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Gradients & Curves: Higher Maths Tutorial

Learn how to find gradients, differentiate functions, and optimize using calculus techniques for higher-level mathematics.

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Gradients & Curves: Higher Maths Tutorial

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  1. Higher Higher Unit 1Applications 1.4 Finding the gradient for a polynomial Increasing / Decreasing functions Differentiating Easy Functions Max / Min and inflexion Points Differentiating Harder Functions Curve Sketching Differentiating with Leibniz Notation Max & Min Values on closed Intervals Equation of a Tangent Line Optimization Mind Map of Chapter www.mathsrevision.com

  2. Gradients & Curves Higher On a straight line the gradient remains constant, however with curves the gradient changes continually, and the gradient at any point is in fact the same as the gradient of the tangent at that point. The sides of the half-pipe are very steep(S) but it is not very steep near the base(B). S Demo B

  3. Gradients & Curves Higher Gradient of tangent = gradient of curve at A A B Gradient of tangent = gradient of curve at B

  4. To find the gradient at any point on a curve we need to modify the gradient formula Gradients & Curves Higher For the function y = f(x) we do this by taking the point (x, f(x)) and another “very close point” ((x+h), f(x+h)). Then we find the gradient between the two. ((x+h), f(x+h)) Approx gradient (x, f(x)) True gradient

  5. Gradients & Curves Higher The gradient is not exactly the same but is quite close to the actual value We can improve the approximation by making the value of h smaller This means the two points are closer together. ((x+h), f(x+h)) Approx gradient (x, f(x)) True gradient

  6. Gradients & Curves Higher We can improve upon this approximation by making the value of h even smaller. So the points are even closer together. Demo ((x+h), f(x+h)) Approx gradient True gradient (x, f(x))

  7. Higher We have seen that on curves the gradient changes continually and is dependant on the position on the curve. ie the x-value of the given point. Derivative Finding the GRADIENT Differentiating The process of finding the gradient is called Finding the rate of change DIFFERENTIATING or FINDING THE DERIVATIVE (Gradient)

  8. Derivative Higher If the formula/equation of the curve is given by f(x) Then the derivative is called f '(x) - “f dash x” There is a simple way of finding f '(x) from f(x). f(x) f '(x) 2x2 4x 4x2 8x 5x10 50x9 6x7 42x6 x3 3x2 x5 5x4 x99 99x98

  9. Derivative Rule for Differentiating Higher It can be given by this simple flow diagram ... multiply by the power reduce the power by 1 If f(x) = axn n n -1 ax then f '(x) = NB: the following terms & expressions mean the same GRADIENT, DERIVATIVE, RATE OF CHANGE, f '(x)

  10. Derivative Rule for Differentiating Higher To be able to differentiate it is VERY IMPORTANT that you are comfortable using Fractions and Surds & Indices rules

  11. xm. xn = xm+n Surds & Indices

  12. Do YOU need extra help or revision then do Surds & Indices HHM page 340 Ex8 HHM page 342 Ex9

  13. Higher (I) f(x) = ax (Straight line function) Special Points Index Laws x0 = 1 If f(x) = ax = ax1 then f '(x) = 1 X ax0 = a X 1 = a So if g(x) = 12x then g '(x) = 12 Also using y = mx + c The line y = 12x has gradient 12, and derivative = gradient !!

  14. Higher Special Points (II) f(x) = a, (Horizontal Line) Index Laws x0 = 1 If f(x) = a = a X 1 = ax0 then f '(x) = 0 X ax-1 = 0 So if g(x) = -2 then g '(x) = 0 Also using formula y = c , (see outcome 1 !) The line y = -2 is horizontal so has gradient 0 !

  15. Differentiation techniques Differentiation = Gradient Differentiation = Rate of change Drill Name :

  16. Calculus Revision Drill Differentiate

  17. Calculus Revision Drill Differentiate

  18. Calculus Revision Drill Differentiate

  19. Derivative Higher HHM Ex6D , Ex6E and Ex6F Even Numbers only

  20. Derivative Higher Example 1 A curve has equation f(x) = 3x4 Find the formula for its gradient and find the gradient when x = 2 Its gradient is f '(x) = 12x3 f '(2) = 12 X 23 = 12 X 8 = 96 Example 2 A curve has equation f(x) = 3x2 Find the formula for its gradient and find the gradient when x = -4 Its gradient is f '(x) = 6x At the point where x = -4 the gradient is f '(-4) = 6 X -4 = -24

  21. Derivative Higher Example 3 If g(x) = 5x4 - 4x5 then find g '(2) . g '(x) = 20x3 - 20x4 g '(2) = 20 X 23 - 20 X 24 = 160 - 320 = -160

  22. Derivative Higher Example 6 : Find the points on the curve f(x) = x3 - 3x2 + 2x + 7 where the gradient is 2. NB: gradient = derivative = f '(x) Now using original formula We need f '(x) = 2 ie 3x2 - 6x + 2 = 2 f(0) = 7 or 3x2 - 6x = 0 ie 3x(x - 2) = 0 f(2) = 8 -12 + 4 + 7 ie 3x = 0 or x - 2 = 0 = 7 Points are (0,7) & (2,7) so x = 0 or x = 2

  23. Calculus Revision Drill Differentiate

  24. Calculus Revision Drill Differentiate Straight line form Differentiate

  25. Calculus Revision Drill Differentiate Straight line form Differentiate

  26. Calculus Revision Drill Differentiate Straight line form Chain Rule Simplify

  27. Calculus Revision Drill Differentiate Straight line form Differentiate

  28. Calculus Revision Drill Differentiate Straight line form Differentiate

  29. Calculus Revision Drill Differentiate Straight line form Differentiate

  30. Differentiation techniques Differentiation = Gradient Differentiation = Rate of change Name :

  31. Calculus Revision Drill Differentiate Multiply out Differentiate

  32. Calculus Revision Drill Differentiate multiply out differentiate

  33. Calculus Revision Drill Differentiate Straight line form multiply out Differentiate

  34. Calculus Revision Drill Differentiate multiply out Differentiate

  35. Calculus Revision Drill Differentiate multiply out Simplify Straight line form Differentiate

  36. Calculus Revision Drill Differentiate Multiply out Straight line form Differentiate

  37. Calculus Revision Drill Differentiate Split up Straight line form Differentiate

  38. Higher Leibniz Notation is an alternative way of expressing derivatives to f'(x) , g'(x) , etc. Leibniz Notation If y is expressed in terms of x then the derivative is written as dy/dx . eg y = 3x2 - 7x so dy/dx = 6x - 7 . Example 19 Q = 9R2 - 15 R3 Find dQ/dR = 18R + 45 R4 NB: Q = 9R2 - 15R-3 So dQ/dR = 18R + 45R-4

  39. Leibniz Notation Higher Example 20 A curve has equation y = 5x3 - 4x2 + 7 . Find the gradient where x = -2 ( differentiate ! ) gradient = dy/dx = 15x2 - 8x if x = -2 then gradient = 15 X (-2)2 - 8 X (-2) = 60 - (-16) = 76

  40. Derivative Higher HHM Ex6H Q1 – Q3 HHM Ex6G Q1,4,7,10,13,16,19,22,25

  41. Real Life Example Physics Higher Newton’s 2ndLaw of Motion s = ut + 1/2at2 where s = distance & t = time. Finding ds/dt means “diff in dist”  “diff in time” ie speed or velocity so ds/dt = u + at but ds/dt = v so we get v = u + at and this is Newton’s 1st Law of Motion

  42. Derivative Higher HHM Ex6H Q4 – Q6

  43. Equation of Tangents y = mx +c Higher y = f(x) Demo A(a,b) tangent NB: at A(a, b) gradient of line = gradient of curve gradient of line = m (from y = mx + c ) gradient of curve at (a, b) = f (a) it follows that m = f (a)

  44. Straight line so we need a point plus the gradient then we can use the formula y - b = m(x - a) . Equation of Tangents Higher Demo Example 21 Find the equation of the tangent line to the curve y = x3 - 2x + 1 at the point where x = -1. Point: if x = -1 then y = (-1)3 - (2 X -1) + 1 = -1 - (-2) + 1 = 2 point is (-1,2) Gradient:dy/dx = 3x2 - 2 when x = -1 dy/dx = 3 X (-1)2 - 2 m = 1 = 3 - 2 = 1

  45. Equation of Tangents Higher Now using y - b = m(x - a) point is (-1,2) m = 1 we get y - 2 = 1( x + 1) or y - 2 = x + 1 or y = x + 3

  46. Equation of Tangents Higher Example 22 Find the equation of the tangent to the curve y = 4 x2 at the point where x = -2. (x  0) Also find where the tangent cuts the X-axis and Y-axis. Point: when x = -2 then y = 4 (-2)2 = 4/4 = 1 point is (-2, 1) Gradient: y = 4x-2 so dy/dx = -8x-3 = -8 x3 when x = -2 then dy/dx = -8 (-2)3 = -8/-8 = 1 m = 1

  47. Equation of Tangents Higher Now using y - b = m(x - a) we get y - 1 = 1( x + 2) or y - 1 = x + 2 or y = x + 3 Axes Tangent cuts Y-axis when x = 0 so y = 0 + 3 = 3 at point (0, 3) Tangent cuts X-axis when y = 0 so 0 = x + 3 or x = -3 at point (-3, 0)

  48. Equation of Tangents Higher Example 23 - (other way round) Find the point on the curve y = x2 - 6x + 5 where the gradient of the tangent is 14. gradient of tangent = gradient of curve dy/dx = 2x - 6 so 2x - 6 = 14 2x = 20 x = 10 Put x = 10 into y = x2 - 6x + 5 Point is (10,45) Giving y = 100 - 60 + 5 = 45

  49. Derivative Higher HHM Ex6J

  50. Higher Increasing & Decreasing Functions and Stationary Points Consider the following graph of y = f(x) ….. y = f(x) + 0 0 + - + + a b c d e f - X + 0

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