Application of ...

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 volume.

The curve $$\displaystyle y=ax^{3}+bx^{2}+cx+5$$ touches the $$x$$ - axis at $$P(-2, 0)$$ and cuts the $$y$$-axis at a point $$Q$$, where its gradient is $$3$$. Find $$a, b, c$$.

The slope of the normal to the curve $$\displaystyle x=a\left ( \theta -\sin \theta  \right ),\: \: y=a\left ( 1-\cos \theta  \right )$$ at point $$\displaystyle \theta =\dfrac{\pi }2$$ is

Let h be a twice continuously differentiable positive function on an open interval $$J$$. Let $$\displaystyle g\left ( x \right )=ln\left ( h(x) \right ) $$ for each $$\displaystyle x\epsilon J $$
Suppose $$\displaystyle \left ( h'\left (  x \right )\right )^{2}> h''\left ( x \right )h\left ( x \right )$$ for each $$\displaystyle x\epsilon J$$ Then

Let $$f$$ be a continuous, differentiable and bijective function. If the tangent to $$y=f\left( x \right) $$ at $$x=b$$, then there exists at least one $$c\in \left( a,b \right) $$ such that 

The slope of normal to the curve $$\displaystyle y^{2}=4ax$$ at a point $$\displaystyle \left ( at^{2},2at \right )$$ is

The slope of the tangent to the curve $$xy + ax - by = 0$$ at the point $$(1, 1)$$ is $$2$$ then values of $$a$$ and $$b$$ are respectively -

At what point the tangent line to the curve $$\displaystyle y=\cos \left ( x+y \right ),\left ( -2\pi \leq x\leq 2\pi  \right )$$ is parallel to $$x + 2y = 0$$

The point at which the tangent to the curve $$\displaystyle y=x^{3}+5$$ is perpendicular to the line $$x + 3y = 2$$ are

At what point of the curve $$\displaystyle y=2x^{2}-x+1$$ tangent is parallel to $$y = 3x + 4$$