Application of Derivatives

Find the equations of tangents to the curve 3x² – y² = 8, which pass through the point (4/3, 0).            

A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the rectangle that will produce the largest area of the window.

Find the equation of tangent to the curve x = Sin 3t, y = Cos 2t at t = π/4. 

Find the equation of the tangent to the curve  = y = dy/dx = (x² - 1) (1) + (x – 2) (2x) = 3x² - 4x- 1 Which is parallel to the line 4x – 2y + 5 = 0

Find the points on the curve y = x³ at which the slope of the tangent is equal to the y-coordinate of point. 

Prove that [1, 8/5] is an increasing function of θ in [0, π/2]

For the curve y = 4x³ – 2x⁵, find all the points at which the tangent passes through the origin. 

Show that the total surface area of a closed cuboid with square base and given volume, is minimum, when it is a cube.

Find the points on the curve x² + y² – 2x – 3 = 0 at which the tangents are parallel to the x-axis. 

Manufacturer can sell x items at a price of rupees 

 each. The cost price of x items is Rs 

 Find the number of items he should sell to earn maximum profit.                                

Find the equation of the normal at the point (am², am³) for the curve ay² = x³. 

An open box with a square base is to be made out of a given quantity of metal sheet of area c². Show that the maximum volume of the box is dz/dx = 0.       

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 α is one-third that of the cone and the greatest volume of cylinder is 

Prove that the curves x = y² and x y = k cut at right angles if 8k² = 1.

Show that the right circular cone of least curved surface and given volume has an altitude equal to √(y²-x² ) times the radius of the base

The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 13x² + 26x + 15.  Find the marginal revenue when x = 7.

If length of three sides of a trapezium other than base are equal to 10 cm, then find the area of the trapezium when it is maximum.        

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle. 

If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is π/3.

Find the point on the curve y = x³ – 11x + 5 at which the equation of the tangent is y=x–11.