Lecture one Quadratic Maxima and Minima (1).pptx

Quadratic Functions

1. Quadratic Functions The quadratic function is of the form The quadratic equation has a solution given by  

2. To plot a quadratic or a polynomial function on an interval [ ], w e proceed as follows; Find when , Find when . D etermine the turning points, stationary point / critical points of the curve. At the Turning Point, and find values of . Or find values of for which  

3. 4. Determine the nature of the turning point To do this find I , then TP is a minimum TP. If , then TP is a maximum TP. If , then we have a point of inflection or its indeterminate. After determining the turning points we can now plot the graph. Go to example at the back of this staple.  

4. Example 2 Plot the graph of the function . when , when is a point on the curve When  

5. At the turning point When So turning point is (-2, -3). . Hence minimum TP.  

6. Example 3 When At the TP so TP  

7. Graphs of Quadratic functions Nature of TP minimum TP When  

8. P When When , or  

9. Graphs of Polynomials Polynomial Functions Plot the graph of When  

10. Graphs of Polynomials So TP when When  

11. Graphs of Polynomials TP are and Nature of TP When When  

12. Polynomials Example 2. Determine the coordinates of the TP of When  

13. Graphs of Polynomials  

14. Graphs of Polynomials When Now When  

15. Graphs of Polynomials When is a point on the curve. Nature of TP  

16. Examples a. Find the stationary points on the graph of the polynomial function Distinguish between the stationary points and sketch the graph of the function in the domain ( Solution Given , At the stationary points , that is . or To determine nature of stationary points we examine the values of for . When , .. Hence is a minimum stationary point .  

17. When , . Hence is a maximum stationary point. When , . Hence minimum stationary point is When , . Hence maximum stationary point is . To sketch the curve we find value of at . when , Hence is a point on the curve.  

18. Applications of maxima minima Example 1 A rectangular structure is to be constructed such that it is long and wide. If you are required to cut out identical squares at the four corners of the structure, assuming that the height of the structure equals the length of the corners to be cut out, find out how much you need to cut so as to give the maximum volume of the structure. Find out the dimensions of the structure. (Length of each corner cannot be more than )  

19. Let be the width of the of the square to be cut out of the corners and be the volume of the resulting structure, then For maximum volume or But and so . For maximum volume , Hence Length of room = Breath of room = Height of room =  

20. Example 2 A construction firm estimates that the cost of operating a truck for aggregates is a function of the speed at which the truck is driven. If the operating cost is estimated to be cedis/mile when the truck is driven at a speed of miles/hr., and the driver is paid 1400 cedis /hr., what speed should the truck be driven to minimize the cost of delivery to a city miles away.  

21. Let be the total cost of driving the truck for miles. Then + For minimum cost, A speed of 53miles per hour will give minimum operating cost.  

22. Example 3 The payload of a truck delivering aggregates to a site is defined by Where is the length of chassis of the truck. Show that for maximum payload the chassis length must equal  

23. For maximum power  

24. implies