# 11x1 t08 02 first principles

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- 1. The Slope of a Tangent to a Curve

2. The Slope of a Tangent to a Curve yy f x x 3. The Slope of a Tangent to a Curveyy f xPx k 4. The Slope of a Tangent to a Curveyy f xQP x k 5. The Slope of a Tangent to a Curveyy f x Slope PQ is an estimateQ for the slope of line k.P x k 6. The Slope of a Tangent to a Curveyy f x Slope PQ is an estimateQ for the slope of line k.PQ: Where do we position Qx to get the best estimate? k 7. The Slope of a Tangent to a Curveyy f x Slope PQ is an estimateQ for the slope of line k.PQ: Where do we position Qx to get the best estimate? A: As close to P as possible.k 8. The Slope of a Tangent to a Curveyy f x Slope PQ is an estimateQ for the slope of line k.PQ: Where do we position Qx to get the best estimate? A: As close to P as possible.k QP 9. The Slope of a Tangent to a Curveyy f x Slope PQ is an estimateQ for the slope of line k.PQ: Where do we position Qx to get the best estimate? A: As close to P as possible.kQP x, f x 10. The Slope of a Tangent to a Curveyy f x Slope PQ is an estimateQ for the slope of line k.PQ: Where do we position Qx 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 Curve yy f x Slope PQ is an estimateQ for the slope of line k.PQ: Where do we position Q xto get the best estimate? A: As close to P as possible. k x h, f x h f x h f xQ mPQ xhx f x h f x hP x, f x 12. The Slope of a Tangent to a Curve yy f x Slope PQ is an estimateQ for the slope of line k.PQ: Where do we position Qx to get the best estimate? A: As close to P as possible. k x h, f x h f x h f xQ mPQ xhx f x h f xh To find the exact value of the slope of k, weP x, f x calculate the limit of the slope PQ as h gets closer to 0. 13. f x h f x slope of tangent = limh0h 14. f x h f xslope of tangent = lim h0h This is known as the derivative of y with respect to x and is symbolised; 15. f x h f xslope of tangent = lim h0h This is known as the derivative of y with respect to x and is symbolised; dy dx 16. f x h f xslope of tangent = lim h0h This is known as the derivative of y with respect to x and is symbolised; dy , y dx 17. f x h f xslope of tangent = lim h0h This is known as the derivative of y with respect to x and is symbolised; dy , y , f x dx 18. f x h f xslope of tangent = lim h0h This is known as the derivative of y with respect to x and is symbolised; dy , y , f x , d f x dx dx 19. f x h f x slope of tangent = limh0h This is known as the derivative of y with respect to x and is symbolised; dy , y , f x , d f x the derivative dx dx measures the rateof somethingchanging 20. f x h f xslope of tangent = lim h0h This is known as the derivative of y with respect to x and is symbolised; dy , y , f x , d f x the derivative dx dx measures the rate f x h f x of something f x lim h0hchanging 21. f x h f xslope of tangent = lim h0h This is known as the derivative of y with respect to x and is symbolised; dy , y , f x , d f x the derivative dx dx measures the rate f x h f x of something f x lim h0hchanging The process is called differentiating from first principles 22. f x h f xslope of tangent = lim h0h This is known as the derivative of y with respect to x and is symbolised; dy , y , f x , d f x the derivative dx dx measures the rate f x h f x of something f x lim h0hchanging The process is called differentiating from first principles e.g. i Differentiate y 6 x 1 by using first principles. 23. f x h f xslope of tangent = lim h0h This is known as the derivative of y with respect to x and is symbolised; dy , y , f x , d f x the derivative dx dx measures the rate f x h f x of something f x lim h0hchanging The process is called differentiating from first principles e.g. i Differentiate y 6 x 1 by using first principles.f x 6x 1 24. f x h f xslope of tangent = lim h0h This is known as the derivative of y with respect to x and is symbolised; dy , y , f x , d f x the derivative dx dx measures the rate f x h f x of something f x lim h0hchanging The process is called differentiating from first principles e.g. i Differentiate y 6 x 1 by using first principles. f x 6x 1 f x h 6 x h 1 6 x 6h 1 25. f x h f x slope of tangent = limh0h This is known as the derivative of y with respect to x and is symbolised; dy , y , f x , d f x the derivative dx dx measures the rate f x h f x of something f x lim h0hchangingThe process is called differentiating from first principles e.g. i Differentiate y 6 x 1 by using first principles. dy f x h f xf x 6x 1 limdx h0 h f x h 6 x h 1 6 x 6h 1 26. f x h f xslope of tangent = lim h0h This is known as the derivative of y with respect to x and is symbolised; dy , y , f x , d f x the derivative dx dx measures the rate f x h f x of something f x lim h0hchangingThe process is called differentiating from first principles e.g. i Differentiate y 6 x 1 by using first principles. dy f x h f xf x 6x 1 limdx h0 h 6 x 6h 1 6 x 1 f x h 6 x h 1 lim h0 h 6 x 6h 1 27. f x h f xslope of tangent = lim h0h This is known as the derivative of y with respect to x and is symbolised; dy , y , f x , d f x the derivative dx dx measures the rate f x h f x of something f x lim h0hchangingThe process is called differentiating from first principles e.g. i Differentiate y 6 x 1 by using first principles. dyf x h f xf x 6x 1 limdx h0h6 x 6h 1 6 x 1 f x h 6 x h 1 limh0 h 6 x 6h 1 lim6hh 0 h 28. f x h f xslope of tangent = lim h0h This is known as the derivative of y with respect to x and is symbolised; dy , y , f x , d f x the derivative dx dx measures the rate f x h f x of something f x lim h0hchangingThe process is called differentiating from first principles e.g. i Differentiate y 6 x 1 by using first principles. dyf x h f xf x 6x 1 limdx h0h6 x 6h 1 6 x 1 f x h 6 x h 1 limh0 h 6 x 6h 1 lim6hh 0 h lim 6 h0 29. f x h f xslope of tangent = lim h0h This is known as the derivative of y with respect to x and is symbolised; dy , y , f x , d f x the derivative dx dx measures the rate f x h f x of something f x lim h0hchangingThe process is called differentiating from first principles e.g. i Differentiate y 6 x 1 by using first principles. dyf x h f xf x 6x 1 limdx h0h6 x 6h 1 6 x 1 f x h 6 x h 1 limh0 h 6 x 6h 1 lim6hh 0 h lim 6 h0 6 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 2 f x h x h 5 x h 2 2 33. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 .f x x2 5x 2 f x h x h 5 x h 2 2 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 2 f x h x h 5 x h 2 2 x 2 2 xh h 2 5 x 5h 2dy f x h f x limdx 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 2 f x h x h 5 x h 2 2 x 2 2 xh h 2 5 x 5h 2dy f x h f x limdx h0h x 2 2 xh h 2 5 x 5h 2 x 2 5 x 2 lim h0h 36. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 .f x x2 5x 2 f x h x h 5 x h 2 2 x 2 2 xh h 2 5 x 5h 2dyf x h f x limdx h0 hx 2 2 xh h 2 5 x 5h 2 x 2 5 x 2 lim h0 h2 xh h 5h 2 limh0h 37. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 .f x x2 5x 2 f x h x h 5 x h 2 2 x 2 2 xh h 2 5 x 5h 2dyf x h f x limdx h0 hx 2 2 xh h 2 5 x 5h 2 x 2 5 x 2 lim h0 h2 xh h 5h 2 limh0h lim 2 x h 5h0 38. ii Find the equation of the tangent to y x 2 5 x 2 at the point 1, 2 .f x x2 5x 2 f x h x h 5 x h 2 2 x 2 2 xh h 2 5 x 5h 2dyf x h f x limdx h0 hx 2 2 xh h 2 5 x 5h 2 x 2 5 x 2 lim h0 h2 xh h 5h 2 limh0h lim 2 x h 5h0 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 2 f x h x h 5 x h 2 2 x 2 2 xh h 2 5 x 5h 2dyf x h f x limdx h0 hx 2 2 xh h 2 5 x 5h 2 x 2 5 x 2 lim h0 h2 xh h 5h 2 limh0h lim 2 x h 5h0 2x 5dywhen x 1, 2 1 5dx 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 2 f x h x h 5 x h 2 2 x 2 2 xh h 2 5 x 5h 2dyf x h f x limdx h0 hx 2 2 xh h 2 5 x 5h 2 x 2 5 x 2 lim h0 h2 xh h 5h 2 limh0h lim 2 x h 5h0 2x 5dywhen x 1, 2 1 5dx 3 the slope of the tangent at 1, 2 is 3 41. y 2 3 x 1 42. y 2 3 x 1 y 2 3 x 3 y 3 x 1 43. y 2 3 x 1 y 2 3 x 3 y 3 x 1 Exercise 7B; 1, 2adgi, 3(not iv), 4, 7ab i,v, 12 (just h approaches 0)