NAME OF THE TEACHER | DINESH KUMAR | ||||
CLASS | 10+2 | CHAPTER | 06 | SUBJECT | MATHEMATICS |
TOPIC | APPLICATIONS OF DERIVATIVES | DURATION : 25 Class Meetings | |||
After studying this lesson students should know the
Increasing and Decreasing of Functions
Now teacher will explain the Concept of increasing and decreasing and their graphical representation. Also teacher will help the students in finding the interval in which the given function is increasing or decreasing with the help of Baby Curve Method.
a) f is strictly increasing in (a, b) if f '(x) > 0 for each x ϵ (a, b)
b)f is strictly decreasing in (a, b) if f '(x) < 0 for each x ϵ (a, b)
Slope and Tangent of a line or curve (Deleted)
First of all teacher will explain the term slope of a line and explain different methods of finding the slopes of lines or curves. Teacher will also explain the method of finding slope of tangent and equation of tangent by taking some examples.
Slope of a line (Deleted)
It is the tangent of the angle made by the line with the positive direction of x - axis measured anti-clockwise direction. Slope of any line or curve can be find by finding dy / dx
Equation of Tangent (Deleted)
Equation of the tangent at (x1, y1) is given by
y – y1 = m(x - x1) where m is the slope of the tangent
Slope and Normal of a line or curve (Deleted)
Here teacher will explain the Method of finding the slope of normal and equation of normal. Also explain the relationship between slope of tangent and normal.
Equation of Normal (Deleted)
Equation of the Normal at (x1, y1) is given by
y – y1 = m(x - x1) where m is the slope of the normal
Method of Approximation (Deleted)
Teacher will explain the method of approximation, approximate change in surface area and volumes and approximate error. Explain the implementation of the formula used in the method of approximation in different problems.
f(x + Δx) = f(x) + f '(x) x Δx
Maximum Value and Minimum Value
First of all teacher will provide the definition of maxima and minima and then explain the definition by giving examples. Teacher will explain the concept of maxima and minima graphically and geometrically so that a clear picture will be developed in the mind of the students.
Definition of Maximum Value:
Let f(x) be a real function defined on an interval I. Then f(x) is said to have the maximum value in I, if there exist a point c in I such that f(x) ≤ f(c) for all x ϵ I
In this case f(c) is called the maximum value
Definition of Minimum Value :
Let f(x) be a real function defined on an interval I. Then f(x) is said to have the maximum value in I, if there exist a point c in I such that f(x) ≥ f(c) for all x ϵ I
In this case f(c) is called the minimum value
Critical Points
Now explain the concept of Critical points, local maxima and local minima and its geometrical meaning and different methods of finding local maxima and local minima as given below.
Explain the Method of finding maxima and minima by using first derivative test.
Algorithms used in First Derivative Test
i) Put y = f(x)
ii) Find dy/dx and put dy/dx = 0 to find the critical points.
iii) Place all the critical points on the horizontal line and write signs ( +ve or -ve) as in baby curve method.
iv) The critical point at which dy/dx changes its sign from positive to negative is called the point of local maximum and the function have maximum value at this point.
v) The critical point at which dy/dx changes its sign from negative (- ve) to positive (+ ve) is called the point of local minimum and the function have local minimum value at this point.
vi) The critical point at which dy/dx does not change its sign is called the point of inflexion.
Explain the Method of finding maxima and minima by using second derivative test.
Algorithms used in second derivative test
i) Put y = f(x)
ii) Find f '(x) and put f '(x) = 0 to find the critical points.
iii) Find f ''(x) and find the value of f''(x) at the critical points.
iv) If f ''(c) < 0 , then f(x) has local maximum value at x = c
v) If f ''(c) > 0, then f(x) has local minimum value at x = c
vi) If f '' (c) = 0 then test fail.
Now explain the concept of Absolute maximum value and absolute minimum value and the method of finding these.
Explain the Implementations of Concept of maxima and minima in daily life word problems.
REFLECTION OF ACTIVITY
1) Rate of change of quantities.
2) Increasing, Decreasing,
strictly increasing and strictly decreasing of functions.
3) Method of finding the maxima
and minima of functions.
4) Local maxima and local minima
of functions.
5) Absolute maxima and absolute
minima of functions.
Teacher will ask few questions related to the above topics and note the reflection of the students.
IMMEDIATE FEEDBACK
After completing the above
activities students will be able to
1) Identify the rate of change
of different quantities with respect to different variables.
2) Identify different methods of
finding the increasing, decreasing, strictly increasing and strictly decreasing
of functions.
3) Able to find the maximum value,
minimum value, local maximum value, local minimum value, absolute maximum value
and absolute minimum value.
4) Understand the difference
between the first derivative and second derivative test.
nice
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