Application of Derivatives

If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find the approximate error in calculating its surface area.

Find the intervals in which the function f given by f (x) = Sin x + Cos x, 0 ≤ x ≤ 2π  is strictly increasing or strictly decreasing.

Find the approximate change in the surface area of a cube of side x meters caused by decreasing the side by 1%.

Using differentials, find the approximate value of f (2.01), where f (x) = 4x³ + 5x² + 2.

Given the sum of the perimeter of a circle and square. Show that the sum of their areas is least when the side of square is equal to diameter of the circle.

Using differentials, find the approximate value of each of the following up to 3 places of decimal.

Find the equation of the tangent line to the curve y = x² – 2x +7 which is

(a) Parallel to the line 2x – y + 9 = 0

(b) Perpendicular to the line 5y – 15x = 13.

A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?

Find the intervals in which the following functions are strictly increasing or decreasing:

(a) – 2x³ – 9x² – 12x + 1 

(b) (x + 1)³ (x – 3)³

Show that the right circular cylinder of given volume, open at the top, has minimum total surface area if its height is equal to the radius of the base.

Find the area of the greatest rectangle that can be inscribed in an ellipse 

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/√3. Also find the maximum volume.

Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.   

A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. find the dimensions of the window to admit maximum light through the whole opening.

Find the equation of the normal to the curve x² = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.        

Find the equations of tangents to the curve v= 1/3 πr²h at the points where the curve cuts the x-axis.

Show that the equation of the tangent to the parabola y² = 4ax at (x₁, y₁) is yy₁ = 2a(x + x₁).

Find the equation of the normals to the curve y = x³ + 2x + 6 which are parallel to line x + 14y + 4 = 0.

Find the intervals in which the function θ=cos^–1 (x/y) = cos^–1 (1/2) = π/3 is (a) increasing, (b) deceasing