Graphical Interpretation of Derivative

Q.

The area (in sq.units) of the region enclosed by the curves $y=x^2-1$ and $y=1-x^2$ is equal to

1. $\frac{4}{3}$

2. $\frac{7}{2}$

3. $\frac{16}{3}$

4. $\frac{8}{3}$

Q. 2. Prove that the function f given by f(x)=|x-1| , x R is not differentiable at x=1 .

Q.

The area enclosed by $y=3x-5,y=0,x=3$ and $x=5$ is

1. $12\mathrm{sq}\mathrm{units}$

2. $13\mathrm{sq}\mathrm{units}$

3. $\frac{131}{2}\mathrm{sq}\mathrm{units}$

4. $14\mathrm{sq}\mathrm{units}$

Q. If the derivative of a function f(x) is negative at a particular point, then about that point, the value of the function as the x-value increases.

1. decreases
2. increases
3. remains the same

Q. If the derivative of a function f(x) is positive at a particular point, then about that point, the value of the function as the x-value increases.

1. increases
2. decreases
3. remains the same

Q.

Show that the function f(x)=|x+1|+|x-1| for all x belongs to R, is not differentiable at points x= -1 and x=1.

Q. f(x) is a differentiable function and g(x) is a double differentiable function such that |f(x)|≤1 and f(x)=g(x) . If f2(0)+g2(0)=9 . Prove that there exists some c∈(–3, 3) such that g(c).g(c)<0.

Q. The set of all points, where the function $f\left(x\right)=\frac{x}{1+\left|x\right|}$ is differentiable is

1. $\left(-\infty ,\infty \right)$
2. $[0,\infty )$
3. $\left(-\infty ,0\right)\cup \left(0,\infty \right)$
4. $\left(0,\infty \right)$

Q.

Find the equations of the tangent and normal to the hyperbola at the point.

Q. If $f\left(x\right)=x|x^2-3x|e^{|x-3|}+[\frac{3+sgn(x^2-1-5x)}{4+x^2}]+\{\frac{1}{3}+[\frac{x^2-2x}{|x|+1}]\}$ is non differentiable at $x=x_1,x_2,x_3.....x_n$ then ${\sum }_{i=1}^n{x}_i$ equals
( where $[.],\{.\},sgn(.)$, denotes greatest integer function, fractional part function and signum function respectively)

1. \N
2. 3
3. 4
4. 5

Q.

1. k = 0

2. k=-1

3. k = 2

4. k = 1

Q. If the derivative of the function f(x) is positive at a particular point, then about that point the value of the function is decreasing as the x-value increases.

1. True
2. False

Q.

1. k = 1

2. k = 1/x

3. k = 2

4. k = 0

Q. The set of points where the function f (x) given by f (x) = |x − 3| cos x is differentiable, is
(a) R
(b) R − {3}
(c) (0, ∞)
(d) none of these

Q.
The derivative of the curve

1. Will be same at all the points
2. Will be positive at all points
3. Will be negative at all points
4. Will be positive at some points and negative at some points and zero at some

Q. The derivative value is same at all the points on a straight line.

1. True
2. False

Q. If the derivative of the function f(x) is positive at a particular point, then about that point the value of the function is decreasing as the x-value increases.

1. True
2. False

Q. The set of all points, where the function $f\left(x\right)=\frac{x}{1+\left|x\right|}$ is differentiable is

1. $\left(-\infty ,\infty \right)$
2. $[0,\infty )$
3. $\left(-\infty ,0\right)\cup \left(0,\infty \right)$
4. $\left(0,\infty \right)$

Q. For the curve y = (2x + 1)^3 find the rate of change of slope at x = 1

Q.
The derivative of the curve

1. Will be same at all the points
2. Will be positive at all points
3. Will be negative at all points
4. Will be positive at some points and negative at some points and zero at some

Q. If the derivative of the function f(x) is positive at a particular point, then about that point the value of the function is decreasing as the x-value increases.

1. True
2. False

Q. If the derivative of the function f(x) is positive at a particular point, then about that point the value of the function is decreasing as the x-value increases.

1. True
2. False