RD Sharma solutions for Class 12 Maths chapter 18 - Maxima and Minima [Latest edition]

Chapters Chapter 18: Maxima and Minima

Exercise 18.1Exercise 18.2Exercise 18.3Exercise 18.4Exercise 18.5Others
Exercise 18.1 [Page 7]

RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and MinimaExercise 18.1 [Page 7]

Exercise 18.1 | Q 1 | Page 7

f(x) = 4x2 + 4 on R .

Exercise 18.1 | Q 2 | Page 7

f(x) = - (x-1)2+2 on R ?

Exercise 18.1 | Q 3 | Page 7

f(x)=| x+2 | on R .

Exercise 18.1 | Q 4 | Page 7

f(x)=sin 2x+5 on R .

Exercise 18.1 | Q 5 | Page 7

f(x) = | sin 4x+3 | on R ?

Exercise 18.1 | Q 6 | Page 7

f(x)=2x3 +5 on R .

Exercise 18.1 | Q 7 | Page 7

f (x) = $-$ | x + 1 | + 3 on R .

Exercise 18.1 | Q 8 | Page 7

f(x) = 16x2 $-$ 16x + 28 on R ?

Exercise 18.1 | Q 9 | Page 7

f(x) = x$-$ 1 on R .

Exercise 18.2 [Page 16]

RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and MinimaExercise 18.2 [Page 16]

Exercise 18.2 | Q 1 | Page 16

f(x) = (x $-$ 5)4.

Exercise 18.2 | Q 2 | Page 16

f(x) = x$-$ 3x .

Exercise 18.2 | Q 3 | Page 16

f(x) = x3  (x $-$ 1).

Exercise 18.2 | Q 4 | Page 16

f(x) =  (x $-$ 1) (x+2)2

Exercise 18.2 | Q 5 | Page 16

f(x) = $\frac{1}{x^2 + 2}$ .

Exercise 18.2 | Q 6 | Page 16

f(x) =  x$-$ 6x2 + 9x + 15 .

Exercise 18.2 | Q 7 | Page 16

f(x) = sin 2x, 0 < x < $\pi$ .

Exercise 18.2 | Q 8 | Page 16

f(x) =  sin x $-$ cos x, 0 < x < 2$\pi$ .

Exercise 18.2 | Q 9 | Page 16

f(x) =  cos x, 0 < x < $\pi$ .

Exercise 18.2 | Q 10 | Page 16

f(x)=sin2x-x, -pi/2<=x<=pi/2

Exercise 18.2 | Q 11 | Page 16

f(x)=2sinx-x, -pi/2<=x<=pi/2

Exercise 18.2 | Q 12 | Page 16

f(x) =$x\sqrt{1 - x} , x > 0$.

Exercise 18.2 | Q 13 | Page 16

f(x) = x3 (2x $-$ 1)3.

Exercise 18.2 | Q 14 | Page 16

f(x) =$\frac{x}{2} + \frac{2}{x} , x > 0$ .

Exercise 18.3 [Page 31]

RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and MinimaExercise 18.3 [Page 31]

Exercise 18.3 | Q 1.01 | Page 31

f(x) = x4 $-$ 62x2 + 120x + 9.

Exercise 18.3 | Q 1.02 | Page 31

f(x) = x3$-$ 6x2 + 9x + 15

Exercise 18.3 | Q 1.03 | Page 31

f(x) = (x - 1) (x + 2)2.

Exercise 18.3 | Q 1.04 | Page 31

f(x) = 2/x - 2/x^2,  x>0

Exercise 18.3 | Q 1.05 | Page 31

f(x) = xex.

Exercise 18.3 | Q 1.06 | Page 31

f(x) = x/2+2/x, x>0 .

Exercise 18.3 | Q 1.07 | Page 31

f(x) = (x+1) (x+2)^(1/3), x>=-2 .

Exercise 18.3 | Q 1.08 | Page 31

f(x)=xsqrt(32-x^2),  -5<=x<=5 .

Exercise 18.3 | Q 1.09 | Page 31

f(x) = $x^3 - 2a x^2 + a^2 x, a > 0, x \in R$ .

Exercise 18.3 | Q 1.1 | Page 31

f(x) = $x + \frac{a2}{x}, a > 0,$ , x ≠ 0 .

Exercise 18.3 | Q 1.11 | Page 31

f(x) = $x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}$ .

Exercise 18.3 | Q 1.12 | Page 31

f(x) = $x + \sqrt{1 - x}, x \leq 1$ .

Exercise 18.3 | Q 2.1 | Page 31

f(x) = (x $-$ 1) (x $-$ 2)2.

Exercise 18.3 | Q 2.2 | Page 31

f(x)=xsqrt(1-x),  x<=1 .

Exercise 18.3 | Q 2.3 | Page 31

f(x) = $- (x - 1 )^3 (x + 1 )^2$ .

Exercise 18.3 | Q 3 | Page 31

The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?

Exercise 18.3 | Q 4 | Page 31

Show that $\frac{\log x}{x}$ has a maximum value at x = e ?

Exercise 18.3 | Q 5 | Page 31

Find the maximum and minimum values of the function f(x) = $\frac{4}{x + 2} + x .$

Exercise 18.3 | Q 6 | Page 31

Find the maximum and minimum values of y = tan $x - 2x$ .

Exercise 18.3 | Q 7 | Page 31

If f(x) = x3 + ax2 + bx + c has a maximum at x = $-$ 1 and minimum at x = 3. Determine a, b and c ?

Exercise 18.3 | Q 8 | Page 31

Prove that f(x) = sinx + $\sqrt{3}$ cosx has maximum value at x = $\frac{\pi}{6}$ ?

Exercise 18.4 [Page 37]

RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and MinimaExercise 18.4 [Page 37]

Exercise 18.4 | Q 1.1 | Page 37

f(x) = 4x $-$ $\frac{x^2}{2}$ in [ $-$ 2,4,5] .

Exercise 18.4 | Q 1.2 | Page 37

f(x) = (x $-$ 1)2 + 3 in [ $-$ 3,1] ?

Exercise 18.4 | Q 1.3 | Page 37

f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3] .

Exercise 18.4 | Q 1.4 | Page 37

f(x) = (x $-$ 2) $\sqrt{x - 1} \text { in }[1, 9]$ .

Exercise 18.4 | Q 2 | Page 37

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

Exercise 18.4 | Q 3 | Page 37

Find the absolute maximum and minimum values of the function of given by $f(x) = \cos^2 x + \sin x, x \in [0, \pi]$ .

Exercise 18.4 | Q 4 | Page 37

Find the absolute maximum and minimum values of a function f given by f(x) = 12 x^(4/3) - 6 x^(1/3) , x in [ - 1, 1] .

Exercise 18.4 | Q 5 | Page 37

Find the absolute maximum and minimum values of a function f given by $f(x) = 2 x^3 - 15 x^2 + 36x + 1 \text { on the interval } [1, 5]$ ?

Exercise 18.5 [Pages 72 - 74]

RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and MinimaExercise 18.5 [Pages 72 - 74]

Exercise 18.5 | Q 1 | Page 72

Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.

Exercise 18.5 | Q 2 | Page 72

Divide 64 into two parts such that the sum of the cubes of two parts is minimum.

Exercise 18.5 | Q 3 | Page 72

How should we choose two numbers, each greater than or equal to -2, whose sum______________ so that the sum of the first and the cube of the second is minimum?

Exercise 18.5 | Q 4 | Page 72

Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.

Exercise 18.5 | Q 5 | Page 72

Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm3, which has the minimum surface area?

Exercise 18.5 | Q 6.1 | Page 72

A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by $M = \frac{WL}{2}x - \frac{W}{2} x^2$ .

Find the point at which M is maximum in a given case.

Exercise 18.5 | Q 6.2 | Page 72

A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by $M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}$ .

Find the point at which M is maximum in a given case.

Exercise 18.5 | Q 7 | Page 72

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 lengths of the two pieces so that the combined area of the circle and the square is minimum?

Exercise 18.5 | Q 8 | Page 72

A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the we should be cut so that the sum of the areas of the square and triangle is minimum?

Exercise 18.5 | Q 9 | Page 72

Given the sum of the perimeters of a square and a circle, show that the sum of there areas is least when one side of the square is equal to diameter of the circle.

Exercise 18.5 | Q 10 | Page 73

Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.

Exercise 18.5 | Q 11 | Page 73

Two sides of a triangle have lengths 'a' and 'b' and the angle between them is $\theta$. What value of $\theta$ will maximize the area of the triangle? Find the maximum area of the triangle also.

Exercise 18.5 | Q 12 | Page 73

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 a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.

Exercise 18.5 | Q 13 | Page 73

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners 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 maximum possible?

Exercise 18.5 | Q 14 | Page 73

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

Exercise 18.5 | Q 15 | Page 73

A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.

Exercise 18.5 | Q 16 | Page 73

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

Exercise 18.5 | Q 17 | Page 73

Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is $\frac{2R}{\sqrt{3}} .$

Exercise 18.5 | Q 18 | Page 73

A rectangle is inscribed in a semi-circle of radius r with one of its sides on diameter of semi-circle. Find the dimension of the rectangle so that its area is maximum. Find also the area ?

Exercise 18.5 | Q 19 | Page 73

Prove that a conical tent of given capacity will require the least amount of  canavas when the height is $\sqrt{2}$ times the radius of the base.

Exercise 18.5 | Q 20 | Page 73

Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to $\frac{2}{3}$ of the diameter of the sphere.

Exercise 18.5 | Q 21 | Page 73

Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is $\cot^{- 1} \left( \sqrt{2} \right)$ .

Exercise 18.5 | Q 22 | Page 73

An isosceles triangle of vertical angle 2 $\theta$ is inscribed in a circle of radius a. Show that the area of the triangle is maximum when $\theta$ = $\frac{\pi}{6}$ .

Exercise 18.5 | Q 23 | Page 73

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is $6\sqrt{3}$r.

Exercise 18.5 | Q 24 | Page 73

Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible when revolved about one of its sides ?

Exercise 18.5 | Q 25 | Page 73

Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?

Exercise 18.5 | Q 26 | Page 73

A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?

Exercise 18.5 | Q 27 | Page 73

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .$

Exercise 18.5 | Q 28 | Page 74

Show that among all positive numbers x and y with x2 + y2 =r2, the sum x+y is largest when x=y=r $\sqrt{2}$ .

Exercise 18.5 | Q 29 | Page 74

Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?

Exercise 18.5 | Q 30 | Page 74

Find the point on the curve y2 = 4x which is nearest to the point (2,$-$ 8).

Exercise 18.5 | Q 31 | Page 74

Find the point on the curve x2 = 8y which is nearest to the point (2, 4) ?

Exercise 18.5 | Q 32 | Page 74

Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?

Exercise 18.5 | Q 33 | Page 74

Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x $-$ 3 ?

Exercise 18.5 | Q 34 | Page 74

Find the point on the curvey y2 = 2x which is at a minimum distance from the point (1, 4).

Exercise 18.5 | Q 35 | Page 74

Find the maximum slope of the curve y = $- x^3 + 3 x^2 + 2x - 27 .$

Exercise 18.5 | Q 36 | Page 74

The total cost of producing x radio sets per  day is Rs $\left( \frac{x^2}{4} + 35x + 25 \right)$ and the price per set  at which they may be sold is Rs. $\left( 50 - \frac{x}{2} \right) .$ Find the daily output to maximum the total profit.

Exercise 18.5 | Q 37 | Page 74

Manufacturer can sell x items at a price of rupees $\left( 5 - \frac{x}{100} \right)$ each. The cost price is Rs  $\left( \frac{x}{5} + 500 \right) .$ Find the number of items he should sell to earn maximum profit.

Exercise 18.5 | Q 38 | Page 74

An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead with be least, if depth is made half of width.

Exercise 18.5 | Q 39 | Page 74

A box of constant volume c is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?

Exercise 18.5 | Q 40 | Page 74

The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube.

Exercise 18.5 | Q 41 | Page 74

A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is $\pi : (\pi + 2)$ .

Exercise 18.5 | Q 42 | Page 74

The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a ?

Exercise 18.5 | Q 43 | Page 74

A straight line is drawn through a given point P(1,4). Determine the least value of the sum of the intercepts on the coordinate axes ?

Exercise 18.5 | Q 44 | Page 74

The total area of a page is 150 cm2. The combined width of the margin at the top and bottom is 3 cm and the side 2 cm. What must be the dimensions of the page in order that the area of the printed matter may be maximum?

Exercise 18.5 | Q 45 | Page 74

The space s described in time by a particle moving in a straight line is given by S = $t5 - 40 t^3 + 30 t^2 + 80t - 250 .$ Find the minimum value of acceleration.

Exercise 18.5 | Q 46 | Page 74

A particle is moving in a straight line such that its distance at any time t is given by  S = $\frac{t^4}{4} - 2 t^3 + 4 t^2 - 7 .$  Find when its velocity is maximum and acceleration minimum.

[Page 80]

RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima [Page 80]

Q 1 | Page 80

Write necessary condition for a point x = c to be an extreme point of the function f(x).

Q 2 | Page 80

Write sufficient conditions for a point x = c to be a point of local maximum.

Q 3 | Page 80

If f(x) attains a local minimum at x = c, then write the values of f' (c) and f'' (c).

Q 4 | Page 80

Write the minimum value of f(x) = $x + \frac{1}{x}, x > 0 .$

Q 5 | Page 80

Write the maximum value of f(x) = $x + \frac{1}{x}, x > 0 .$

Q 6 | Page 80

Write the point where f(x) = x log, x attains minimum value.

Q 7 | Page 80

Find the least value of f(x) = $ax + \frac{b}{x}$, where a > 0, b > 0 and x > 0 .

Q 8 | Page 80

Write the minimum value of f(x) = xx .

Q 9 | Page 80

Write the maximum value of f(x) = x1/x.

Q 10 | Page 80

Write the maximum value of f(x) = $\frac{\log x}{x}$, if it exists .

[Pages 80 - 82]

RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima [Pages 80 - 82]

Q 1 | Page 80

The maximum value of x1/x, x > 0 is __________ .

• e^(1/e)

• (1/e)^e

• 1

• none of these

Q 2 | Page 81

If $ax + \frac{b}{x} \frac{>}{} c$ for all positive x where a,b,>0, then _______________ .

• ab<c^2/4

• ab>=c^2/4

• ab>=c/4

Q 3 | Page 81

The minimum value of $\frac{x}{\log_e x}$ is _____________ .

• e

• 1/e

• 1

• none of these

Q 4 | Page 81

For the function f(x) = $x + \frac{1}{x}$

• x = 1 is a point of maximum

• x = $-$ 1 is a point of minimum

• maximum value > minimum value

• maximum value < minimum value

Q 5 | Page 81

Let f(x) = x3+3x$-$ 9x+2. Then, f(x) has _________________ .

• a maximum at x = 1

• a minimum at x = 1

• neither a maximum nor a minimum at x = - 3

• none of these

Q 6 | Page 81

The minimum value of f(x) = $x4 - x2 - 2x + 6$ is _____________ .

• 6

• 4

• 8

• none of these

Q 7 | Page 81

The number which exceeds its square by the greatest possible quantity is _________________ .

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{3}{4}$

• none of these

Q 8 | Page 81

Let f(x) = (x $-$ a)2 + (x $-$ b)2 + (x $-$ c)2. Then, f(x) has a minimum at x = _____________ .

• $\frac{a + b + c}{3}$

• $\sqrt{abc}$

• $\frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}$

• none of these

Q 9 | Page 81

The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .

• $\frac{1}{4}$

• $\frac{1}{2}$

• $\frac{1}{8}$

• none of these

Q 10 | Page 81

The function f(x) = $\sum^5_{r = 1}$ (x $-$ r)2 assumes minimum value at x = ______________ .

• 5

• 5/2

• 3

• 2

Q 11 | Page 81

At x= $\frac{5\pi}{6}$ f(x) = 2 sin 3x + 3 cos 3x is ______________ .

• 0

• maximum

• minimum

• none of these

Q 12 | Page 81

If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .

• 3

• 3/4

• 1

• none of these

Q 13 | Page 81

The least value of the function f(x) = $x3 - 18x2 + 96x$ in the interval [0,9] is _____________ .

• 126

• 135

• 160

• 0

Q 14 | Page 81

The maximum value of f(x) = $\frac{x}{4 - x + x^2}$ on [ $-$ 1, 1] is _______________ .

• $\frac{1}{4}$

• $- \frac{1}{3}$

• $\frac{1}{6}$

• $\frac{1}{5}$

Q 15 | Page 81

The point on the curve y2 = 4x which is nearest to, the point (2,1) is _______________ .

• $1, 2\sqrt{2}$

• (1, 2)

• (1, -2)

• ( -2,1)

Q 16 | Page 82

If x+y=8, then the maximum value of xy is ____________ .

• 8

• 16

• 20

• 24

Q 17 | Page 82

The least and greatest values of f(x) = x3$-$ 6x2+9x in [0,6], are ___________ .

• 3, 4

• 0, 6

• 0, 3

• 3, 6

• 0, 54

Q 18 | Page 82

f(x) = $\sin + \sqrt{3} \cos x$ is maximum when x = ___________ .

• $\frac{\pi}{3}$

• $\frac{\pi}{4}$

• $\frac{\pi}{6}$

• 0

Q 19 | Page 82

If a cone of maximum volume is inscribed in a given sphere, then the ratio of the height of the cone to the diameter of the sphere is ______________ .

• $\frac{3}{4}$

• $\frac{1}{3}$

• $\frac{1}{4}$

• $\frac{2}{3}$

Q 20 | Page 82

The minimum value of $\left( x^2 + \frac{250}{x} \right)$ is __________ .

• 75

• 50

• 25

• 55

Q 21 | Page 82

If(x) = x+$\frac{1}{x}$,x > 0, then its greatest value is _______________ .

• -2

• 0

• 3

• none of these

Q 22 | Page 82

If(x) = $\frac{1}{4x^2 + 2x + 1}$ then its maximum value is _________________ .

• $\frac{4}{3}$

• $\frac{2}{3}$

• 1

• $\frac{3}{4}$

Q 23 | Page 82

Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .

• 1

• 2

• $2\frac{1}{2}$

• $3\frac{1}{3}$

Q 24 | Page 82

f(x) = 1+2 sin x+3 cos2x, 0<=x<=(2pi)/3 is ________________ .

• Minimum at x =$\frac{\pi}{2}$

• Maximum at x = sin $- 1$ ( $\frac{1}{\sqrt{3}}$)

• Minimum at x = $\frac{\pi}{6}$

• Maximum at sin^-1(1/6)

Q 25 | Page 82

The function f(x) = $2 x^3 - 15 x^2 + 36x + 4$ is maximum at x = ________________ .

• 3

• 0

• 4

• 2

Q 26 | Page 82

The maximum value of f(x) = $\frac{x}{4 + x + x^2}$ on [ $-$ 1,1] is ___________________ .

• $- \frac{1}{4}$

• $- \frac{1}{3}$

• $\frac{1}{6}$

• $\frac{1}{5}$

Q 27 | Page 82

Let f(x) = 2x3$-$ 3x2$-$ 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .

• -2

• -1

• 2

• 4

Q 28 | Page 82

The minimum value of x loge x is equal to ____________ .

• e

• 1/e

• -1/e

• 2/e

• -e

Q 29 | Page 82

The minimum value of the function f(x)=2x^3-21x^2+36x-20 is ______________ .

• -128

• -126

• -120

• none of these

Chapter 18: Maxima and Minima

Exercise 18.1Exercise 18.2Exercise 18.3Exercise 18.4Exercise 18.5Others RD Sharma solutions for Class 12 Maths chapter 18 - Maxima and Minima

RD Sharma solutions for Class 12 Maths chapter 18 (Maxima and Minima) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Maths chapter 18 Maxima and Minima are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies Or Quantities, Introduction to Applications of Derivatives.

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