1 / 23

Looking closely at a function

Looking closely at a function. Chapter 9. Quick Review of Parameterization. y = 3x 2 + 4x + 5 Point P(6,2) x = 6 + u y = 2 + v 2 + v = 3 ( 6 + u ) 2 + 4( 6 + u ) + 5 2 + v = 3 ( 36 +12u + u 2 ) + 24 + 4 u + 5

clarke-chan
Télécharger la présentation

Looking closely at a function

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. Looking closely at a function Chapter 9

  2. Quick Review of Parameterization y = 3x2 + 4x + 5 Point P(6,2) x = 6 + u y = 2 + v 2 + v = 3 (6 + u)2 + 4(6 + u) + 5 2 + v = 3 (36 +12u + u2) + 24 + 4u + 5 2 + v = 108 +36u + 3u2+ 24 + 4u + 5 2 + v = 108 +36u+ 3u2+ 24 +4u + 5 v = 135 + 40u + 3u2 x -6 = u y - 2 = v y – 2 = 135 + 40 (x - 6) + 3 (x2 – 12x + 36) ....

  3. Linear Approximation using Polynomial Parameterization (1,2) y = 4x2 - 2

  4. y = 4x2 – 2 (1,2) • What do we do first parameterize the function?

  5. Now how about a circle… • x2 + y2 = 25 • What is the equation of the tangent line at (3,4)

  6. Now how about a circle… • x2 + y2 = 25 (3,4)

  7. Writing the equation of a line… • What do you need to know? • Then what do you do?

  8. Find the tangent line when x = -2 • y = 2x2 + 3x + 1

  9. Find the tangent line when x = -2 • y = 2x2 + 3x + 1 if x = -2 • y = 2(-2)2 + 3(-2) + 1 • y = 8 – 6 + 1 • y = 3 (-2, 3) x = u – 2 y = v + 3 • v + 3 = 2(u - 2)2 + 3(u – 2) + 1 • v + 3 = 2u2 – 8u + 8 + 3u – 6 + 1 • v = -5u + 2u2 • y – 3 = -5 (x + 2) • y = -5x - 7

  10. y • As indicated in the diagram (which is not to scale) the tangent line to the graph of f(x) = x2+5x-24 at x = 12 meets the x-axis at an angle AOB whose tangent is . The angle AOB measures radians.

  11. How can you find angle measures?

  12. Use trigonometry... • What do you need to use trig? • How can you find it given this situation?

  13. Looking at the tangent line... • f(x) = x2+5x-24 • at x = 12 • What information can you find using this...

  14. Looking at the tangent line... • f(x) = x2+5x-24 • at x = 12 • Find the tangent line

  15. Looking at the tangent line... • f(12) = x2+5x-24 • f(12) = (12)2+5(12)-24 • f(12) = 144 + 70 – 24 • f(12) = 190 • A(12,190)

  16. Using parameters... f(x) = x2+5x-24 A(12,190) x=12+u y=190+v • 190+v= (12+u)2+5(12+u)-24 • 190+v=144+24u+u2+70+5u-24 • v-29u-u2=0 x - 12=u y- 190=v

  17. Using parameters... • v-29u-u2=0 • y – 190 – 29(x-12) = 0 • y – 190 – 29x + 348=0 • y = 29x - 158 x-12=u y-190=v

  18. Using parameters... • v-29u-u2=0 • y – 190 – 29(x-12) = 0 • y – 190 – 29x + 348=0 • y = 29x - 158 - 158 x-12=u y-190=v

  19. Looking at the tangent line... Diagram is not to scale. Now what do you know? (12,190) - 158

  20. Looking at the tangent line... Diagram is not to scale. (12,190) 348 - 158 12

  21. Looking at the tangent line... Diagram is not to scale. Now you have 2 sides – you can find the desired angle. (12,190) 348 - 158 12

  22. Looking at the tangent line... Diagram is not to scale. tan q = 348/12 (must find degree to change to radians) tan q = tan-1(348/12) . . . (12,190) 348 - 158 12

  23. Another way to find the tangent line? • Derivative? • What is the Derivative?

More Related