Application of differentiation
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The document discusses the application of differentiation in analyzing increasing and decreasing functions, as well as finding stationary points and their nature using second differentiation. It includes methods for solving real-life problems, such as optimizing the design of an open tank with a square base to minimize metal usage while maintaining a specific volume. Additionally, it covers the use of the chain rule to find rates of change in various scenarios involving geometric figures.
Application of differentiation
- 1. APPLICATION OF DIFFERENTIATION INCREASING AND DECREASING FUNCTION MINIMUM & MAXIMUM VALUES RATE OF CHANGE
- 2. Increasing & Decreasing function 2 ND D I F F E R E N T I A T I O N
- 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 -80 The function decreases when The function increases when
- 4. The nature of stationary point 2 ND D I F F E R E N T I A T I O N
- 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. 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. 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. How do we apply these concepts? Find the coordinates of the stationary points on the curve y = x3 3x + 2 and determine the nature of these points. Hence, sketch the graph of y = x3 3x + 2 and determine the set values of x in which the function increases and decreases. What are the strategies to solve this question?
- 9. 5 y 4 3 2 1 –6 –4 –2 2 –1 –2 –3
- 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 thin sheet of metal. Find the length of the square base and the height of the tank so that the least amount of metal is used to make 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. An open tank with a square base is to be made from a thin sheet of metal. Find the length of the square base and the height of the tank so that the least amount of metal is used to make a tank with a capacity of 8 m3. h The Volume shows relationship between x the 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. Rate of Change CHAIN RULE
- 13. What the symbol means A radius of a circle increases at a rate of 0.2 cm/ s A water drops at a rate of 0.5 cm3/ s The side of a metal cube expands at a rate of 0.0013 mm/ s
- 14. The radius of a circle increases at a rate of 3 cms-1. Find the rate increase of the area when a) the radius is 5 cm, b) the area is 4π cm2 Apply CHAIN RULE