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# Application of differentiation

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This presentation explains how the differentiation is applied to identify increasing and decreasing functions,identifying the nature of stationary points and also finding maximum or minimum values.

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### Application of differentiation

1. 1. APPLICATION OF DIFFERENTIATIONINCREASING AND DECREASING FUNCTION MINIMUM & MAXIMUM VALUES RATE OF CHANGE
2. 2. Increasing & Decreasing function 2 ND D I F F E R E N T I A T I O N
3. 3. Determine set values of x in which the function is increasing and decreasing y 40 20 x -6 -4 -2 2 4 -20 -40 -60 -80The function decreases whenThe function increases when
4. 4. The nature of stationary point 2 ND D I F F E R E N T I A T I O N
5. 5. 10 y Find the point on the curve when8 its tangent line has a gradient of 0. 6 4 2 x-10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Stationary point is a point where its tangent line is either horizontal or vertical. How is this related to 2nd differentiation?
6. 6. 10 y 8 6 4 2 x-10 -8 -6 -4 -2 2 4 6 8 10 -2 Find the point on the curve when its -4 tangent line has a gradient of 0. -6 -8 -10 How is this related to 2nd differentiation?
7. 7. Find the point on the curve when its y tangent line has a gradient of 0. 5.5 5 4.5 4 3.5 3 2.5 2 1.5 x-2 -1.5 -1 -0.5 1 0.5 1 1.5 2 What is the nature of this point? This point is neither maximum nor minimum point and its called STATIONARY POINT OF INFLEXION
8. 8. How do we apply these concepts?Find the coordinates of the stationary points on the curvey = x3  3x + 2 and determine the nature of these points.Hence, sketch the graph of y = x3  3x + 2 and determine the setvalues of x in which the function increases and decreases. What are the strategies to solve this question?
9. 9. 5 y 4 3 2 1–6 –4 –2 2 –1 –2 –3
10. 10. How do we apply these concepts to solve real-life problems?An open tank with a square base is to be made from a thinsheet of metal. Find the length of the square base and theheight of the tank so that the least amount of metal is used tomake a tank with a capacity of 8 m3. What are the strategies to solve h this question? x x • Derive a function from surface area and/ or volume area. • Express the function in one single term (x) • Use the function to identify maximum or minimum value.
11. 11. An open tank with a square base is to be made from athin sheet of metal. Find the length of the square baseand the height of the tank so that the least amount ofmetal is used to make a tank with a capacity of 8 m3. hThe Volume shows relationship between xthe height (h) and length (x) of the tank x Since the amount of the metal needed depends on the surface area of the tank, the area of metal needed is Express S in terms of x
12. 12. Rate of Change CHAIN RULE
13. 13. What the symbol meansA radius of a circleincreases at a rate of0.2 cm/ sA water drops at arate of 0.5 cm3/ sThe side of a metalcube expands at arate of 0.0013 mm/ s
14. 14. The radius of a circle increases at a rate of 3 cms-1. Find therate increase of the area whena) the radius is 5 cm, b) the area is 4π cm2 Apply CHAIN RULE
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