Graphical Interpretation of Differentiability

Q.

What is the use of graphs?

Q.

What is the parabolic curve?

Q. Let $f^{\prime }\left(x\right)=\frac{192x^3}{2+{\sin }^4\pi x}$ for all $x\in R$ with $f\left(\frac{1}{2}\right)=0.$ If $m\le \underset{\frac{1}{2}}{\overset{1}{\int }}f\left(x\right)dx\le M,$ then the possible values of $m$ and $M$ are

1. $m=13,M=24$
2. $m=\frac{1}{4},M=\frac{1}{2}$
3. $m=-11,M=0$
4. $m=1,M=12$

Q. Let $S$ be the set of all points in $(-\pi ,\pi )$ at which the function, $f\left(x\right)$ = min $\{\sin x,\cos x\}$ is not differentiable. Then $S$ is a subset of which of the following?

1. $\{-\frac{3\pi }{4},-\frac{\pi }{4},\frac{3\pi }{4},\frac{\pi }{4}\}$
2. $\{-\frac{\pi }{2},-\frac{\pi }{4},\frac{\pi }{4},\frac{\pi }{2}\}$
3. $\{-\frac{3\pi }{4},-\frac{\pi }{2},\frac{\pi }{2},\frac{3\pi }{4}\}$
4. $\{-\frac{\pi }{4},0,\frac{\pi }{4}\}$

Q. Consider the function $f\left(x\right)=\frac{x^3}{4}-\sin \pi x+3$

1. $f\left(x\right)$ does not attain value within the interval $[-2,2]$
2. $f\left(x\right)$ takes on the value $2\frac{1}{3}$ in the interval $[-2,2]$
3. $f\left(x\right)$ takes on the value $3\frac{1}{4}$ in the interval $[-2,2]$
4. $f\left(x\right)$ takes no value $p,1<p<5$ in the interval $[-2,2]$

Q. Let $f:(-1,1)\to R$ be a function defined by $f\left(x\right)=max\{-|x|,-\sqrt{1-x^2}\}$. If $K$ be the set of all points at which $f$ is not differentiable, then $K$ has exactly

1. one element
2. two elements
3. three elements
4. five elements

Q.

If $g\left(x\right)$ is a polynomial satisfying $g\left(x\right)g\left(y\right)=g\left(x\right)+g\left(y\right)+g\left(xy\right)-2$ for all real $x$ and $y$ and $g\left(2\right)=5$, then $\underset{x\to 3}{\lim }g\left(x\right)=$

1. $9$

2. $10$

3. $25$

4. $20$

Q.

How do you tell if a parabola opens left or right$?$

Q. What is the perfect graph for action potential.?

Q. Which of the following curve(s) is(are) symmetric about $y=x$?

1. $x^2-y^2=16$
2. $x^2+y^2=16$
3. $xy=16$
4. $3x^2+4y^2=12$

Q.

The horizontal asymptote of the curve $y=e^{\frac{1}{x}}$ is

1. $y=0$
2. $y=1$
3. $y=-1$
4. $y=2$

Q. If $f\left(x\right)$ is a differentiable everywhere then

1. $|f\left(x\right)|$ is differentiable everywhere
2. $|f{|}^2$ is differentiable everywhere
3. $f|f|$ is not differentiable at some point
4. None of these

Q.

Select the correct statement for the function $f\left(x\right)=x{e}^{1-x}$.

1. Strictly increases in the interval $\left(\frac{1}{2},2\right)$

2. Increases in the interval $\left(0,\infty \right)$

3. Decreases in the interval $\left(0,2\right)$

4. Strictly decreases in the interval $\left(1,\infty \right)$

Q. Which of the following curve has no asymptote:

1. $y=5^x$
2. $y={\log }_{e}x$
3. $y=x^2-3x+2$
4. $x^2-4y^2=4$

Q. Give an example of a function which is continuous everywhere but not differentiable at a point.

Q. $f\left(x\right)=min\{\cos x,1-\sin x\},-\pi \le x\le \pi .$ Then which among the following options is/are correct

1. $f\left(x\right)$ is not differentiable at $x=0$
2. $f\left(x\right)$ is differentiable at $x=\frac{\pi }{2}$
3. $f\left(x\right)$ is discontinuous at $x=0$
4. $f\left(x\right)$ is continuous at $x=\frac{\pi }{2}$

Q. Let $f\left(x\right)=\{\begin{array}{cc}max\{e^x,e^{-x},2\},& x\le 0\\ min\{e^x,e^{-x},2\},& x>0\end{array}$.
Which of the following statements is (are) CORRECT?

1. $f\left(x\right)$ is discontinuous at $x=0$.
2. $f\left(x\right)$ is non-differentiable at exactly two points.
3. $f\left(x\right)$ has non-removable type of discontinuity at $x=0$ with jump of discontinuity equal to $2$.
4. $f\left(x\right)$ is continuous but non-differentiable at $x=\ln \frac{1}{2}$.

Q. The area of the region $(x,y)$ : $xy\le 8,1\le y\le x^2$ is

1. $16{\log }_e^2-\frac{14}{3}$
2. $8{\log }_e^2-\frac{14}{3}$
3. $16{\log }_e^2-6$
4. $8{\log }_e^2-\frac{7}{3}$

Q. Let $f:R\to R$ be twice continuously differentiable. Let $f\left(0\right)=f\left(1\right)=f^{\prime }\left(0\right)=0$. Then

1. $f^{\prime \prime }\left(x\right)\ne 0$ for all $x$
2. $f^{\prime \prime }\left(c\right)=0$ for some $c\in R$
3. $f^{\prime \prime }\left(x\right)\ne 0$ if $x\ne 0$
4. $f^{\prime }\left(x\right)>0$ for all $x$

Q. Let $f:R\to R$ be a function defined by $f\left(x\right)=max\{x,x^3\}.$ Then the set of all points where $f$ is not differentiable, is

1. $\{-1,0\}$
2. $\{-1,0,1\}$
3. $\{0,1\}$
4. $\{-1,1\}$

Q. The number of intersection points of the function $f\left(x\right)=\sin x\&y=0.5\text{in}$
$\left(a\right).x\in (0,3\pi )$
$\left(b\right).x\in [-6\pi ,6\pi ]$ respectively are:

1. $6,10$
2. $4,10$
3. $4,12$
4. $6,12$

Q. The motion of a body is given by equation dv/dt=a-bv. The velocity of particle varies with time as. Draw the graph.

Q. If $f\left(x\right)={\sin }^{-1}\left(\cos x\right){\cos }^{-1}\left(\sin x\right)\forall x\in [0,2\pi ],$ then the number of point(s) of non differentiability is

1. $0$
2. $1$
3. $3$
4. $4$

Q. What is the nature of the graph: $y=2x^2$

1. parabola passing through origin
2. Hyperbola but not passing through origin
3. Hyperbola passing through origin
4. ellipse passing through origin

Q.

Determine if the graph is symmetric about the $x$-axis, the $y$-axis, or the origin.$\mathrm{r}=4\cos 3\mathrm{\theta }$.

Q. Let $f\left(x\right)$ be a polynomial function satisfying the following conditions
$\underset{x\to \infty }{lim}\frac{f\left(x\right)}{|x{|}^3}=0,\underset{x\to \infty }{lim}(\sqrt{f\left(x\right)}-x)=-1andf\left(0\right)=0$
(where, [.] denotes greatest integer function) then which of the following is/are correct?

1. The number of points of discontinuity of $g\left(x\right)=\left[f\right(x\left)\right]$ in [0, 3] is 3
2. The number of points of discontinuity of $g\left(x\right)=\left[f\right(x\left)\right]$ in [0, 3] is 5
3. The number of points of non-derivability of $h\left(x\right)=|f\left(|x|\right)|$ is 4
4. The number of points of non-derivability of $h\left(x\right)=|f\left(|x|\right)|$ is 3

Q. If $f\left(x\right)=\{\begin{array}{cc}{x}^2+2,& x<0\\ -2e^{-x},& 0\le x<2,\\ \frac{-2}{e^2}(x-3),& x\ge 2\end{array}$ then

1. $|f\left(x\right)|$ is discontinuous at $0$ points
2. $|f\left(x\right)|$ is discontinuous at $2$ points
3. $|f\left(x\right)|$ is not differentiable at $2$ points
4. $|f\left(x\right)|$ is not differentiable at $3$ points

Q.

Use the method of symmetry to find the extreme value of each quadratic function and the value of $x$ for which it occurs. $g\left(x\right)=(5-x)(2x+3)$ pls give it in this form:

The $x$-intercepts of the graph of the function are (_, _), (_, _). The midpoint of the $x$-intercepts is ___. The extreme value is (a maximum or a minimum).

$g\left(\right)$=_____

Q. Assertion :Let function $f:R\to R$ is such that $f\left(x\right)f\left(y\right)-f\left(xy\right)=x+y$ for all $x,y\in R$
$f\left(x\right)$ is a Bijective function. Reason: $f\left(x\right)$ is a linear function.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct

Q. What is the nature of the graph : $y=-4x^2+6$

1. Hyperbola not passing through origin
2. Ellipse not passing through origin
3. it is not a conic
4. parabola not passing through origin