11X1 T09 02 first principles (2011)

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  • 1. The Slope of a Tangent to a Curve

2. The Slope of a Tangent to a Curvey y f xx 3. The Slope of a Tangent to a Curve y y f x Px k 4. The Slope of a Tangent to a Curve y y f x Q P x k 5. The Slope of a Tangent to a Curve y y f xSlope PQ is an estimate Qfor the slope of line k. P x k 6. The Slope of a Tangent to a Curve y y f xSlope PQ is an estimate Qfor the slope of line k. PQ: Where do we position Q x to get the best estimate? k 7. The Slope of a Tangent to a Curve y y f xSlope PQ is an estimate Qfor the slope of line k. PQ: Where do we position Q x to get the best estimate?A: As close to P as possible. k 8. The Slope of a Tangent to a Curve y y f xSlope PQ is an estimate Qfor the slope of line k. PQ: Where do we position Q x to get the best estimate?A: As close to P as possible. k QP 9. The Slope of a Tangent to a Curve y y f xSlope PQ is an estimate Qfor the slope of line k. PQ: Where do we position Q x to get the best estimate?A: As close to P as possible. kQP x, f x 10. The Slope of a Tangent to a Curve y y f xSlope PQ is an estimate Qfor the slope of line k. PQ: Where do we position Q x to get the best estimate?A: As close to P as possible. k x h, f x h Q P x, f x 11. The Slope of a Tangent to a Curvey y f xSlope PQ is an estimate Qfor the slope of line k. PQ: Where do we position Qxto get the best estimate?A: As close to P as possible.k x h, f x h f x h f xQmPQ xhxf x h f x h P x, f x 12. The Slope of a Tangent to a Curvey y f xSlope PQ is an estimate Qfor the slope of line k. PQ: Where do we position Q x to get the best estimate?A: As close to P as possible.k x h, f x h f x h f xQmPQ xhxf x h f x hTo find the exact value of the slope of k, we P x, f x calculate the limit of the slope PQ as h getscloser to 0. 13. f x h f xslope of tangent = lim h0h 14. f x h f x slope of tangent = limh0hThis is known as the derivative of y with respect to x and issymbolised; 15. f x h f x slope of tangent = limh0hThis is known as the derivative of y with respect to x and issymbolised; dydx 16. f x h f x slope of tangent = limh0hThis is known as the derivative of y with respect to x and issymbolised; dy , ydx 17. f x h f x slope of tangent = limh0hThis is known as the derivative of y with respect to x and issymbolised; dy , y , f x dx 18. f x h f x slope of tangent = limh0hThis is known as the derivative of y with respect to x and issymbolised; dy , y , f x , d f x dx dx 19. f x h f xslope of tangent = lim h0hThis is known as the derivative of y with respect to x and issymbolised; dy , y , f x , d f x the derivativedx dxmeasures the rate of something changing 20. f x h f x slope of tangent = limh0hThis is known as the derivative of y with respect to x and issymbolised; dy , y , f x , d f x the derivativedx dxmeasures the ratef x h f x of somethingf x limh0hchanging 21. f x h f x slope of tangent = limh0hThis is known as the derivative of y with respect to x and issymbolised; dy , y , f x , d f x the derivativedx dxmeasures the ratef x h f x of somethingf x limh0hchangingThe process is called differentiating from first principles 22. f x h f x slope of tangent = limh0hThis is known as the derivative of y with respect to x and issymbolised; dy , y , f x , d f x the derivativedx dxmeasures the ratef x h f x of somethingf x limh0hchangingThe process is called differentiating from first principlese.g. i Differentiate y 6 x 1 by using first principles. 23. f x h f x slope of tangent = limh0hThis is known as the derivative of y with respect to x and issymbolised; dy , y , f x , d f x the derivativedx dxmeasures the ratef x h f x of somethingf x limh0hchangingThe process is called differentiating from first principlese.g. i Differentiate y 6 x 1 by using first principles.f x 6x 1 24. f x h f x slope of tangent = limh0hThis is known as the derivative of y with respect to x and issymbolised; dy , y , f x , d f x the derivativedx dxmeasures the ratef x h f x of somethingf x limh0hchangingThe process is called differentiating from first principlese.g. i Differentiate y 6 x 1 by using first principles. f x 6x 1f x h 6 x h 1 6 x 6h 1 25. f x h f xslope of tangent = lim h0hThis is known as the derivative of y with respect to x and issymbolised; dy , y , f x , d f x the derivativedx dxmeasures the ratef x h f x of somethingf x limh0hchanging The process is called differentiating from first principlese.g. i Differentiate y 6 x 1 by using first principles.dy f x h f x f x 6x 1 lim dx h0 hf x h 6 x h 1 6 x 6h 1 26. f x h f x slope of tangent = limh0hThis is known as the derivative of y with respect to x and issymbolised; dy , y , f x , d f x the derivativedx dxmeasures the ratef x h f x of somethingf x limh0hchanging The process is called differentiating from first principlese.g. i Differentiate y 6 x 1 by using first principles.dy f x h f x f x 6x 1 lim dx h0 h6 x 6h 1 6 x 1f x h 6 x h 1 limh0 h 6 x 6h 1 27. f x h f x slope of tangent = limh0hThis is known as the derivative of y with respect to x and issymbolised; dy , y , f x , d f x the derivativedx dxmeasures the ratef x h f x of somethingf x limh0hchanging The process is called differentiating from first principlese.g. i Differentiate y 6 x 1 by using first principles.dyf x h f x f x 6x 1 lim dx h0h 6 x 6h 1 6 x 1f x h 6 x h 1 lim h0 h 6 x 6h 1 lim 6h h 0 h 28. f x h f x slope of tangent = limh0hThis is known as the derivative of y with respect to x and issymbolised; dy , y , f x , d f x the derivativedx dxmeasures the ratef x h f x of somethingf x limh0hchanging The process is called differentiating from first principlese.g. i Differentiate y 6 x 1 by using first principles.dyf x h f x f x 6x 1 lim dx h0h 6 x 6h 1 6 x 1f x h 6 x h 1 lim h0 h 6 x 6h 1 lim 6h h 0 h lim 6h0 29. f x h f x slope of tangent = limh0hThis is known as the derivative of y with respect to x and issymbolised; dy , y , f x , d f x the derivativedx dxmeasures the ratef x h f x of somethingf x limh0hchanging The process is called differentiating from first principlese.g. i Differentiate y 6 x 1 by using first principles.dyf x h f x f x 6x 1 lim dx h0h 6 x 6h 1 6 x 1f x h 6 x h 1 lim h0 h 6 x 6h 1 lim 6h h 0 h lim 6h06 30. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 . 31. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 . f x x2 5x 2 32. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 . f x x2 5x 2f x h x h 5 x h 22 33. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 . f x x2 5x 2f x h x h 5 x h 22 x 2 2 xh h 2 5 x 5h 2 34. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 . f x x2 5x 2f x h x h 5 x h 22 x 2 2 xh h 2 5 x 5h 2 dy f x h f x lim dx h0h 35. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 . f x x2 5x 2f x h x h 5 x h 22 x 2 2 xh h 2 5 x 5h 2 dy f x h f x lim dx h0hx 2 2 xh h 2 5 x 5h 2 x 2 5 x 2 limh0h 36. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 . f x x2 5x 2f x h x h 5 x h 22 x 2 2 xh h 2 5 x 5h 2 dyf x h f x lim dx h0 h x 2 2 xh h 2 5 x 5h 2 x 2 5 x 2 limh0 h 2 xh h 5h2 lim h0h 37. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 . f x x2 5x 2f x h x h 5 x h 22 x 2 2 xh h 2 5 x 5h 2 dyf x h f x lim dx h0 h x 2 2 xh h 2 5 x 5h 2 x 2 5 x 2 limh0 h 2 xh h 5h2 lim h0h lim 2 x h 5 h0 38. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 . f x x2 5x 2f x h x h 5 x h 22 x 2 2 xh h 2 5 x 5h 2 dyf x h f x lim dx h0 h x 2 2 xh h 2 5 x 5h 2 x 2 5 x 2 limh0 h 2 xh h 5h2 lim h0h lim 2 x h 5 h0 2x 5 39. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 . f x x2 5x 2f x h x h 5 x h 22 x 2 2 xh h 2 5 x 5h 2 dyf x h f x lim dx h0 h x 2 2 xh h 2 5 x 5h 2 x 2 5 x 2 limh0 h 2 xh h 5h2 lim h0h lim 2 x h 5 h0 2x 5 dy when x 1, 2 1 5 dx 3 40. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 . f x x2 5x 2f x h x h 5 x h 22 x 2 2 xh h 2 5 x 5h 2 dyf x h f x lim dx h0 h x 2 2 xh h 2 5 x 5h 2 x 2 5 x 2 limh0 h 2 xh h 5h2 lim h0h lim 2 x h 5 h0 2x 5 dy when x 1, 2 1 5 dx 3 the slope of the tangent at 1, 2 is 3 41. y 2 3 x 1 42. y 2 3 x 1y 2 3 x 3y 3 x 1 43. y 2 3 x 1y 2 3 x 3y 3 x 1Exercise 7B; 1, 2adgi, 3(not iv), 4,7ab i,v, 12 (just h approaches 0)