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Topic 7

Topic 7. Rates of Change II. Rates of Change. Rules for differentiation including. evaluation of the derivative of a function at a point (SLE 3,4) interpretation of the derivative as the instantaneous rate of change (SLE 1,2,5,10)

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Topic 7

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  1. Topic 7 Rates of Change II

  2. Rates of Change Rules for differentiation including evaluation of the derivative of a function at a point (SLE 3,4) interpretation of the derivative as the instantaneous rate of change (SLE 1,2,5,10) interpretation of the derivative as the gradient function (SLE 1,2,4,6) practical applications of instantaneous rates of change (SLE 1-5,7-11)

  3. Derivative of y = axn When we did differentiation from first principles, we found that if Are you seeing a pattern here?

  4. Derivative of y = axn When we did differentiation from first principles, we found that if Can you see what’s happening? Calculus Phobe Videos

  5. Model: Find the derivative of y = 3x4 – 2x3 + 5x2 + 7x + 8 y = 3x4 – 2x3 + 5x2 + 7x + 8 = 12x3 – 6x2 + 10x + 7 dy dx

  6. Model: Find the derivative of y = 3x2 + 2x -5 at x = (a) 1 (b) -1 y = 3x2 + 2x - 5 = 6x + 2 (a) When x = 1, = 6x1 + 2 = 8 (b) When x = -1, = 6x-1 + 2 = -4 dy dx dy dx dy dx y = 3x2+2x-5

  7. Gradient is the same as derivative When you hear “gradient”, you think you think “gradient” When you hear

  8. Exercise NewQ Set 8.2 Page 262 No. 2 Set 8.3 Page 266 No.4,7-9

  9. Model Differentiate each of the following:

  10. Exercise NewQ Set 8.3 Page 266 No.1 Set 8.4 Page 270 No. 1-2

  11. The Chain Rule

  12. Let u = x2 + 3x – 6 ∴ y = u2 • To be used to differentiate a function of a function • Recall our last function

  13. Try these… a) y = (3x2 + 5x – 2)2b)y = (5x3 – 6x2 + 3x + 5)3 a) (12x + 10)(3x2 + 5x – 2) b) (45x2 – 36x + 9)(5x3 – 6x2 + 3x + 5)2

  14. Exercise NewQ Set 8.4 Page 270 No. 5(a&b), 6(i-t) Set 8.5 Page 273 No. 1(a,e,f)

  15. The Product Rule

  16. If y = (3x + 4) (5x2 – 2x + 1) Let u = 3x+ 4 and v = 5x2 – 2x + 1 To be used when differentiating a function multiplied by a function

  17. Exercise NewQ Set 8.5 Page 273 No. 2-3

  18. Answers 8.5 number 3 • 15(3x-2)4(x+8)2+ 60(3x-2)3(x+8)3 • 4(4t-5)3(2t-7) + 12(4t-5)2(2t-7)2 • 48(m-6)(4m+3)7+ 6(4m+3)8 • 21(2k+9)4(3k-5)6+ 8(2k+9)3(3k-5)7 • 24(3g-4)5(5g-4)5+ 60(3g-4)4(5g-4)6 • 112(12-5t)9(4t-11)3+ 315(12-5t)8(4t-11)4

  19. The Quotient Rule

  20. Exercise Set 8.5 Page 273 No. 4-5

  21. f(x) = 5x – 3x2 + 8 f’(x) = 5 – 6x f’(2) = 5 – 6 × 2 m = -7 At x = 2, y = 6, m = -7 ∴ y = mx + c 6 = -7 × 2 + c c = 20  y = -7x + 20 Application of Derivatives Model No. 1 If the equation of a parabola is y = 5x – 3x2 + 8, at the point where x = 2, find: a)The gradient of the tangentb)the equation of the tangent

  22. Model No. 2 The volume of liquid in an underground tank being gravity filled from a tanker is given by V = 1200 + 1020t – 17t2 where the volume, V, is in litres and time, t, is in minutes. a) What is the filling rate after 5 minutes? b) What is the flow rate in the pipe after 10 minutes?

  23. The volume of liquid in an underground tank being gravity filled from a tanker is given by V = 1200 + 1020t – 17t2 where the volume, V, is in litres and time, t, is in minutes. a) What is the filling rate after 5 minutes? b) What is the flow rate in the pipe after 10 minutes?  Filling rate and flow rate in the pipe are both given by the instantaneous rate of change of the volume. V = 1200 + 1020t – 17t2 V’ = 1020 – 34t a) @ 5 min V’ = 1020 - 34×5 = 850 L/min b) @ 10 min V’ = 1020 - 34×10 = 680 L/min

  24. Model No. 3 The tangent, AB, touches the curve f(x) = 12 + 4x – x2 at point A, and cuts the x-axis at point B. If point A has an x-coordinate of 3, find i) The length AB and ii) The acute angle AB makes with the x-axis A B

  25. The tangent, AB, touches the curve f(x) = 12 + 4x – x2 at point A, and cuts the x-axis at point B. If point A has an x-coordinate of 3, find • The length AB and • The acute angle AB makes with the x-axis •  Find the equation of AB •  find • (i) coordinates at A and • (ii) the gradient at A, then • (iii) equation of AB A (3, 15) B f(x) = 12 + 4x – x2 f(3) = 12 + 4×3 – 32 = 15 f’(x) = 4 – 2x f’(3) = 4 – 2×3 = -2 = m ∴ y = mx + c 15 = -2×3 + c c = 21 y = -2x + 21

  26. The tangent, AB, touches the curve f(x) = 12 + 4x – x2 at point A, and cuts the x-axis at point B. If point A has an x-coordinate of 3, find • The length AB and • The acute angle AB makes with the x-axis • Now find where AB cuts the x-axis A (3, 15) B (10.5, 0) y = -2x + 21 Cuts x-axis at y = 0 ∴ 0 = -2x + 21 x = 10.5

  27. The tangent, AB, touches the curve f(x) = 12 + 4x – x2 at point A, and cuts the x-axis at point B. If point A has an x-coordinate of 3, find • The length AB and • The acute angle AB makes with the x-axis • Now find the length of AB A (3, 15) B (10.5, 0)

  28. The tangent, AB, touches the curve f(x) = 12 + 4x – x2 at point A, and cuts the x-axis at point B. If point A has an x-coordinate of 3, find • The length AB and • The acute angle AB makes with the x-axis A (3, 15) 15 units B (10.5, 0) θ 10.5 – 3 = 7.5 units

  29. Exercise Set 8.6 Page 277 No. 1-3, 8, 11-13

  30. 1. If y = x2 + Bx + 6 has a stationary point at (2,k) , find B and k. 2. Sketch a curve for which, when x< -2 when -2<x<1 when x>1

  31. 3. For the first 4 seconds of its motion, a particle moving in a straight line has a velocity at time t seconds, given by v = t3 - 5t2 + 6t + 1 m/s. Find the greatest and least velocity of the particle in this time. 4. A man standing on a cliff above the ocean throws a ball directly upward. The height of the ball above the water t seconds after release is h metres where h = 50+70t-10t2. How high above the water will the ball go? What is the time of flight? Comment upon any unusual results? 5. Find, with reasons, the greatest and least values of the function f(x) = x3 – 2x2 in the interval -1 x  1

  32. 6.Show that the curves whose equations are given by y = and y = have the same slope at their point of intersection. Find the equation of the common tangent at this point. 7. Find the equation of the tangent to the curve y = x3 – 5 at the point (1,-4). Where does this tangent cut the x-axis? 8. A manufacturer makes a batch of N articles, the cost of each one being N2 – 6N + 35 cents. If he sells each article for 50 cents, find an expression representing his profit, P cents, for the entire batch. How many articles should be in the batch to maximise P? Is it true that P is greatest when the cost per article is least?

  33. 9. Draw a neat sketch of the gradient function for each of the following.

  34. 10. The concentration of adrenalin in the bloodstream immediately after stimulus is given by the function where t is time in seconds and concentration is in ml/L. Find how long it takes for the concentration to reach its maximum and what that concentration is.

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