Mathematics Lab Activity-06 Class XII

Mathematics Laboratory Activities on Applications of Derivatives for class XI Non-Medical students with complete observation tables strictly according to the CBSE syllabus.

Chapter - 06 

applications of derivatives

Activity - 06

Objective

To understand the concepts of decreasing and increasing functions.

Material Required

Piece of wire of different lengths, piece of plywood of suitable size, white paper, adhesive, geometry box, trigonometric tables.

Procedure

1. Take a piece of plywood of convenient size and paste a white paper on it.

2. Take two pieces of wires of length say 20 cm each and fix them on the white paper to represent  x-axis and y- axis.

3. Take two more pieces of wire each of suitable length and bend them in the shape of curves representing two functions and fix them on the paper as shown in figure 6.1.

4. Take two straight wires each of suitable length for the purpose of showing tangents to the curves at different points on them.

5. Take one straight wire and place it on the curve (on the left) such that it is tangent to the curve at the point say P₁ and making an angle 𝛂₁ with the positive direction of x – axis.

6. As 𝛂₁ is an obtuse angle, so tan 𝛂₁ is negative, i.e. the slope of the tangent at P₁ is negative.

Figure 6.1

7. Take another two points say P₂ and P₃ on the same curve, and make tangents, using the same wire, at P₂ and P₃ making angles 𝛂₂ and 𝛂₃ , respectively with the positive direction of x – axis.

8. Here again 𝛂₂ and 𝛂₃ are obtuse angles and therefore slopes of the tangents tan𝛂₂ and tan𝛂₃ are both negative i.e., derivatives of the function at P₂ and P₃ are negative.

9. The function given by the curve (on the left hand side) is a decreasing function.

10. On the right hand side of the curve, take three points Q₁, Q₂ , Q₃ , and using the other straight wires, form tangents at each of these points making angles β₁ , β₂ , β₃ , respectively with the positive direction  of x – axis, as shown in the figure 6.1

11. Here β₁ , β₂ , β₃  are all acute angles so the values of tan β_1, tan β_{2 ,} tan β_{3 all are  positive.}

12. So, the derivatives of the function at these points are positive. Thus, the function given by this curve is an increasing function.

Observations

1.) 𝛂₁  = 130^o  > 90^o ⇒ tan𝛂₁ = tan130^o  (negative)  ,

      𝛂₂  = 120^o  > 90^o ⇒ tan𝛂₂ = tan120^o  (negative),

       𝛂₃  = 110^o  > 90^o ⇒ tan𝛂₃ = tan110^o  (negative)

2.)  β₁  = 60^o < 90^o ⇒ tanβ₁  = tan60^o   (Positive), 

      β₂  = 70^o  < 90^o ⇒ tanβ₂  = tan70^o   (Positive), 

      β₃  = 80^o  < 90^o ⇒ tanβ₃  = tan80^o   (Positive).

Result

Slope of the tangent is negative to the left hand side ⇒ curve is decreasing to the left side. 

Slope of the tangent is positive to the right hand side ⇒ curve is increasing to the right side.

Applications

This activity is very useful in explaining the concept of increasing and decreasing of a function.

VIVA – VOICE

Q1. What is the monotonically increasing function ?
Ans. A function f(x) is said to be monotonically increasing on [a, b] if the value of f(x) increasing or decreasing with increase or decrease in x .

Q2. What is monotonic decreasing function ?

Ans. A function f(x) is said to be monotonically increasing on [a, b] if the value of f(x) increasing or decreasing with decrease or increase in x .

Q3. Is f(x) increasing function for x₁ < x₂ ⇒ f(x₁) ≥  f(x₂) ?

Ans. No, it is a decreasing function.

Q4. Is f(x) decreasing function for x₁ < x₂ ⇒ f(x₁) ≤  f(x₂) ?

Ans. No,  it is an increasing function.

Q5. When a function f(x) is said to be strictly increasing function ?

Ans. A function is said to be strictly increasing on (a, b) if x₁ < x₂ ⇒ f(x₁) <  f(x₂) 

Q6. When a function f(x) is said to be strictly decreasing function ?

Ans. A function is said to be strictly decreasing on (a, b) if x₁ < x₂ ⇒ f(x₁) >  f(x₂) 

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