Chapter-6. Application of Derivatives

The points on the curve 9y² = x³, where the normal to the curve makes equal intercepts with the axes are

(A) (B) 

(C) (D) 

The normal to the curve x² = 4y passing (1, 2) is

(A) x + y = 3 (B) x - y = 3

(C) x + y = 1 (D) x - y = 1

The normal at the point (1, 1) on the curve 2y + x² = 3 is

(A) x + y = 0 (B) x - y = 0

(C) x + y + 1 = 0 (D) x - y = 1

The line y = mx + 1 is a tangent to the curve y²= 4x if the value of m is

(A) 1 (B) 2 (C) 3 (D) 

The slope of the tangent to the curveat the point (2, - 1) is

(A)  (B)  (C)  (D) 

A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic mere per hour. Then the depth of the wheat is increasing at the rate of

(A) 1 m/h (B) 0.1 m/h

(C) 1.1 m/h (D) 0.5 m/h

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 istan2α.

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

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is.

Find the absolute maximum and minimum values of the function f given by

Find the points at which the function f given byhas

(i) local maxima (ii) local minima

(ii) point of inflexion

A point on the hypotenuse of a triangle is at distance a and bfrom the sides of the triangle.

Show that the minimum length of the hypotenuse is 

A window is in the form of 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.

The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs Rs 70 per sq meters for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?

Find the maximum area of an isosceles triangle inscribed in the ellipse with its vertex at one end of the major axis.

Find the intervals in which the function f given byis

(i) increasing (ii) decreasing

Find the intervals in which the function f given by

is (i) increasing (ii) decreasing

Show that the normal at any point θ to the curve

 is at a constant distance from the origin

Find the equation of the normal to curve y² = 4x at the point (1, 2).

The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?

Show that the function given byhas maximum at x = e.

Using differentials, find the approximate value of each of the following.

(a)  (b) 

The maximum value of  is

(A)  (B) 

(C) 1 (D) 0

For all real values of x, the minimum value of  is

(A) 0 (B) 1

(C) 3 (D) 

point on the curve x² = 2y which is nearest to the point (0, 5) is

(A)

  (B) 

(C) (0, 0) (D) (2, 2)

Show that semi-vertical angle of right circular cone of given surface area and maximum volume is .

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is.

Show that the right circular cone of least curved surface and given volume has an altitude equal to  time the radius of the base.

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is of the volume of the sphere.

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?

Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?

Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?

A square piece of tin of side 18 cm is to 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 two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

Find two positive numbers x and y such that their sum is 35 and the product x²y⁵ is a maximum

Find two positive numbers x and y such that their sum is 35 and the product x²y⁵ is a maximum

Find two positive numbers x and y such that x + y = 60 and xy³ is maximum.

Find two numbers whose sum is 24 and whose product is as large as possible.

Find the maximum and minimum values of x + sin 2x on [0, 2π].

It is given that at x = 1, the function x⁴- 62x² + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.

Find the maximum value of 2x³ - 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [-3, -1].

What is the maximum value of the function sin x + cos x?

At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?

Find both the maximum value and the minimum value of

3x⁴ - 8x³ + 12x² - 48x + 25 on the interval [0, 3]

Find the maximum profit that a company can make, if the profit function is given by

p(x) = 41 - 72x - 18x²

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:

(i)  (ii) 

(iii) 

(iv) 

Prove that the following functions do not have maxima or minima:

(i) f(x) = e^x (ii) g(x) = logx

(iii) h(x) = x³ + x² + x + 1

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

(i). f(x) = x²           
(ii). g(x) = x³ - 3x
(iii). h(x) = sinx + cosx, 0 <0 <

(iv). f(x) = sinx - cos x, 0 < x < 2π

(v). f(x) = x³ - 6x² + 9x + 15

(vi).  

(vii). 

(viii). 

Find the maximum and minimum values, if any, of the following functions given by

(i) f(x) = |x + 2| - 1 (ii) g(x) = - |x + 1| + 3

(iii) h(x) = sin(2x) + 5 (iv) f(x) = |sin 4x + 3|

(v) h(x) = x + 4, x ( - 1, 1)

Find the maximum and minimum values, if any, of the following functions given by

(i) f(x) = (2x - 1)² + 3        (ii) f(x) = 9x² + 12x + 2

(iii) f(x) = -(x - 1)² + 10    (iv) g(x) = x³ + 1

The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is

A. 0.06 x³ m³ B. 0.6 x³ m³ C. 0.09 x³ m³ D. 0.9 x³ m³

If f (x) = 3x² + 15x + 5, then the approximate value of f (3.02) is

A. 47.66 B. 57.66 C. 67.66 D. 77.66

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 in surface area.

If the radius of a sphere is measured as 7 m with an error of 0.02m, then find the approximate error in calculating its volume.

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

Find the approximate change in the volume V of a cube of side x metres caused by increasing side by 1%.

Find the approximate value of f (5.001), where f (x) = x³ - 7x² + 15.

Find the approximate value of f (5.001), where f (x) = x³ - 7x² + 15.

Find the approximate value of f (2.01), where f (x) = 4x² + 5x + 2

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

(i)  (ii)  (iii) 

(iv)  (v)  (vi) 

(vii)  (viii)  (ix) 

(x) 

The line y = x + 1 is a tangent to the curve y² = 4x at the point

(A) (1, 2) (B) (2, 1) (C) (1, -2) (D) (-1, 2)

The slope of the normal to the curve y = 2x² + 3 sin x at x = 0 is

(A) 3 (B)  (C) - 3 (D) 

Find the equation of the tangent to the curve  which is parallel to the line 4x - 2y + 5 = 0.

Find the equations of the tangent and normal to the hyperbola at the point.

Prove that the curves x = y² and xy = k cut at right angles if 8k² = 1. [Hint: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]

Find the equations of the tangent and normal to the parabola y² = 4ax at the point (at², 2at).

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

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

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

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

For the curve y = 4x³ - 2x⁵, find all the points at which the tangents passes through the origin

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

Show that the tangents to the curve y = 7x³ + 11 at the points where x = 2 and x = -2 are parallel.

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.

Find the equations of the tangent and normal to the given curves at the indicated points:

(i) y = x⁴ - 6x³ + 13x² - 10x + 5 at (0, 5)

(ii) y = x⁴ - 6x³ + 13x² - 10x + 5 at (1, 3)

(iii) y = x³ at (1, 1)

(iv) y = x² at (0, 0)

 (v) x = cos t, y = sin t at 

Find points on the curve  at which the tangents are

(i) parallel to x-axis (ii) parallel to y-axis

Find the equations of all lines having slope 0 which are tangent to the curve .

Find the equation of all lines having slope 2 which are tangents to the curve.

Find the equation of all lines having slope - 1 that are tangents to the curve .

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

Find a point on the curve y = (x - 2)² at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

Find points at which the tangent to the curve y = x³ - 3x² - 9x + 7 is parallel to the x-axis.

Find the slope of the normal to the curve x = 1 - a sin θ, y = b cos²θ at .

Find the slope of the normal to the curve x = acos³θ, y = asin³θ at.

Find the slope of the tangent to the curve y = x³ - 3x + 2 at the point whose x-coordinate is 3.

Find the slope of the tangent to curve y = x3 - x + 1 at the point whose x-coordinate is 2.

Find the slope of the tangent to the curve, x ≠ 2 at x = 10.

Find the slope of the tangent to the curve y = 3x4 - 4x at x = 4.

The interval in which  is increasing is

(A)  (B) ( - 2, 0) (C)  (D) (0, 2)

Prove that the function given by is increasing in R.

Prove that the function f given by f(x) = log cos x is strictly decreasing on  and strictly increasing on

Prove that the function f given by f(x) = log sin x is strictly increasing on and strictly decreasing on

Let I be any interval disjoint from ( - 1, 1). Prove that the function f given by

 is strictly increasing on I.

Find the least value of a such that the function f given is strictly increasing on (1, 2).

:  

On which of the following intervals is the function f given by  strictly decreasing?

(A)  (B) 

(C)  (D) None of these

Which of the following functions are strictly decreasing on?

(A) cos x (B) cos 2x (C) cos 3x (D) tan x

Prove that the function f given by f(x) = x² - x + 1 is neither strictly increasing nor strictly decreasing on (-1, 1).

Prove that the logarithmic function is strictly increasing on (0, ∠ž).

Prove that  is an increasing function of θ in.

Find the values of x for whichis an increasing function.

Show that, is an increasing function of x throughout its domain.

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

(a) x² + 2x - 5 (b) 10 - 6x - 2x²

(c) -2x³ - 9x² - 12x + 1 (d) 6 - 9x - x²

(e) (x + 1)³ (x - 3)³

Find the intervals in which the function f given by f(x) = 2x³ - 3x² - 36x + 7 is

(a) strictly increasing (b) strictly decreasing

Find the intervals in which the function f given by f(x) = 2x² - 3x is

(a) strictly increasing (b) strictly decreasing

Show that the function given by f(x) = sin x is

(a) strictly increasing in  (b) strictly decreasing in 

(c) neither increasing nor decreasing in (0, π)

Show that the function given by f(x) = e^2x is strictly increasing on R.

Show that the function given by f(x) = 3x + 17 is strictly increasing on R.

The total revenue in Rupees received from the sale of x units of a product is given by

. The marginal revenue, when  is

(A) 116 (B) 96 (C) 90 (D) 126

The rate of change of the area of a circle with respect to its radius rat r = 6 cm is

(A) 10À (B) 12À (C) 8À (D) 11À

The total revenue in Rupees received from the sale of x units of a product is given by

Find the marginal revenue when x= 7.

The total cost C (x) in Rupees associated with the production of xunits of an item is given by

Find the marginal cost when 17 units are produced.

Sand is pouring from a pipe at the rate of 12 cm³/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

A balloon, which always remains spherical, has a variable diameter  Find the rate of change of its volume with respect to x.

The radius of an air bubble is increasing at the rate of  cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

A particle moves along the curve. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.

The length xof 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.

The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?

An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm?

Find the rate of change of the area of a circle with respect to its radius r when

(a) r= 3 cm (b) r= 4 cm