1 / 27

Understanding Stationary Points and Their Nature: A Comprehensive Guide

Learn how to identify, find, and analyze stationary points in calculus. Discover the difference between stationary points and values, and master differentiation to determine the nature of these critical points.

wwoodford
Download Presentation

Understanding Stationary Points and Their Nature: A Comprehensive Guide

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Stationary Points

  2. What is to be learned • How to use differentiation to identify SPs • What SPs are!!!!!! • What the difference between a stationary point and a stationary value • What SVs are • How to find SP/SVs • How to finf the nature of SP/SVs

  3. m ~ 3 m ~ ½

  4. m ~ ½ m ~ 3 m ~ ½

  5. m ~ ½ m ~ 3 m ~ ½

  6. m = 0 m ~ ½ m ~ 3 m ~ ½

  7. m = 0 m ~ ½ m ~ 3 m ~ ½

  8. At Turning Points the gradient = 0 Turning points are known as stationary points (or values) (SPs or SVs) So for SPs the derivative = 0

  9. Ex y = x2 -8x + 10 SPs? For SPs dy/dx = 0

  10. Ex y = x2 -8x + 10 SPs? For SPs dy/dx = 0

  11. Ex y = x2 -8x + 10 SPs? For SPs dy/dx = 0 dy/dx = 2x – 8 2x – 8 = 0 2x = 8 x = 4 y?

  12. Ex y = x2 -8x + 10 SPs? For SPs dy/dx = 0 dy/dx = 2x – 8 2x – 8 = 0 2x = 8 x = 4 y? at x = 4 y = 42 – 8(4) + 10 = - 6 SP is (4 , -6)

  13. (0 , 10) and (2 , 2) Ex y = 2x3 -6x2 + 10 SPs? For SPs dy/dx = 0 dy/dx = 6x2 – 12x 6x2 – 12x = 0 6x(x – 2) = 0 6x = 0 or x – 2 = 0 x = 0 or x = 2 x = 0 y = 10 x = 2 y =2(2)3 – 6(2)2 + 10 = 2 Quadratic Equation Factorise

  14. Stationary Points • Max TPs, Min TPs or……. • For SPs gradient = 0 • For SPs dy/dx = 0 Tactics Find derivative Solve equation → x y → sub x into original equation = 0

  15. Ex y = 1/3x3 - 2x2 - 12x SPs? For SPs dy/dx = 0 dy/dx = x2 – 4x – 12 x2 – 4x – 12 = 0 (x – 6)(x + 2) = 0 x–6= 0 or x+2 = 0 x = 6 or x = -2 Quadratic Equation Factorise

  16. x = 6 y = 1/3(6)3 – 2(6)2 – 12(6) = - 72 Ex y = 1/3x3 - 2x2 - 12x SPs? For SPs dy/dx = 0 dy/dx = x2 – 4x – 12 x2 – 4x – 12 = 0 (x – 6)(x + 2) = 8 x–6= 0 or x+2 = 0 x = 6 or x = -2 x = -2 y = 1/3(-2)3 – 2(-2)2 – 12(-2) = 132/3 SPs are (-2 , 132/3) and (6 , -72)

  17. Finding the nature

  18. m negative m positive m = 0 for nature need to know gradient just before and after SP

  19. Making a nature table

  20. Making a Nature Table y = x2 -8x + 10 For SPs dy/dx = 0 dy/dx = 2x – 8 2x – 8 = 0 2x = 8 x = 4 y? at x = 4 y = 42 – 8(4) + 10 = - 6 SP is (4 , -6)

  21. SP is (4 , -6) Making a Nature Table y = x2 -8x + 10 dy/dx = 2x – 8 x 5 3 4 - 0 + dy/dx = 2x - 8 Slope Min TP at (4 , -6)

  22. SPs (0 , 10) and (2 , 2) Making a Nature Table y = 2x3 -6x2 + 10 dy/dx = 6x2 – 12x x 1 3 -1 2 0 + 0 + dy/dx = 6x2 – 12x 0 - Slope Min TP Max TP at (2 , 2) at (0 , 10)

  23. Nature Table • Used to find nature of SPs • Show gradient just before and after SPs • Use derivative

  24. SPs (-2, 132/3) and (6 , -72) Making a Nature Table y = 1/3x3 - 2x2 - 12x dy/dx = x2 – 4x - 12 x 0 7 -3 6 -2 + 0 + dy/dx = x2 – 4x - 12 0 - Slope Min TP Max TP at (6 , -72) at (-2, 132/3)

  25. SPs (-2, 132/3) and (6 , -72) Making a Nature Table y = 1/3x3 - 2x2 - 12x dy/dx = x2 – 4x - 12 x 0 7 -3 6 -2 + 0 + dy/dx = x2 – 4x - 12 0 - Slope Min TP Max TP at (6 , -72) at (-2, 132/3)

  26. SPs (-2, 132/3) and (6 , -72) Making a Nature Table y = 1/3x3 - 2x2 - 12x dy/dx = x2 – 4x - 12 x 0 7 -3 6 -2 - X - 0 dy/dx = (x - 6)(x + 2) 0 - X + + X + = + = - = + Slope Min TP Max TP at (6 , -72) at (-2, 132/3)

  27. SPs (-2, 132/3) and (6 , -72) Making a Nature Table y = 1/3x3 - 2x2 - 12x dy/dx = x2 – 4x - 12 x 0 7 -3 6 -2 * + 0 + dy/dx = x2 – 4x - 12 0 - Slope *could use factorised derivative Min TP Max TP at (6 , -72) at (-2, 132/3)

More Related