**12 class Maths Notes Chapter 6**

**Application of Derivatives**

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**Application of Derivatives**

**Notes class 12 maths**

**CBSE Revision Notes for CBSE Class 12 Mathematics Application of Derivatives**Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

Class 12 Maths Chapter-6

**Application of Derivatives**

**Quick Revision Notes Free Pdf**

**ðŸ”· Chapter - 6 ðŸ”·**

**ðŸ‘‰ Application of Derivatives ðŸ‘ˆ**

**✳️ DERIVATIVE AS RATE OF CHANGE**

In various fields of applied mathematics one has the quest to know the rate at which one variable is changing, with respect to other,. The rate of change naturally refers to time. But we can have rate of change with respect to other variables also.

An economist may want to study how the investment changes with respect to variations in interest rates.

A physician may want to know, how small changes in dosage can affect the body's response to a drug.

A physicist may want to know the rate of change of distance with respect to time.

All questions of the above type can be interpreted and represented using derivatives.

**✳️ Definition :**

The

The instantaneous rate of change of f with respect to x is defined as

✴️

To use the word 'instantaneous', x may not be representing time. We usually use the word 'rate of change' to mean 'instantaneous rate of change'.

(I) The value of the derivative at P (x, y,) gives the slope of the tangent to the curve at P. Symbolically

(II) Equation of tangent at (x1, y1) is ;

(III) Equation of normal at (x1, y1) is ;

✴️

1. The point P (x₁, y₁) will satisfy the equation of the curve & the equation of tangent & normal line.

2. If the tangent at any point P on the curve is parallel to X-axis then dy/dx = 0 at the point P.

3. If the tangent at any point on the curve is parallel to Y-axis, then dy/dx = ∞ or dx/dy=0.

4. If the tangent at any point on the curve is equally inclined to both the axes then dy/dx +1.

5. If the tangent at any point makes equal intercept on the coordinate axes then dy/dx = +1.

6. Tangent to a curve at the point P (x₁, y₁) can be drawn even though dy/dx at P does not exist,

7. If a curve passing through the origin be given by a rational integral algebraic equation, the equation of the tangent (or tangents) at the origin is obtained by equating to zero the terms of the lowest degree in the equation.

(V) Differential :

The differential of a function is equal to its derivative multiplied by the differential of the independent variable.

Thus if, y = tan x then dy = sec²x dx.

In general dy =f ' (x) dx.

✴️

d (c) =0 where 'c' is a constant.

d (u +v - w) = du + dv- dw

d (uv) = udv+ vdu

* The relation dy = f ' (x) dx can be written as

Given a point P (a, b) which does not lie on the curve y=f(x), then the equation of possible tangents to the curve y=f(x), passing through (a, b) can be found by solving for the point of contact Q.

Angle between two intersecting curves is defined as the acute angle between their tangents or the normals at the point of intersection of two curves.

✴️

(i) The angle is defined between two curves if the curves are intersecting. This can be ensured by finding their point of intersection or by graphically.

(ii) If the curves intersect at more than one point then angle between curves is found out with respect to the point of intersection.

(iii) Two curves are said to be

Shortest distance between two non-intersecting differentiable curves is always along their common normal. (Wherever defined)

Let y=f(x)

From definition of derivative,

1. A function f(x) is called an

f (a+h) >f(a)

f (a - h) <f(a)

Similarly

f(a+h) <f(a)

f(a-h) >f(a)

Above statements hold true irrespective of whether f is non derivable or even discontinuous at x = a

2. A differentiable function is called increasing in an interval (a, b) if it is increasing at every point within the interval (but not necessarily at the end points). A function decreasing in an interval (a, b) is similarly defined.

3. A function which in a given interval is increasing or decreasing is called "

4. Tests for increasing and decreasing of a function at a point :

If the derivative f '(x) is positive at a point x = a, then the function f (x) at this point is increasing. If it is negative, then the function is decreasing.

Even if f'(a) is not defined, f can still be increasing or decreasing. (Look at the cases below).

✴️

If f' (a) =0, then for x =a the function may be still increasing or it may be decreasing as shown. It has to be identified by a seperate rule.

Note that, dy/dx = 3x².

✴️

1. If a function is invertible it has to be either increasing or decreasing.

2. If a function is continuous, the intervals in which it rises and falls may be separated by points at which its derivative fails to exist.

3. If f is increasing in [a, b] and is continuous then f (b) is the greatest and f (a) is the least value of f in [a, b]. Similarly if f is decreasing in [a, b] then f(a) is the greatest value and f(b) is the least value.

Let f (x) be a function of x subject to the following conditions :

(i) f (x) is a continuous function of x in the closed interval of a ≤ x ≤b.

(ii) f'(x) exists for every point in the open interval a<x<b.

(iii) f(a)=f(b).

Then there exists at least one point x = c such that a<c<bwhere f' (c)=0.

let f (x) be a function of x subject to the following conditions

(i) f(x) is a continuous function of x in the closed interval of a≤x≤b.

(ii) f' (x) exists for every point in the open interval a<x<b.

Then there exists at least one point x = c such that

Geometrically, the slope of the secant line joining the curve at x = a & x= b is equal to the slope of the tangent line drawn to the curve at x= c.

✴️

Now [f (b) -f (a)] is the change in the function f as x

rate of change of the function over the interval [a, b]. Also f' (c) is the actual rate of change of the function for x=c. Thus, the theorem states that the average rate of change of a function over an interval is also the actual rate of change of the function at some point of the interval. In particular, for instance, the average velocity of a particle over an interval of time is equal to the velocity at some instant belonging to the interval.

This interpretation of the theorem justifies the name "Mean Value" for the theorem.

(c) Application of rolles theorem for isolating the real roots of an equation f(x) = 0

Suppose a & b are two real numbers such that :

(i) f (x) & its first derivative f' (x) are continuous for a≤x≤b.

(ii) f (a) & f (b) have opposite signs.

(iii) f' (x) is different from zero for all values of x between a & b.

Then there is one & only one real root of the equation f(x) = 0 between a & b.

1 . A function f(x) is said to have a local maximum at x =a if f(a) is greater than every other value assumed by f (x) in the immediate neighbourhood of x= a. Symbolically

for a sufficiently small positive h.

Similarly, a function f (x) is said to have a local minimum value at x=b if f(b) is least than every other value assumed by f(x) in the immediate neighbourhood at x=b. Symbolically if

✴️

(i) The local maximum & local minimum values of a function are also known as local/relative maxima or local/relative minima as these are the greatest & least values of the function relative to some neighbourhood of the point in question.

(ii) The term 'extremum' is used both for maxima or a minima.

(iii) A local maximum (local minimum) value of a function may not be the greatest (least) value in a finite interval.

(iv) A function can have several local maximum & local minimum values & a local minimum value may even be greater than a local maximum value.

(V) Maxima & minima of a continuous function occur alternately & between two consecutive maxima there is a minima & vice versa.

If f (x) is a maxima or minima at x=c & if f' (c) exists then f' (c)=0.

✴️

(1) The set of values of x for which f (x)=0 are often called as stationary points. The rate of change of function is zero at a stationary point.

(2) In case f' (c) does not exist f(c) may be a maxima or a minima & in this case left hand and right hand derivatives are of opposite signs.

(3) the greatest (global maxima) and the least (global minima) values of a function f in an interval [a, b] are f(a) or f(b) or are given by the values of x which are critical points.

(4)

✴️

If f (x) does not change sign i.e. has the same sign in a certain complete neighbourhood of c, then f (x) is either strictly increasing or decreasing throughout this neighbourhood implying that f(c) is not an extreme value of f.

(a) f (c) is a minima of the function f, if f' (c) =0 & f' ' (c)>0.

(b) f (c) is a maxima of the function f, if f'(c)=0&f'' (c)<0.

✴️

If f"(c) = 0 then the test fails. Revert back to the first order derivative check for ascertaining the maxima or minima.

But if dy/dx changes sign from negative to zero to positive as x advances through x0, there is a minima. If dy/dx does not change sign, neither a maxima nor a minima. Such points are called

ðŸ‘‰

(In general, check at all Critical Points).

● Volume of a cuboid = lbh.

● Surface area of a cuboid = 2 (lb + bh + hl).

● Volume of a prism = area of the base × height.

● Lateral surface of a prism=perimeter of the base × height.

● Total surface of a prism = lateral surface + 2 area of the base

(Note that lateral surfaces of a prism are all rectangles).

The sign of the 2nd order derivative determines the concavity of the curve. Such point such as C & E on the graph where the concavity of the curve changes are called the points of inflection. From the graph we find that if:

**average rate of change**of a function f(x) with respect to**✳️ Definition :**The instantaneous rate of change of f with respect to x is defined as

✴️

**Note..**To use the word 'instantaneous', x may not be representing time. We usually use the word 'rate of change' to mean 'instantaneous rate of change'.

**✳️ EQUATIONS OF TANGENT & NORMAL**(I) The value of the derivative at P (x, y,) gives the slope of the tangent to the curve at P. Symbolically

(II) Equation of tangent at (x1, y1) is ;

(III) Equation of normal at (x1, y1) is ;

✴️

**Note.**1. The point P (x₁, y₁) will satisfy the equation of the curve & the equation of tangent & normal line.

2. If the tangent at any point P on the curve is parallel to X-axis then dy/dx = 0 at the point P.

3. If the tangent at any point on the curve is parallel to Y-axis, then dy/dx = ∞ or dx/dy=0.

4. If the tangent at any point on the curve is equally inclined to both the axes then dy/dx +1.

5. If the tangent at any point makes equal intercept on the coordinate axes then dy/dx = +1.

6. Tangent to a curve at the point P (x₁, y₁) can be drawn even though dy/dx at P does not exist,

**e.g.**x=0 is a tangent to y=x¹/² at (0,0).7. If a curve passing through the origin be given by a rational integral algebraic equation, the equation of the tangent (or tangents) at the origin is obtained by equating to zero the terms of the lowest degree in the equation.

**e.g.**If the equation of a curve be x² -y² + x³ + 3x²y-y³ = 0, the tangents at the origin are given by x² - y² = 0 i.e. x + y = 0 and x-y = 0.(V) Differential :

The differential of a function is equal to its derivative multiplied by the differential of the independent variable.

Thus if, y = tan x then dy = sec²x dx.

In general dy =f ' (x) dx.

✴️

**Note.**d (c) =0 where 'c' is a constant.

d (u +v - w) = du + dv- dw

d (uv) = udv+ vdu

* The relation dy = f ' (x) dx can be written as

**✳️ TANGENT FROM AN EXTERNAL POINT**Given a point P (a, b) which does not lie on the curve y=f(x), then the equation of possible tangents to the curve y=f(x), passing through (a, b) can be found by solving for the point of contact Q.

**✳️ ANGLE BETWEEN THE CURVES**Angle between two intersecting curves is defined as the acute angle between their tangents or the normals at the point of intersection of two curves.

✴️

**Note.**(i) The angle is defined between two curves if the curves are intersecting. This can be ensured by finding their point of intersection or by graphically.

(ii) If the curves intersect at more than one point then angle between curves is found out with respect to the point of intersection.

(iii) Two curves are said to be

**orthogonal**if angle between them at each point of intersection is right angle i.e. m₁ m₂ =-1.**✳️ SHORTEST DISTANCE BETWEEN TWO CURVES**Shortest distance between two non-intersecting differentiable curves is always along their common normal. (Wherever defined)

**✳️ ERRORS AND APPROXIMATIONS****(a) Errors**Let y=f(x)

**(b) Approximations**From definition of derivative,

**✳️ DEFINITIONS**1. A function f(x) is called an

**Increasing Function**at a point x = a if in a sufficiently small neighbourhood around X=a we havef (a+h) >f(a)

f (a - h) <f(a)

Similarly

**Decreasing Function**iff(a+h) <f(a)

f(a-h) >f(a)

Above statements hold true irrespective of whether f is non derivable or even discontinuous at x = a

2. A differentiable function is called increasing in an interval (a, b) if it is increasing at every point within the interval (but not necessarily at the end points). A function decreasing in an interval (a, b) is similarly defined.

3. A function which in a given interval is increasing or decreasing is called "

**Monotonic**" in that interval.4. Tests for increasing and decreasing of a function at a point :

If the derivative f '(x) is positive at a point x = a, then the function f (x) at this point is increasing. If it is negative, then the function is decreasing.

**✴️ Note.**Even if f'(a) is not defined, f can still be increasing or decreasing. (Look at the cases below).

✴️

**Note.**If f' (a) =0, then for x =a the function may be still increasing or it may be decreasing as shown. It has to be identified by a seperate rule.

**e.g.**f(x)=x³ is increasing at every point.Note that, dy/dx = 3x².

✴️

**Note.**1. If a function is invertible it has to be either increasing or decreasing.

2. If a function is continuous, the intervals in which it rises and falls may be separated by points at which its derivative fails to exist.

3. If f is increasing in [a, b] and is continuous then f (b) is the greatest and f (a) is the least value of f in [a, b]. Similarly if f is decreasing in [a, b] then f(a) is the greatest value and f(b) is the least value.

**5. (a) ROLLE'S Theorem:**Let f (x) be a function of x subject to the following conditions :

(i) f (x) is a continuous function of x in the closed interval of a ≤ x ≤b.

(ii) f'(x) exists for every point in the open interval a<x<b.

(iii) f(a)=f(b).

Then there exists at least one point x = c such that a<c<bwhere f' (c)=0.

**(b) LMVT Theorem:**let f (x) be a function of x subject to the following conditions

(i) f(x) is a continuous function of x in the closed interval of a≤x≤b.

(ii) f' (x) exists for every point in the open interval a<x<b.

Then there exists at least one point x = c such that

Geometrically, the slope of the secant line joining the curve at x = a & x= b is equal to the slope of the tangent line drawn to the curve at x= c.

**Note the following:**Rolles theorem is a special case of LMVT since✴️

**Note .****Physical Enterpretation of LMVT:**Now [f (b) -f (a)] is the change in the function f as x

rate of change of the function over the interval [a, b]. Also f' (c) is the actual rate of change of the function for x=c. Thus, the theorem states that the average rate of change of a function over an interval is also the actual rate of change of the function at some point of the interval. In particular, for instance, the average velocity of a particle over an interval of time is equal to the velocity at some instant belonging to the interval.

This interpretation of the theorem justifies the name "Mean Value" for the theorem.

(c) Application of rolles theorem for isolating the real roots of an equation f(x) = 0

Suppose a & b are two real numbers such that :

(i) f (x) & its first derivative f' (x) are continuous for a≤x≤b.

(ii) f (a) & f (b) have opposite signs.

(iii) f' (x) is different from zero for all values of x between a & b.

Then there is one & only one real root of the equation f(x) = 0 between a & b.

**✳️ HOW MAXIMA & MINIMA ARE CLASSIFIED**1 . A function f(x) is said to have a local maximum at x =a if f(a) is greater than every other value assumed by f (x) in the immediate neighbourhood of x= a. Symbolically

for a sufficiently small positive h.

Similarly, a function f (x) is said to have a local minimum value at x=b if f(b) is least than every other value assumed by f(x) in the immediate neighbourhood at x=b. Symbolically if

✴️

**Note.**(i) The local maximum & local minimum values of a function are also known as local/relative maxima or local/relative minima as these are the greatest & least values of the function relative to some neighbourhood of the point in question.

(ii) The term 'extremum' is used both for maxima or a minima.

(iii) A local maximum (local minimum) value of a function may not be the greatest (least) value in a finite interval.

(iv) A function can have several local maximum & local minimum values & a local minimum value may even be greater than a local maximum value.

(V) Maxima & minima of a continuous function occur alternately & between two consecutive maxima there is a minima & vice versa.

**A necessary condition for maxima & minima**If f (x) is a maxima or minima at x=c & if f' (c) exists then f' (c)=0.

✴️

**Note**.(1) The set of values of x for which f (x)=0 are often called as stationary points. The rate of change of function is zero at a stationary point.

(2) In case f' (c) does not exist f(c) may be a maxima or a minima & in this case left hand and right hand derivatives are of opposite signs.

(3) the greatest (global maxima) and the least (global minima) values of a function f in an interval [a, b] are f(a) or f(b) or are given by the values of x which are critical points.

(4)

**Critical points**are those where:

**✳️ Sufficient condition for extreme values First Derivative Test**✴️

**Note.**If f (x) does not change sign i.e. has the same sign in a certain complete neighbourhood of c, then f (x) is either strictly increasing or decreasing throughout this neighbourhood implying that f(c) is not an extreme value of f.

**✳️ Use of second order derivative in ascertaining the maxima or minima**(a) f (c) is a minima of the function f, if f' (c) =0 & f' ' (c)>0.

(b) f (c) is a maxima of the function f, if f'(c)=0&f'' (c)<0.

✴️

**Note.**If f"(c) = 0 then the test fails. Revert back to the first order derivative check for ascertaining the maxima or minima.

**✳️ Summary-working rule****ðŸ‘‰ First :**When possible, draw a figure to illustrate them problem & label those parts that are important in the problem. Constants & variables should be clearly distinguished.**ðŸ‘‰ Second :**Write an equation for the quantity that is to be maximised or minimised. If this quantity is denoted by 'y', it must be expressed in terms of a single independent variable x. This may require some algebraic manipulations.**ðŸ‘‰ Third :**If y=f(x) is a quantity to be maximum or minimum, find those values of x for which dy/dx =f' (x) = 0.**ðŸ‘‰ Fourth:**Test each values of x for which f" (x)=0 to determine whether it provides a maxima or minima or neither. The usual tests are :But if dy/dx changes sign from negative to zero to positive as x advances through x0, there is a minima. If dy/dx does not change sign, neither a maxima nor a minima. Such points are called

**INFLECTION POINTS.**

**ðŸ‘‰ Fifth :**If the function y = f(x) is defined for only a limited range of values a ≤x ≤b then examine x = a & x = b for possible extreme values.ðŸ‘‰

**Sixth :**If the derivative fails to exist at some point, examine this point as possible maxima or minima.(In general, check at all Critical Points).

**✳️ Useful formulae of mensuration to remember**● Volume of a cuboid = lbh.

● Surface area of a cuboid = 2 (lb + bh + hl).

● Volume of a prism = area of the base × height.

● Lateral surface of a prism=perimeter of the base × height.

● Total surface of a prism = lateral surface + 2 area of the base

(Note that lateral surfaces of a prism are all rectangles).

**✳️ Significance of the sign of 2nd order derivative and points of inflection**The sign of the 2nd order derivative determines the concavity of the curve. Such point such as C & E on the graph where the concavity of the curve changes are called the points of inflection. From the graph we find that if: