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PROGRAMME F11

PROGRAMME F11. DIFFERENTIATION = slope finding. Programme F11: Differentiation. The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials

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PROGRAMME F11

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  1. PROGRAMME F11 DIFFERENTIATION = slope finding

  2. Programme F11: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Second derivatives Newton-Raphson iterative method [optional]

  3. Programme F11: Differentiation The gradient of a straight-line graph The gradient of the sloping line straight line in the figure is defined as: the vertical distance the line rises and falls between the two points P and Q the horizontal distance between P and Q

  4. Programme F11: Differentiation The gradient (slope) of a straight-line graph The gradient of the sloping straight line in the figure is given as:

  5. Programme F11: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Second derivatives Newton-Raphson iterative method

  6. Programme F11: Differentiation The AVERAGE gradient of a curve in a region around a given point P What you could call the average gradient of a curve between two points P and Q will depend on the points chosen:

  7. The gradient of a curve AT a given point The gradient of a curve at a point P is defined to be the gradient of the tangent at that point [= the straight line that intersects the curve only at P, when the curve is not itself a straight line around P - JAB ]: NOTE: If the curve is a straight line around P, the tangent is just that line. QUESTION: Does a graph always have well-defined tangent at a given point?? Consider e.g. some graphs you’ve drawn in exercises involving the floor function, etc. [JAB]

  8. Programme F11: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function (Second derivatives –MOVED to a later set of slides) Newton-Raphson iterative method

  9. Programme F11: Differentiation Algebraic determination of the gradient of a curve The gradient of the chord PQ is and the gradient of the tangent at P is

  10. Programme F11: Differentiation Algebraic determination of the gradient of a curve As Q moves to P so the chord rotates. When Q reaches P the chord is coincident with the tangent.

  11. Programme F11: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

  12. Programme F11: Differentiation Derivatives of powers of x Two straight lines Two curves

  13. Programme F11: Differentiation Derivatives of powers of x Two straight lines (a)

  14. Programme F11: Differentiation Derivatives of powers of x Two straight lines (b) QUESTION: what about a vertical line, x = d ?? [JAB]

  15. Programme F11: Differentiation General definition of the derivative dy/dx = limit of y/x as  x  0 (from either side)

  16. Programme F11: Differentiation Derivatives of powers of x Two curves (a) so

  17. Programme F11: Differentiation Derivatives of powers of x Two curves (b) so

  18. Derivatives of powers of x A clear pattern is emerging: EXERCISE: Prove this general result, using a result about (a+b)nthat we saw when studying combinations. [JAB]

  19. Algebraic determination of the gradient of a curve y = 2x2 + 5 At Q: So As Therefore called the derivative of y with respect to x.

  20. Programme F11: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

  21. Programme F11: Differentiation Differentiation of polynomials To differentiate a polynomial, we differentiate each term in turn:

  22. Programme F11: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

  23. Programme F11: Differentiation Derivatives – an alternative notation The double statement: can be written as:

  24. Programme F11: Differentiation Towards derivatives of trigonometric functions (JAB) Limiting value of is 1 [NB:  expressed in RADIANS] [in lecture: a rough argument for this] Following slide includes most of a rigorous argument.

  25. Programme F11: Differentiation Area of triangle POA is: Area of sector POA is: Area of triangle POT is: Therefore: That is ((using fact that the cosine tends to 1 -- JAB)):

  26. Programme F11: Differentiation Derivatives of trigonometric functions and … The table of standard derivatives can be extended to include trigonometric and the exponential functions: [JAB:] The trig cases use the identities for finding sine and cosine of the sum of two angles, and an approximation for the cosine of a small angle (in RADIANS): cos x is approximately 1 – x2/2

  27. Programme F11: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

  28. Programme F11: Differentiation Differentiation of products of functions Given the product of functions of x: then: This is called the product rule.

  29. Programme F11: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method

  30. Programme F11: Differentiation Differentiation of a quotient of two functions Given the quotient of functions of x: then: This is called the quotient rule. [BUT I find it easier to use the PRODUCT rule, replacing v by 1/v and using the chain rule below. -- JAB]

  31. Programme F11: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function [i.e. compositions of functions] Newton-Raphson iterative method

  32. Programme F11: Differentiation Functions of a function (compositions) To differentiate a composition wou we employ the chain rule. If y is a function of u which is itself a function of x so that: y = w(u(x)) e.g. y = sin (x2 + 1) or y = cos 2 x First, think of this as y = w(u), e.g. y = sin u, with u =x2 + 1 Then: This is called the chain rule.

  33. Programme F11: Differentiation Compositions Many functions of a function can be differentiated at sight by a slight modification to the list of standard derivatives (F is the u of previous slide):

  34. Some Clarifications [JAB] For any (differentiable) functions f(x) and g(x), d/dx (f(x) + g(x)) = df(x)/dx + dg(x)/dx d/dx (f(x) - g(x)) = df(x)/dx — dg(x)/dx [and similarly for additions and subtractions of any number of functions] d/dx kf(x) = k df(x)/dxwhere k is any constant. d/dx xp = p xp-1 where p is any non-zero constant (not just when it is a pos. integer) d/dx (u/v) = d/dx (u.v -1) and you can deal with this by the product rule and the power rule just above, instead of remembering the quotient rule separately.

  35. Programme F11: Differentiation The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method [optional]

  36. Programme F11: Differentiation Newton-Raphson iterative method [OPTIONAL] Tabular display of results Given that x0 is an approximate solution to the equation f(x) = 0 then a better solution is given as x1, where: This gives rise to a series of improving solutions by iteration using: A tabular display of improving solutions can be produced in a spreadsheet.

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