Application of Derivatives
Find the points on the curve y=x3at which the slope of the tangent is equal to the y-coordinate of the point.
If the curve y=ax2−6x+b pass through (0,2) and has its tangent parallel to the x-axis at x=23, then find the values of aandb˙
Let x be the length of one of the equal sides of an isosceles triangle, and let θ be the angle between them. If x is increasing at the rate (1/12) m/h, and θ is increasing at the rate of 180π radius/h, then find the rate in m3 / h at which the area of the triangle is increasing when x=12mandthη=π/4.
Let P(x) be a polynomial with real coefficients, Let a,b∈R,a<b, be two consecutive roots of P(x) . Show that there exists c such that a≤c≤bandPprime(c)+100P(c)=0.
If the line xcosθ+ysinθ=P is the normal to the curve (x+a)y=1, then show θ∈(2nπ+2π,(2n+1)π)∪(2nπ+23π,(2n+2)π),n∈Z
At the point P(a,an) on the graph of y=xn,(n∈N) , in the first quadrant, a normal is drawn. The normal intersects the y−aξs at the point (0,b) . If (lim)a0=21, then n equals _____.
Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle is one-third that of the cone and the greatest volume of cylinder is 274πh3tan2α˙