The amount of pollution content added in air in a city due to X diesel vehicles is given by P(x)=0.005x³+0.02x² +30x.Find the marginal increase   in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question?

The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue(marginal revenue).If the total revenue(in rupees)received from the sale of x units of a product is given by R9x)=3x²+36+5,find the marginal revenue,when=5,and write which value does the question indicate?

The total cost C(x) associated with provision of free mid-day meals to x students of a school in primary classes is given by C(x)=0.005x³-0.02x²+30x+50
If the marginal cost is given by rate of change \(dC\over dX\)of total cost,write the marginal cost of food for 300students.What value is shown here?

The total revenue received from the sale of x souvenirs in connection with 'PEACE DAY'is given by R(x)=3x²+40x+10.
Find the marginal revenue when 100souvenirs were sold.What is the importance of celebrating Peace Day in our life?

A balloon which always remains spherical has a variable diameter \({3\over2}(2x+1)\). Find the rate of change of its volume with respect to x.

The total cost C(x) in Rupees, associated with the production of x units of an item is given by
C(x) = 0.005 x³ – 0.02 x² + 30x + 5000
Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

The total revenue received from the sale of x units of a product is given by R(x)=3x²+36x+5.Find the marginal revenue when x=5.

Let f be a function defined on an interval I and c ∈ I. (Second Derivative Test)

For what value of m,The function f(x) = mx + c, is decreasing for x \(\in\) R.

Show that the function y = \({3\over x}+7\) is strictly decreasing for x \(\in\)R (x \(\neq \) 0).

Find the least value of a such that the function f(x)=x+ax+1 is strictly increasing on(1,2).

Prove that the tangents on the curve y=x³+6 at the points (-1,5) and (1,7) are parallel.

Find the point(s) on the curve \(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } =1\),where tangent is parallel to the x-axis.

Find the points on the curve \({x^2\over4}+{y^2\over 25}=1\)at which the  tangents are
(i) parallel to the x-axis.
(ii)parallel to the y-axis.

If radius of a circle is increased from  5cm to 5.1cm.Find the approximate increase in area.

Find the point on the curve \(y^2 =8x + 3\)  for which the y-coordinate changes 4 times more than coordinate of x.

Show that the tangents to the curve \(y=x^2-7x+18\) at (3, 0) and (4, 0) are at right angles.

The function \(f(x)=x^{ 2 },x\in R\) has no maximum value (True/False)

Find the slope  of the tangent to the curve:
\(y=x^{ 3 }-x\quad at\quad x=2\)

The radius of a circle is increasing at the rate of 0.7cm/s. What is the rate of increasing of its circumference?

The total cost C(x) in Rupees associated with the production of x units of an item is given by
\(C(x)=0.007x^{ 3 }-0.003x^{ 2 }+15x+4000\)
Find the marginal cost when 17 units are produced.

Show that for \(a\ge 1\) \(f(x)=\sqrt { 3 } sinx-cosx-2ax+4\)  is increasing in R.

Prove that the curves \(x=y^{ 2 }\) and xy=k cut at right angles if \(8k^{ 2 }=1\)

Show that the equation of normal at any point \('t'\) on the curve:
\(x=3cost-cos^{ 3 }t\) and y=3 \(sint-sin^{ 3 }t\) is : \(4(ycos^{ 3 }t-xsin^{ 3 }t)=3sin4t\)

Prove that \(y={4sin\theta\over2+cos\theta}-\theta\) is an increasing function of \(\theta\) in \([0,{\pi\over2}]\).

Find the equation of the tangent line to the curve \(y=x^2-2x+7\) which is
(i) parallel to the line \(2x-y+9=0\)
(ii) perpendicular to the line \(5y-15x=13\)

If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of triangle is maximum when the angle between them is \(60^\circ\)
\((i.e., \frac{\pi}{3})\)