Practicle application of maxima and minima
26 likes42,799 views
The document discusses maxima and minima, which are the highest and lowest values that a function can reach over a closed interval. It provides examples of finding maxima and minima by taking derivatives and setting them equal to zero. Practical applications are given in chemistry, physics, economics, meteorology, theme park revenue modeling, and space shuttle engineering where maxima and minima help optimize values like temperature, revenue, and pressure resistance.
Practicle application of maxima and minima
- 1. CALCULUS PRACTICLE APPLICATION OF MAXIMA AND MINIMA
- 3. GROUP MEMBERS HAMZA SHAUKAT I.A.S DEPARTMENT PUNJAB UNIVERSITY LAHORE PAKISTAN
- 5. MAXIMA AND MINIMA DEFINITION : The maximum of a function is the highest value that it reaches over a closed interval. Similarly, the minimum of a function is the lowest value that it reaches over a closed interval.
- 7. HOW TO FIND MAXIMA AND MINIMA EXAMPLE Find maxima or minima of following function F (x) = x² - 6x +1 Taking first derivative F (x) = 2x – 6 = 0 2x = 6 X=3 Now take the second derivative F (x) = 2 > 0 So the function is at minima when x =3. After x = 3 function starts increasing so at x = 3 function is at minimum point.
- 9. EXAMPLE NO 1 The profit function of acrosonic company is given by
- 10. SOLUTION
- 11. EXAMPLE NO 2 An economy’s consumer price index (CPI) is described by the function.
- 12. SOLUTION
- 13. EXAMPLE NO 3 Find the dimension of rectangle fence of length 100 ft that maximize the area ?
- 14. SOLUTION
- 15. ANSWER X = 50 ft AND Y = 25 ft
- 16. EXAMPLE NO 4 Find where to cut the wire of the length 12 inches such that the sum of area of the square and circle is minimize ?
- 17. SOLUTION
- 20. PRACTICLE APPLICATION OF MAXIMA AND MINIMA In CHEM , we have used the maxima of wave function and radial probability distribution functions to determine where an electron is most likely to be found in any given orbital. In PHYS, the maximum (or minimum) displacement of a wave is known as its amplitude, and is occasionally found graphically. We have also solved equations to determine the maxima of velocity and acceleration functions for waves, using other physical principles (such as the Law of Conservation of Energy).
- 21. In Economics maxima and minima are used to maximize beneficial values (profit, efficiency, output, etc.) and to minimize negative ones (expenses, effort, etc.). A meteorologist creates a model that predicts temperature variance with respect to time. The absolute maximum and minimum of this function over any 24-hour period are the forecasted high and low temperatures.
- 22. The director of a theme park works with a model of total revenue as a function of admission price. The location of the absolute maximum of this function represents the ideal admission price (i.e., the one that will generate the most revenue). A NASA engineer working on the next generation space shuttle studies a function that computes the pressure acting on the shuttle at a given altitude. The absolute maximum of this function represents the pressure that the shuttle must be designed to sustain.
- 24. NOW QUESTIONS