Application of Derivatives

INDER PAL MATHEMATICS CLASSES

(Class:10+2, Non-medical/Inter-Arts)

Application of Derivative

OUTLINE LECTURE NOTES

APPLICATION OF DERIVATIVE

Rate of change of quantities:

Working Rules to solve problems

Rule 1: Identify the related variables, say, and express one of these in terms of the other to get

Rule II: Differentiate w.r.t and get

Rule III: Substitute the value of in the equation

and get the value of

Rule IV: If the calculated value of is not a constant

Then substitute the value of where the rate of change ofis desired.

ROBOM: 1 If the rate of change of a variable is postive,then the value of the variable increases With the increase in the value of the independent variable

2. If the rate of change of a variable is negative, then the value of the variable Decreases with the increase in the value of the independent variable

Tangents of Circles

QN

P

T Tangent line

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Let P and Q be two points on the circle as shown in figure. As the point Q moves along the curve towards P; the second line ; turns about P and so on. and when Q approaches to P,the line PQ coincides with the limiting line PT.This limiting PT is called the tangent line to the curve at the point P.

The tangent line touch or contact at only a single point of any curve

Equation of Tangent:We know point slope form

Let be any point on the graph of derivable function The slope of tangent at P is

equal to the value of the derivative at P,i.e slope of tangent at P equals .

The equation of the tangent at P is

Comparing 1 and 2 we get

Equation of Normal:

Normal to the curve at any point P is a line through the point P and to the tangent at P to the curve. P is called the foot of Normal.

Slope of normal =

Remarks: Case 1. If

In this case the tangent is parallel to . Therefore the normal at P will be parallel to and would be passing through P

Case 2. If In this case, let the slope of normal at P be m.


ROBOM





Working Rules for solving problems

Step No.1: Identify the equation of the curve and the point(s) where the equations of tangents/normal are to be found out.

Step No.2: Differentiate the equation of curve and find .Find the slope of the tangent by using the

result: Slope of tangent at a point = value of at that point

Step No.3: Find the slope of normal by using the result:Slope of normal at a point = negative reciprocal of the slope of tangent at that point. Also remember that if the tangent at a point is horizental, then the normal at that point would be vertical and its slope would not be defined.

Step No.4: Find the equation of tangent/normal by using the point slope form, which states that if a

line passes through and having slope m, then its equation is

Also remember that if a line passing through is vertical,

then its equation is

Example No.1: Find the slopes of the tangents and normal to the curve

Solution: We have ,

Slope of tangent when = value of when

Let slope of normal be m.

MENSURATION

1. Rectangle: If a and b are sides then Perimeter = 2(a + b) and Area = ab2. Square: Perimeter = 4a and Area =3. Circle: If r be the radius of circle, then its Area = and circumference = 4. Cuboid: If be length, b be breadth and h be the height of cuboid, then its

Curved surface Area= ,Volume = Total surface Area =

5. Cube: If a is side of cube, then itsCurved surface Area= , Volume = Total surface Area =

6. Right Circular Cylinder: if r be the radius and h be height of cylinder then itsCurved Surface Area = , Total Surface Area: = Volume=

7. Sphere: Let r be radius of the sphere, then its:

Total Surface Area: = , Volume :

8. Right Circular Cone: Volume :

Angle of Intersections of curves






Point of intersection

Slope of tangent to curve (1) =

Slope of tangent to curve (2) =

Since slopes of tangents are same, the angle between curves is The curves touches each other

Increasing function

A function is said to be increasing function on an interval ,if






Strictly increasing function

A function is said to be strictly increasing function on an interval if

Decreasing function

A function is said to be decreasing function on an interval ,if

Strictly decreasing function

A function is said to be strictly decreasing function on an interval ,if






Monotonic Function

A Function, which is either strictly increasing or strictly decreasing in its domain, is called monotonic function

Non Monotonic Function

A function, which is increasing in some interval and decreasing in some intervals, is called a non-monotonic function

Here is a strictly increasing function in the intervals (a,c) and (d,e) while it is strictly decreasing function in the intervals (c,d0 and (e,b)

Test for monotonicity of functions

Two methods to solve the problems of increasing and decreasing function

1. Check symbolically 2. By derivative method

Check symbolically:In this system take all the options i.e and check the

position of functions according to above mentioned.

Example: Show that the following function are strictly increasing on R

Solution: Let be real numbers such that

By derivative method:This method based on two theorems






Two Theorems

1. A function is strictly increasing on an open interval where its derivative is positive or non-negative

2. A function is strictly decreasing on an open interval where its derivative is negative or non-positive

Method: Take the derivative of the function and put the values of intervals in derivative function if the then function is strictly increasing otherwise if

then the function is strictly decreasing

Example1: Show that the following function are strictly Increasing

Solution: We have

Now according to question If we put in 2x then is always positive and greater than 0.

is strictly increasing in interval

Method to find the intervals where the function strictly increasing or decreasing

Working rules for solving problems

Step no1: Differentiate the given function and solve the equation to find out the critical value for .

Step no2: Arrange these critical values in the ascending order of Magnitude and partition the domain of into various intervals using critical values.

Step no3: Study the sign of on the corresponding intervals.Step no4: If any particular open interval is +ve, then the

Function is strictly increasing on that interval. On the Other hand, if is –ve,then f(x) is strictly decreasing

on that interval.






Example No 1: For what value of x (or find the interval) in which

the function is strictly increasing or strictly

decreasing

Solution: We have

Step No:1:

Now

Step No:2: The critical values for in ascending orderare

Step No:3:

is strictly increasing and strictly decreasing on

Roll’s Theorem

Statement of Roll’s Theorem

Check points for verifying the Roll’s theoremIf a function f(x) such that :

1. f(x) is continuous on the closed interval [a,b]2. f(x) is derivable on the open interval (a,b)3. f(a)=f(b)4. Then there is some point C (at least one value) of x lying between open interval (a,b)

(For this such that

ROBOM1. Every polynomial function is continuous on R(Real Number)2. The functions are continuous on R3. The function logx is continuous on 4. If f and g are continues function on [a,b], then the function f+g, f-g, f.g are also

continues function on [a,b]and is also continues function on [a,b] provided

on [a,b]

Example: is not continuous at x=7






5. If the derivative of a function has finite and unique value of an interval, then thefunction is derivable on that interval

6. The function is continuous on R and derivable on Working Rules for solving Problems

Step No.1 Show that f(x) is continuous in the closed interval [a,b].The result regarding the continuity of standard function should be made use.

Step No.2 Find and see if it is non-defined at any point in the open interval (a,b),then we say that f(x) is derivable on (a,b)

Step No.3 Find f(a) and f(b) and check these are equalIf all the three conditions mentioned in the above steps are satisfied, then Roll’s theorem is applicable. Even one of the conditions fails to hold then the conclusion of Roll’s theorem may not be applicable

Step No.4 If all the conditions of Roll’s theorem are satisfied, then solve the equation and show that at least one of roots lies in (a,b).

This verifies Roll’s Theorem.

Lagrange’s Theorem

Statement of Lagrange’s Theorem

Check points for verifying the Lagrange,s theoremIf a function f(x) such that :

1. f(x) is continuous on the closed interval [a,b]2. f(x) is derivable on the open interval (a,b)3. Then there exists at least one C such that

Working Rules for solving Problems

Step No.1 Show that f(x) is continuous in the closed interval [a,b].The result regarding the continuity of standard function should be made use.

Step No.2 Find and see if it is non-defined at any point in the open interval (a,b),then we say that f(x) is derivable on (a,b)

Step No.3 If both the conditions mentioned in the above steps are satisfied, then LMV theorem is applicable. Even one of the conditions fails to hold then the conclusion of Roll’s theorem may not be applicable

Step No.4 If both the conditions of the L.M.V theorem are satisfied, the solve the equation and show that at least one of the roots lies in (a,b).This verifies the Lagrange’s mean value theorem.

Example: Verify L.M.V theorem: X(x-2) on interval [1,2]






Local max.or local Minima:Working Rules to solve problems

Rule 1: The given function is differentiated and the equation =0is solved to get the critical values for

Rule II: Arrange these critical values in the ascending order of magnitude

Rule III: To find local extremum values, either the first derivative method or theSecond derivative method is used. If the method to be used is given inthe problem then we do not have any choice; otherwise the second derivative method should be preferred

Rule IV: For using the first derivative method:

i) If for a particular critical value x = a, the sign of changes from+ ve to –ve as x increases through x = a, then is a local max.value.

ii) If for a particular critical value x = a, the sign of changes from- ve to +ve as x increases through x = a, then is a local min.value

iii) If for a particular critical value x = a, the sign of does not change as x increases through x = a ,then is neither a local

max.value nor a local minimum value.

For using the second derivative method:

For a particular critical value x = a

i) is a local maximum valueii) is a local minimum valueiii) The test fails and first derivative

methodto be used.






Absolute Maximum or Absolute min.value: ROBOM: The interval is must when absolute min.or max.to be check any function


Rule 1: Differentiate the given function and solve the equation to find the critical values for .Let be

the critical values (roots ) of the equation

Rule II: Find the values of . at the points a, ,b(where a and b is the lower and higher value of interval)

Rule III: The greatest and the least values in the set are respectively the

absolute max. and absolute min.values of on

Practical problems on maxima and minimum


Rule 1: If possible draw a neat diagram

Rule II: Express the variable to be maximized (or minimized) in terms of other Variables. Also use given condition, if any, to express the variable ‘y’ to Be maximized (or minimized) in terms of only one convenienent variable, ‘x’

Rule III: Find and solve the equation = 0 to get the critical values for y.

Reject impossible critical values.

Rule IV: Find and check its sign at the critical values. The critical value

For which is negative (respectively positive) gives the maximum

(respectively minimum) value of the variable y.






GEOMETRYWe are familiar with the following solids in different objects.






Measurement of surfaces and volumes of various geometrical figures:

1. Cuboid:

3.

4.






5.

6.


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Application of Derivatives