1 / 15

Rate of change / Differentiation (3)

Learn about the rate of change and differentiation through finding gradients, equations of tangents, and equations of normals. Practice solving problems to improve your understanding.

skay
Download Presentation

Rate of change / Differentiation (3)

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. Rate of change / Differentiation (3) Differentiating Using differentiating

  2. The Key Bit dy dx dy dx dy dx dy dx = nxn-1 dy dx E.g. if y = 5x4 = 5 x 4x3 = 20x3 E.g. if y = x2 = 2x E.g. if y = x3 = 3x2 The general rule (very important) is :- If y = xn

  3. Warm-up dy dx dy dx dy dx dy dx dy dx dy dx Find for these functions :- Gradient at x=-2 = 4x = 10x4 = 10x + 10 = 3x2 + 2x +1 = 8x3 - 8x = -8 = 10x(-2)4=160 = -20 + 10 = -10 = 3(-2)2+2(-2)+1 = 12 -4 +1 = 9 = 8(-2)3 -8(-2) = 8 x-8 – 8 x-2 = -64 +16 = -48 y= 2x2 y = 2x5 y = 5x2 + 10x + 5 y = x3 + x2 + x y = 2x4 - 4x2 + 7

  4. dy dx dy dx = 3ax2 + 8x -12 = 3a + 8 – 12 = 2 When x=1 A differentiating Problem The gradient of y = ax3 + 4x2 – 12x is 2 when x=1 What is a? 3a - 4 = 2 3a = 6 a = 2

  5. dy dx dy dx = 12x2 – 2ax + 10 = 12 -2a +10 = 6 When x=1 Try this The gradient of y = 4x3 - ax2 + 10x is 6 when x=1 What is a? 22 - 2a = 6 16 = 2a a = 8

  6. Rate of change / Differentiation (3 pt2) Equations of tangents Equations of normals

  7. y = x3 – 12x We have seen: if dy dx dy dx = 3x2 - 12 Function Notation Instead of ‘y’ we may use the function notation f(x) f(x)= x3 – 12x If then is replace by f’(x) f’(x)= 3x2 - 12 so f’(x) represent the differential/gradient function

  8. dy dx Increase in y Gradient = Increase in x Linear graphs y - intercept Gradient m and c will always be numbers in your examples y = 5x + 7 y = 2x - 1

  9. Definition Parallel lines are ones with the same slope/gradient. i.e.the number in front of the ‘x’ is the same y = 3x - 11.31 y = 3x + 2/3 y = 3x y = 3x + 8 y = 3x + 84 y = 3x - 3 y = 3x - 21 y = 3x + 1 y = 3x + 43 y = 3x - 1.5

  10. Equations of Tangents The tangent to the curve is the gradient at that point dy dx dy dx (3,9) When x=3; = 2x3 = 6 = 2x y=x2 y x What is the equation of the tangent? y=x2 y = mx + c Substitute gradient: 9 = 6x3 + c c = 9 - 18 = -9 y = 6x - 9

  11. Equations of Tangents - try me dy dx dy dx When x=1; = 6+4 = 10 = 6x + 4 y y=3x2 + 4x + 1 - differentiate - gradient at x =1 - y = mx + c - find c (1,8) x What is the equation of the tangent? y=3x2 + 4x + 1 y = mx + c Substitute gradient: 8 = 10x1 + c c = 8 - 10 = -2 y = 10x - 2

  12. Perpendicular Lines If two lines with gradients m1 and m2 are perpendicular, then:

  13. Equations of Normals The normal is always perpendicular to the tangent normal dy dx mT x mN = -1 When x=3; = 2x3 = 6 y=x2 y (3,9) x What is the equation of the normal? y = mx + c Substitute gradient: 9 = -1/6x 3 + c c = 9 - -3/6 =9 + 1/2 y = -1/6 x + 9 1/2 mT x mN = -1 6 x mN = -1 mN = -1/6

  14. Equations of Normals The normal is always perpendicular to the tangent normal dy dx mT x mN = -1 When x=3; = 2x3 = 6 y=x2 y (3,9) x What is the equation of the normal? y = mx + c Substitute gradient: 9 = -1/6x 3 + c c = 9 + 3/6 =9 + 1/2 y = -1/6 x + 9 1/2 mT x mN = -1 6 x mN = -1 mN = -1/6

  15. Equations of Normal - try me dy dx dy dx When x=1; = 6+4 = 10 = 6x + 4 y y=3x2 + 4x + 1 - differentiate - gradient at x =1 - mt x mn = -1 - y = mnx + c - find c (1,8) normal x What is the equation of the normal? y = mnx + c Substitute gradient: 8 = -1/10x 1 + c c = 8 + 1/10 =8 1/10 y = -1/10 x + 8 1/10 mT x mN = -1 10 x mN = -1 mN = -1/10

More Related