Application of Derivatives

If the radius of a sphere is measured as 9 cm
with an error of 0.03 cm, then find the
approximate error in calculating its surface
area.

Find the positive value of \(a\) for which the equality \(2 \alpha+\beta=8\) holds, where \(\alpha\) and \(\beta\) are the points of maximum and minimum, respectively, of the function \(f(x)=2 x^3-9 a x^2+12 a^2 x+1\).

If \(y=4 x-6\) is a tangent to the curve \(y^2=a x^4+b\) at \((3,6)\), then the values of \(a\) and \(b\) are

If the function \(f(x)=2 x^3-9 a x^2+12 a^2 x+1\) attains its maximum and minimum at \(p\) and \(q\) respectively, such that \(p^2=q\), then \(a\) equals

If \(g(x)=\frac{1}{6} f\left(3 x^2-1\right)+\frac{1}{2} f\left(1-x^2\right), \forall x \in R\), where \(f^{\prime \prime}(x) > 0, \forall x \in R\). Then, \(g(x)\) is increasing in the interval

The volume of a spherical balloon is increasing at the rate of \(30 \mathrm{~cm}^3\) per minute. Find the rate of change of surface area of the balloon, when its radius is \(6 \mathrm{~cm}\).

Find the minimum value of \(2x+3y\), when \(xy=6\).

A spherical iron ball 10 cm in radius is coated
with a layer of ice of uniform thickness, which
melts at a rate of 50 cm\(^3\)
/min. When the
thickness of the ice is 15 cm, the rate at which
the thickness of ice decreases is ........ cm/min...

The distance between the origin and the normal to the curve \(y=e^{2 x}+x^2\) drawn at \(x=0\) is units

The stationary points of the curve \(y=8 x^2-x^4-4\) are

If the error committed in measuring the
radius of a circle is 0.05%, then the
corresponding error in calculating its area
would be

The line which is parallel to X-axis and crosses the curve \(y=\sqrt x\) at an angle of 45\(\Upsilon\) is

If \(f^{\prime \prime}(x)\) is a positive function for all \(x \in R, f^{\prime}(3)=0\) and \(g(x)=f\left(\tan ^2(x)-2 \tan (x)+4\right)\) for \(0 < x <\frac{\pi}{2}\), then the interval in which \(g(x)\) is increasing is

Let \(x\) and \(y\) be the sides of two squares such that, \(y=x-x^2\). The rate of change of area of the second square with respect to area of the first square is

If the curves \(\frac{x^2}{a^2}+\frac{y^2}{4}=1\) and \(y^3=16 x\) intersect at right angles, then \(a^2\) is equal to

Given, \(f(x)=x^3-4x\), if x changes from 2 to 1.99, then the approximate change in the value of \(f(x)\) is