Application of ...

A function $$y=f(x)$$ has a second-order derivative $$f''(x)=6(x-1)$$. If its graph passes through the point $$(2,1)$$ and at the point tangent to the graph is $$y=3x-5$$, then the value of $$f(0)$$ is 

The angle made by the tangent of the curve $$x=a (t+\sin t \cos t)$$, $$y=a(1+sint)^2$$ with the $$x- axis$$ at any point on it is

The value of $$x$$ at which tangent to the curve $$y=x^3-6x^2+9x+4,   0\leq x \leq 5$$ has maximum slope is

The point on the curve $$y^{2} = x ,$$ the tangent at which makes an angle of $$45^{0}$$ with positive direction of $$x -$$ axis will be given by

If the curve represented parametrically by the equations $$x=2 \ln\cot t+1$$ and $$y=\tan t+ \cot t$$

The period of oscillation $$T$$ of a pendulum of length $$l$$ at a place of acceleration due to gravity $$g$$ is given by $$T=2\pi \sqrt {\dfrac {l}{g}}$$. If the calculated length is $$0.992$$ times the actual length and if the value assumed for $$g$$ is $$1.002$$ times its actual value, the relative error in the computed value of $$T$$ is

The focal length of a mirror is given by $$\dfrac {1}{v}-\dfrac {1}{u}=\dfrac {2}{f}$$. If equal errors ($$\alpha$$) are made in measuring $$u$$ and $$v$$, then the relative error in $$f$$ is

The tangent of the acute angle between the curves $$y=|x^2-1| $$ and $$y=\sqrt {7-x^2}$$ at their points of intersection is

Consider the function $$f(x)= \begin{cases} x \sin \displaystyle \frac {\pi}{x}, for  \ x>0\\ 0,                   for \   x=0 \end{cases}$$. Then, the number of points in $$(0,1)$$ where the derivative $$f'(x)$$ vanishes is

The area of a triangle is computed using the formula $$S=\dfrac {1}{2}$$ bc sin A. If the relative errors made in measuring b, c and calculating S are respectively $$0.02$$, $$0.01$$ and $$0.13$$ the approximate error in A when $$A=\pi /6$$ is