investigate\,\,the\,\,\lim it\,\,of\,\,\,\mathop {\lim }\limits_{x \to 0^ + } \,\,\left( {\frac{{\left| x \right|}}{x} + x} \right) = - - - - .

1. 0
2. none\,\,of\,\,these
3. 1\,
4. - 1

Correct answer: 1\,

Q3. Rolle's Theorem says, suppose that f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) and f(a) = f(b).

1. Then f'(c) = 0 for some c in the open interval (a,).
2. Then f'(c) exists for some c in the open interval (a,b).
3. Then f'(c) 0 for some c in the open interval (a,b).
4. Then f'(c) = 0 for some c in the open interval (a,b).

Correct answer: Then f'(c) = 0 for some c in the open interval (a,b).

Q4. \begin{array}{l} The\,series\,\sum\limits_{}^{} {a_n b_n } \,\,\,converges\,\,if\,\,a_{n + 1} \le \,a_{n\,} \,for\,\,n\, - - - k,\,\,\mathop {\lim }\limits_{x \to \infty } \,\,a_n = 0,\,\,and\, \\\ |b_k + \,b_{k + 1} + ... + b_n | \le M,\,\,for\,\,some\,\,constant\,M. \\\ \end{array}

1. <
2. >
3. \,
4. \,

Correct answer: \,

Q5. \begin{array}{l} The\,series\,\sum\limits_{}^{} {a_n b_n } \,\,\,converges\,\,if\,\,a_{n + 1} \le \,a_{n\,} \,for\,\,n\, \ge \,k,\,\,\mathop {\lim }\limits_{x \to \infty } \,\,a_n = - - - - - ,\,\,and\, \\\ |b_k + \,b_{k + 1} + ... + b_n |\,\, \le \,\,M,\,\,for\,\,some\,\,constant\,M. \\\ \end{array}

1. 1
2. 0\,
3. None\,of\,these
4. - 1

Correct answer: 0\,

Q6. Which function has 0/0 form

1. f(x) = 2 x at x=2
2. f(x) = x + x at x =0
3. f(x) = { x}{x-1} at x=1
4. f(x) = {e^{-x}}{x} at x=2

Correct answer: f(x) = { x}{x-1} at x=1

Q7. If f is differentiable at a local extreme point x_0\in D_{f}^{0}, then

1. |f'(x_0)|<~0.
2. f'(x_0) ~0.
3. f'(x_0)=~0.
4. x_0=~0.

Correct answer: f'(x_0)=~0.

Q8. If f(x) = |x| and g(x) = |5x – 2|. Then, fog =

1. f(x)
2. |g(x)|
3. |f(x)|
4. f^2(x)

Correct answer: |g(x)|

Q9. The\,series\,\sum\limits_{}^{} {( - 1)^n a_n } \,\,\, - - - - - - \,\,\,if\,\,0 \le a_{n + 1} \le \,a_{n\,} \,and\,\mathop {\lim }\limits_{x \to \infty } \,\,a_n = 0

1. diverges
2. converges

Correct answer: converges

Q10. For two functions f(x) and g(x), the composit function fog(x) is defined as -------

1. None of these
2. (f-g)(x)
3. g(f(x))
4. f(g(x))

Correct answer: f(g(x))