Continuity in an Interval

Q.

Find the derivative of ${\sin }^{2}x$ using first principles.

Q. If $\left[x\right]$ represents the greatest integer function and $f\left(x\right)=x–\left[x\right]–\cos x$ then $f^{\prime }\left(\frac{\pi }{2}\right)=$

1. $2$
2. $0$
3. $\text{does not exist}$
4. $1$

Q.

The derivative of $a^{secx}$ with respect to $a^{\tan x},\left(a>0\right)$ is

1. $secx{a}^{secx-\tan x}$

2. $\sin x{a}^{\tan x-secx}$

3. $\sin x{a}^{secx-\tan x}$

4. $a^{secx-\tan x}$

Q.

The domain of the function $f\left(x\right)={\log }_{\left(2x-1\right)}\left(x-1\right)$ is

1. $\left(1,\infty \right)$

2. $\left(\frac{1}{2},\infty \right)$

3. $\left(0,\infty \right)$

4. None of these

Q. Let $f\left(x\right)=\sqrt{x+4}$. Then the point $c$ that satisfies the mean value theorem for the function on the interval $[0,5]$, is

1. $\frac{1}{4}$
2. $\frac{25}{4}$
3. $\frac{9}{16}$
4. $\frac{9}{4}$

Q. The function f(x) defined by $f\left(x\right)=(\begin{array}{c}\frac{{\sin }^2\mathrm{ax}}{x^2},x\ne 0\\ 1,x=0\end{array}$ is continuous at x = 0. The value(s) of a is/are

1. \N
2. 1
3. -1
4. 2

Q. Draw the graph of the function $f:R\to R$ defined by $f\left(x\right)=x^3,xϵR$.

Q.

The least value of $k$ for which the function $x^2+kx+1$ is an increasing function in the interval $1<x<2$ is

1. $-4$

2. $-3$

3. $-1$

4. $-2$

Q. If the function f()=3ax+b if x is greater than 1. F(x)=11 if x=1. F()=5ax-2b if x is less than 1 continuous at x=1 find the value of a and b

Q.

The function $\mathrm{f}\left(\mathrm{x}\right)=2\log (\mathrm{x}-2)-{\mathrm{x}}^2+4\mathrm{x}+1$ increases on the interval

1. $(1,2)$

2. $\left(2,3\right)$

3. $\left(\frac{5}{2},3\right)$

4. $\left(2,4\right)$

Q. If the function $f$ defined on $(-\frac{1}{3},\frac{1}{3})$ by $f\left(x\right)=\{\begin{array}{c}\frac{1}{x}{\log }_e\left(\frac{1+3x}{1-2x}\right),\text{when}x\ne 0\\ k,\text{when}x=0\end{array}$
is continuous, then $k$ is equal to

Q. Let $a,b,c\in R$ such that $a+b+c=\pi .$
If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin (a{x}^2+bx+c)}{x^2-1},& \text{if}x<1\\ -1,& \text{if}x=1\\ a\text{sgn}(x+1)\cos (2x-2)+b{x}^2,& \text{if}1<x\le 2\end{array}$
is continuous at $x=1,$ then the value of $\frac{a^2+b^2}{5}$ is
( Here, $\text{sgn}\left(k\right)$ denotes signum function of $k$ )

Q. Let f and g be increasing and decreasing functions respectively from [0, inf) to [0, inf). Let h(x) = f(g(x)) if h(0) = 0, h (x) - h(1) is 1 always zero 2. Always negative 3. Always positive 4 strictly increasing

Q.

What is the midpoint rule formula?

Q.

If $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{(\mathrm{x}+1)}-\log (1+\mathrm{x}),\mathrm{x}>0$then $\mathrm{f}$ is

1. An increasing function

2. A decreasing function

3. Both increasing and decreasing function

4. None of the above

Q. Which of the following function is continuous (where [.] denotes the greatest integer function)

1. $\sin \pi \left[x\right]$
2. $\cos \frac{\pi }{2}\left[x\right]$
3. $\cos \left[x\right]$
4. both b and c

Q. What is value of 2-5i ?

Q. 34. Let f be a continuous function such that f(11)=10 and for all x, f(x) f(f(x)) = 1 then f(9) = A) 9 B) 1/9 C) 10/9 D) 9/10

Q. Let $f\left(x\right)=\{\begin{array}{cc}\sin x,& x\le 0\\ x^2+l,& 0<x<1\\ mx+3,& 1\le x\le 3\end{array}$. If both $\underset{x\to 0}{lim}f\left(x\right)$ and $\underset{x\to 1}{lim}f\left(x\right)$ exist, then the value of $\underset{x\to 5}{lim}(l{x}^2+mx+17)$ is equal to

Q.

Find the integral of $\frac{\sin x-\cos x}{1+\sin x\cos x}$with limits zero to $\frac{\mathrm{\pi }}{2}$.

Q.

Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is $800m^2$? If so, find its length and breadth.

Q. For the function $f\left(x\right)=\frac{1}{x}$, the value of $c$ that satisfy the mean value theorem for $f\left(x\right)$ on the interval $[1,4]$ is:

Q. If $p=\underset{x\to \infty }{lim}(\sin \sqrt{x^2+1}-\sin |x|)$ and $q=\underset{x\to -\infty }{lim}\left[\sin \left(\cos {\left(\sqrt{|x|-x^3}\right)}^{-1}\right)\right]$
(Where [.] denotes greatest integer function), then which of the following is/are correct:

1. $p=0$
2. $q=0$
3. $p=1$
4. $q=-1$

Q. How do we verify rolles theorem for greatest integer functions?

Q. If the function $f\left(x\right)=\{\begin{array}{cc}x+a^2\sqrt{2}sinx,& 0\le x<\frac{\pi }{4}\\ xcotx+b,& \frac{\pi }{4}\le x<\frac{\pi }{2}\\ bsin2x-acos2x,& \frac{\pi }{2}\le x\le \pi \end{array}$ is continuous in the interval $[0,\pi ]$ , then the values of (a, b) are

1. (–1, –1)
2. (0, 0)
3. (–1, 1)
4. (1, -1)

Q.

The function $\mathrm{f}\left(\mathrm{x}\right)={\int }_{-1}^{\mathrm{x}}\mathrm{t}\left({\mathrm{e}}^{\mathrm{t}}-1\right)\left(\mathrm{t}-1\right){\left(\mathrm{t}-2\right)}^3{\left(\mathrm{t}-3\right)}^5\mathrm{dt}$ has a local minimum at $\mathrm{x}=$

1. $0$

2. $1$

3. $2$

4. $3$

5. $3$ and $4$

Q.

What is the integral of a constant?

Q. Discuss the continuity of $f\left(x\right)=\left\{\begin{array}{ll}2x-1& ,x<0\\ 2x+1& ,x\ge 0\end{array}\right.\mathrm{at}x=0$

Q.

If $f\left(x\right)=\left[x\right]\frac{\sin {\displaystyle \frac{\mathrm{\pi }}{\left[x+1\right]}}+\sin \left(\mathrm{\pi }\left[\mathrm{x}+1\right]\right)}{1+\left[x\right]}$, then

1. $f\left(x\right)$ is continuous on $\mathrm{ℝ}$.

2. $f\left(x\right)$ is continuous on $\mathrm{ℝ}$, but not differentiable on $\mathrm{ℝ}$

3. $f\left(x\right)$ is discontinuous at every integer.

4. None of these.

Q. Let $A=\{a,b,c\}$ and $B=\{1,2,3,4\}$. Then the number of elements in the set $C=\{f:A\to B|2\in f(A\left)\text{and}f\text{is not one–one}\right\}$ is