**Question 1**

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

**Question 2**

The volume of a cube is increasing at the rate of 8 cm^{3}/s. How fast is the surface area increasing when the length of an edge is 12 cm?

**Question 3**

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.

**Question 4**

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?

**Question 5**

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?

**Question 6**

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?

**Question 7**

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.

**Question 8**

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?

**Question 9**

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

**Question 10**

Sand is pouring from a pipe at the rate of 12 cm^{3}/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?

**Question 11**

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

Find the marginal cost when 17 units are produced.

**Question 12**

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.

**Question 13**

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

(A) 10π (B) 12π (C) 8π (D) 11π

**Question 14**

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

**Question 15**

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.

**Question 16**

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.

**Question 17**

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.

**Question 18**

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

**Question 19**

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

**Question 20**

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

**Question 21**

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

**Question 22**

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)

**Question 23**

Find the intervals in which the function *f* given by *f*(*x*) = 2*x*^{2} − 3*x* is

(a) strictly increasing (b) strictly decreasing

**Question 24**

Find the intervals in which the function *f* given by *f*(*x*) = 2*x*^{3} − 3*x*^{2} − 36*x* + 7 is

(a) strictly increasing (b) strictly decreasing

**Question 25**

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

(a) *x*^{2} + 2*x* − 5 (b) 10 − 6*x* − 2*x*^{2}

(c) −2*x*^{3} − 9*x*^{2} − 12*x* + 1 (d) 6 − 9*x* − *x*^{2}

(e) (*x* + 1)^{3} (*x* − 3)^{3}

**Question 26**

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

**Question 27**

Find the values of *x* for whichis an increasing function.

**Question 28**

Prove that is an increasing function of *θ* in.

**Question 29**

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

**Question 30**

Prove that the function *f* given by *f*(*x*) = *x*^{2} − *x* + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).

**Question 31**

Which of the following functions are strictly decreasing on?

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

**Question 32**

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

(A) (B)

(C) (D) None of these

**Question 33**

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

**Question 34**

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

is strictly increasing on **I**.

**Question 35**

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

**Question 36**

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

**Question 37**

Prove that the function given by is increasing in **R**.

**Question 38**

The interval in which is increasing is

(A) (B) (−2, 0) (C) (D) (0, 2)

**Question 39**

Find the intervals in which the function *f *given by

is (i) increasing (ii) decreasing

**Question 40**

Find the intervals in which the function *f* given byis

(i) increasing (ii) decreasing

**Question 41**

Find the slope of the tangent to the curve *y* = 3*x*^{4} − 4*x* at *x* = 4.

**Question 42**

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

**Question 43**

Find the slope of the tangent to curve *y* = *x*^{3} − *x *+ 1 at the point whose *x*-coordinate is 2.

**Question 44**

Find the slope of the tangent to the curve *y* = *x*^{3} − 3*x* + 2 at the point whose *x*-coordinate is 3.

**Question 45**

Find the slope of the normal to the curve *x* = *a*cos^{3}*θ*, *y* = *a*sin^{3}*θ* at.

**Question 46**

Find the slope of the normal to the curve *x* = 1 − *a *sin *θ*, *y* = *b *cos^{2}*θ* at .

**Question 47**

Find points at which the tangent to the curve *y* = *x*^{3} − 3*x*^{2} − 9*x* + 7 is parallel to the *x*-axis.

**Question 48**

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).

**Question 49**

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

**Question 50**

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

**Question 51**

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

**Question 52**

Find points on the curve at which the tangents are

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

**Question 53**

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

(i) *y* = *x*^{4} − 6*x*^{3} + 13*x*^{2} − 10*x* + 5 at (0, 5)

(ii) *y* = *x*^{4} − 6*x*^{3} + 13*x*^{2} − 10*x* + 5 at (1, 3)

(iii) *y* = *x*^{3} at (1, 1)

(iv) *y* = *x*^{2} at (0, 0)

(v) *x* = cos *t*, *y* = sin *t* at

**Question 54**

Find the equation of the tangent line to the curve *y* = *x*^{2} − 2*x* + 7 which is

(a) parallel to the line 2*x* − *y* + 9 = 0

(b) perpendicular to the line 5*y* − 15*x* = 13.

**Question 55**

Show that the tangents to the curve *y* = 7*x*^{3} + 11 at the points where *x* = 2 and *x* = −2 are parallel.

**Question 56**

Find the points on the curve *y* = *x*^{3} at which the slope of the tangent is equal to the *y*-coordinate of the point.

**Question 57**

For the curve *y* = 4*x*^{3} − 2*x*^{5}, find all the points at which the tangents passes through the origin.

**Question 58**

Find the points on the curve *x*^{2} + *y*^{2} − 2*x* − 3 = 0 at which the tangents are parallel to the *x*-axis.

**Question 59**

Find the equation of the normal at the point (*am*^{2}, *am*^{3}) for the curve *ay*^{2} = *x*^{3}.

**Question 60**

Find the point on the curve *y* = *x*^{3} − 11*x* + 5 at which the tangent is *y* = *x* − 11.

**Question 61**

Find the equation of the normals to the curve *y* = *x*^{3} + 2*x *+ 6 which are parallel to the line *x* + 14*y* + 4 = 0.

**Question 62**

Find the equations of the tangent and normal to the parabola *y*^{2} = 4*ax* at the point (*at*^{2}, 2*at*).

**Question 63**

Prove that the curves *x* = *y*^{2} and *xy = k* cut at right angles if 8*k*^{2} = 1. [**Hint**: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]

**Question 64**

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

**Question 65**

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

**Question 66**

The slope of the normal to the curve *y* = 2*x*^{2} + 3 sin *x* at *x* = 0 is

(A) 3 (B) (C) −3 (D)

**Question 67**

The line *y* = *x* + 1 is a tangent to the curve *y*^{2} = 4*x* at the point

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

**Question 68**

Find the equation of the normal to curve *y*^{2} = 4*x* at the point (1, 2).

**Question 69**

Show that the normal at any point *θ* to the curve

is at a constant distance from the origin.

**Question 70**

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

(A) (B) (C) (D)

**Question 71**

The line *y* = *mx* + 1 is a tangent to the curve *y*^{2} = 4*x* if the value of *m* is

(A) 1 (B) 2 (C) 3 (D)

**Question 72**

The normal at the point (1, 1) on the curve 2*y* + *x*^{2} = 3 is

(A) *x* + *y* = 0 (B) *x* − *y *= 0

(C) *x* + *y* + 1 = 0 (D) *x *− *y* = 1

**Question 73**

The normal to the curve *x*^{2} = 4*y* passing (1, 2) is

(A) *x* + *y* = 3 (B) *x* − *y* = 3

(C) *x* + *y *= 1 (D) *x* − *y *= 1

**Question 74**

The points on the curve 9*y*^{2} = *x*^{3}, where the normal to the curve makes equal intercepts with the axes are

(A) (B)

(C) (D)

**Question 75**

**1. **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) (xi) (xii)

(xiii) (xiv) (xv)

**Question 76**

Find the approximate value of *f* (2.01), where* f *(*x*) = 4*x*^{2} + 5*x* + 2

**Question 77**

Find the approximate value of *f* (5.001), where *f* (*x*) = *x*^{3} − 7*x*^{2} + 15.

**Question 78**

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

**Question 79**

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

**Question 80**

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.

**Question 81**

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.

**Question 82**

If *f* (*x*) = 3*x*^{2} + 15*x* + 5, then the approximate value of *f *(3.02) is

**A.** 47.66 **B.** 57.66 **C.** 67.66 **D. **77.66

**Question 83**

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

**A.** 0.06 *x*^{3} m^{3} **B.** 0.6 *x*^{3} m^{3} **C.** 0.09 *x*^{3} m^{3} **D. **0.9 *x*^{3} m^{3}

**Question 84**

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

(a) (b)

**Question 85**

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

(v) *h*(*x*) = *x *+ 4, *x* (−1, 1)

**Question 86**

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

(i) *f*(*x*) = (2*x* − 1)^{2} + 3 (ii) *f*(*x*) = 9*x*^{2} + 12*x* + 2

(iii) *f*(*x*) = −(*x* − 1)^{2} + 10 (iv) *g*(*x*) = *x*^{3} + 1

**Question 87**

Find the maximum value of 2*x*^{3} − 24*x* + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].

**Question 88**

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*) = *x*^{2} (ii). *g*(*x*) = *x*^{3} − 3*x*

(iii). *h*(*x*) = sin*x* + cos, 0 < (iv). *f*(*x*) = sin*x* − cos *x*, 0 < *x* < 2π

(v). *f*(*x)* = *x*^{3} − 6*x*^{2 }+ 9*x* + 15

(vi).

(vii).

(viii).

**Question 89**

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

(i) *f*(*x*) = *e*^{x} (ii) *g*(*x*) = log*x*

(iii) *h*(*x*) = *x*^{3} + *x*^{2} + *x* + 1

**Question 90**

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

*p*(*x*) = 41 − 24*x *− 18*x*^{2}

**Question 91**

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

**Question 92**

Find two positive numbers *x* and *y* such that *x* + *y* = 60 and *xy*^{3} is maximum.

**Question 93**

Find two positive numbers *x *and *y *such that their sum is 35 and the product *x*^{2}*y*^{5} is a maximum

**Question 94**

Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

**Question 95**

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?

**Question 96**

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?

**Question 97**

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

**Question 98**

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.

**Question 99**

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?

**Question 100**

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?

**Question 101**

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.

**Question 102**

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

**Question 103**

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

**Question 104**

The point on the curve *x*^{2} = 2*y* which is nearest to the point (0, 5) is

(A) (B)

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

**Question 105**

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

(A) 0 (B) 1

(C) 3 (D)

**Question 106**

The maximum value of is

(A) (B)

(C) 1 (D) 0

**Question 107**

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.

**Question 108**

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

**Question 109**

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?

**Question 110**

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

**Question 111**

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 m^{3}. 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?

**Question 112**

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.

**Question 113**

A point on the hypotenuse of a triangle is at distance *a *and *b* from the sides of the triangle.

Show that the minimum length of the hypotenuse is

**Question 114**

Find the points at which the function *f *given byhas

(i) local maxima (ii) local minima

(ii) point of inflexion

**Question 115**

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

**Question 116**

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.

**Question 117**

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 istan^{2}*α*.

**Question 118**

Find the maximum value of 2*x*^{3} − 24*x* + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].

**Question 119**

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

(i) (ii)

(iii)

(iv)

**Question 120**

Find both the maximum value and the minimum value of

3*x*^{4} − 8*x*^{3} + 12*x*^{2} − 48*x* + 25 on the interval [0, 3]

**Question 121**

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

**Question 122**

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

**Question 123**

It is given that at *x* = 1, the function *x*^{4}− 62*x*^{2} + *ax* + 9 attains its maximum value, on the interval [0, 2]. Find the value of *a*.

**Question 124**

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

**Question 125**

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

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