Week6n7 Applications of Derivative.pptx
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The document explains the concept of derivatives and their importance in understanding the behavior of functions, including identifying stationary points, local maxima and minima, and concavity. It describes how derivatives provide insights into the slope of functions, whether they are increasing or decreasing, and details methods to find and analyze these critical points through examples. Additionally, it highlights the role of the second derivative in determining points of inflection and concavity.
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Week6n7 Applications of Derivative.pptx
- 2. What does the Derivative tell us? The Derivative of a function () gives us another function () which can be referred as SLOPE FUNCTION For Example if = 1 3 3 , ≠ = 2 . () = 2 is itself a function but it tells us the slope of tangent lines to the curve of function at any point .
- 3. What does the Derivative tell us? Let us demonstrate it on a program made on GeoGebra https://www.geogebra.org/m/BDYnGhbt
- 4. What does Derivative tell us? Its tells us about the SLOPE of the Function. However, Slope (rate of change of function) tells us a lot of things: 1. When a function is increasing or decreasing, 2. How rapidly its increasing or decreasing, 3. Or when a function is not changing at all.. 4. Or when it changes its behavior. 5. What are the highest/lowest points of a function
- 6. Using Derivative When is a function not changing? when = 0 = 0 means is not changing w.r.t Such points 0, 0 where 0 = 0 are known as STATIONARY POINTS
- 7. Stationary Points A train station is a railway facility where trains stop to load or unload passengers.
- 8. Stationary Points Similarly, the points where the function momentarily stop or where = 0 are called Stationary Points of the curve.
- 9. Critical Points Stationary points are all critical points. But all critical points are not stationary points. (See definition)
- 10. Not all Critical Points are Stationary Points But we will mostly deal with critical points that are stationary points. We will discuss SMOOTH CURVES
- 11. Why are Stationary(critical points) important? Observe at stationary points, a curve takes a turn.. A U-turn. The slope of the function changes direction. These points are also local MAXIMA or local MINIMA of the curve.
- 12. Local Maxima/Minima Points- Extreme Points
- 13. How to identify whether a stationary point is Maxima or Minima? Two Ways: But lets discuss the first way:
- 14. Example 1 Let = 3 92 48 + 52 defined over the interval (, +) 1. Find the stationary point of the curve. 2. In which intervals of the domain is the function increasing and/or decreasing. 3. Which of the stationary point is the local maxima and minima? 4. Sketch the graph of the function, by using the answers above. Also find the stationary values( function value) at maxima and minima point.
- 15. Example 1 Solution: On white-board
- 16. Do it yourself
- 17. Activity A brochure for a roller coaster says that, for the first 10 seconds of the ride, the height of the coaster can be determined by () = 1 3 3 52 + 21, where t is the time in seconds and h is the height in feet. What was the height of the coaster at = seconds? At what time instant(s) during the first 10 seconds , the coaster momentarily stopped? At which intervals, the coaster was going up and when was it coming down? Calculate the maximum and minimum height the rollercoaster can reach during the first 10 seconds.
- 18. Second Derivative For the function y = Slope Function dy dx = : Tells the rate of change of function Where Function is increasing/decreasing. 2 2 = 霞(): Tell us the rate of change of slopes Where Slopes is increasing/ decreasing. Or in other words, it tells us about CONCAVITY of the graph
- 19. Concavity Observe! At Maxima, the graph is concave down At Minima, the graph is concave up.
- 20. How to identify a Stationary point is Maxima/Minima
- 21. Inflection Point
- 22. Inflection Point IMPORTANT! The Inflection point is the point where most rapid change occurs: Most rapid increases or decrease MOST RAPID INCREASE MOST RAPID DECREASE
- 23. Example 1 Let = 3 92 48 + 52 defined over the interval (, +) 1. Find the stationary point of the curve. 2. In which intervals of the domain is the function increasing and/or decreasing. 3. Which of the stationary point is the local maxima and minima? 4. Sketch the graph of the function, by using the answers above. Also find the stationary values( function value) at maxima and minima point. 5. Redo part 3, by using 2nd derivative test. 6. Find the inflection point.
- 24. Example 2 (c) With help of above answer, try to sketch the graph of the function over the interval 0, 4 .
- 29. Exercise
- 31. 55. (A past paper question) Solution: Lets try to understand the question, it asks for value of where it has greatest slope of tangent line. At inflection point, the graph of the function shows most rapid increase or decrease. Most rapid increase means that the tangent line has the greatest slope at that point. Therefore, to find this point, we find inflection point. At inflection point 霞 () = .
- 33. To find inflection point, we know, 2 2 = 0
- 34. Its not over yet! We got two values of . From these two values, which value of corresponds to the tangent line with greatest slope? Because these values might include a value of that gives least slope(a negative slope) !!! The first derivative could help us! The first derivative tells us about slope, whether its negative or positive. If the first derivative at gives a positive value, then its the point of where there is positive slope and vice versa.
- 35. 55. Solution(cont.) 0.577 < 0 Therefore at = . , the tangent line has negative slope and since it is a inflection point, then it will have the least slope(most rapid decrease) at this point. 0.577 > 0 Therefore at = . , the tangent line has positive slope and since it is a inflection point, then it will have the greatest slope(most rapid increase) at this point. Our Answer: =
- 37. Good Luck for Mids! For Queries: Tuesdays/Thursdays: 9-12 noon saba.naeem@ucp.edu.pk