Short Answer TypeFind the equation of tangents to the curve y= x3+2x-4, which are perpendicular to line x+14y+3 =0.
Long Answer TypeShow that semi-vertical angle of a cone of maximum volume and given slant height is cos-1
.
Short Answer TypeLong Answer TypeIf the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum when the angle between them is 
Show that the height of the cylinder of maximum volume, which can be inscribed in a sphere of radius R is 
 Also find the maximum volume.Â
Find the equation of the normal at a point on the curve x2 = 4y which passes through the point (1, 2). Also, find the equation of the corresponding tangent.
The equation of the given curve is x2 = 4y.
Differentiating w.r.t. x, we get
Let (h, k) be the co-ordinates of the point of contact of the normal to the curve x2 = 4y.
Now, slope of the tangent at (h, k) is given by
Hence, slope of the normal at (h, k) Â =Â 
Therefore, the equation of normal at (h, k) is 
Since, it passes through the point (1, 2) we have
Now, (h, k) lies on the curve x2Â = 4y, so, we have:
            ...(3)Â
Solving (2) and (3), Â we get,
h = 2 Â and k = 1.
From (1), the required equation of the normal is:
Also, slope of the tangent  = 1
 Equation of tangent at (1, 2) is:
Short Answer TypeThe volume of a sphere is increasing at the rate of 3 cubic centimetres per second. Find the rate of increase of its surface area, when the radius is 2 cm.
 is an increasing function.
Switch
