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Topic 6

Topic 6. Rates of Change I. Topic 6: New Q Maths Chapter 6.1 - 6.4, 6.7 Rates of Change I Chapter 8.2. concept of the rate of change calculation of average rates of change in both practical and purely mathematical situations

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Topic 6

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  1. Topic 6 Rates of Change I

  2. Topic 6: New Q Maths Chapter 6.1 - 6.4, 6.7Rates of Change IChapter 8.2 • concept of the rate of change • calculation of average rates of change in both practical and purely mathematical situations • interpretation of the average rate of change as the gradient of the secant • intuitive understanding of a limit (N.B. – Calculations using limit theorems are not required) • definition of the derivative of a function at a point • derivative of simple algebraic functions from first principles

  3. Model : A cyclist travels 315 km in 9 hours. Express this in m/sec 315 km in 9 hours = 35 km in 1 hour = 9.72 m/sec (2dp) Read e.g. 3 Page 187

  4. EXAMPLE 3: page 187

  5. EXAMPLE 3: page 187 Within experimental error, these variables are related by a fixed rate (≈ 0.79 Kg/L)

  6. TI – 83 (Enter data via Stat – Edit) 2nd Stat Plot Turn plot 1 on Choose scatter plot X list: L1 Y list: L2 Set window Graph TI – 89 (Enter data via APPS – option 6 – option 1). You may need to set up a variable if you’ve never used this function before. F2 (plot setup) F1 (define) Plot type → scatter x: C1 y: C2 Frequency: no Enter to save You’ll return to this screen (ESC) Set window Calculator Steps for Linear Regression

  7. TI – 83 Turn on DiagnosticOn (via catalog) Stat – Calc 4: LinReg LinReg L1, L2, Y1 Enter (examine stats) Graph TI – 89 F5: calc Calc type → 5: LinReg x: C1 y: C2 Store regEQ → y1 Freq → no Enter to save Graph Add a Regression Line

  8. Exercise NewQ P 188 Exercise 6.1

  9. Rates of Change The rate of change of a second quantity w.r.t. a first quantity is the quotient of their differences: Read e.g. 4 Page 190 (Do on GC) N.B. If the rate of change is constant, the graph will be a straight line.

  10. Consider the following situation: A car travels from Bundaberg to Miriamvale (100 km) at 50 km/h. How fast must he travel coming home to average 100 km/h for the entire trip? N.B. Average speed = total distance total time

  11. Consider the following situation: A car travels from Bundaberg to Miriamvale (100 km) at 80km/h. How fast must he travel coming home to average 100 km/h for the entire trip?

  12. Exercise NewQ P 193 Exercise 6.2

  13. Use CBR to emulate motion graphs

  14. Use your GC to find the rate of change ofy = 2 + 4x – 0.25x2 from x = 3 to x = 5 Draw graph Find VALUES Rate of change = 4/2 = 2 (5 , 15.75) (3 , 11.75)

  15. Exercise NewQ P 198 Exercise 6.3 No. 1, 2, 4, 6(a&b), 7

  16. Exercise NewQ P 204, 212 Exercise 6.4 no. 2, 5, 6 & 7 6.6 no. 1-3, 6, 9

  17. y = x2 + 2

  18. y = x3 –x2 -4x + 4

  19. Finding Tangents Differentiation 1: (11B) • Tangent applet • An algebraic approach • Differentiation by First Principles

  20. Let P[ x, f(x) ] be a point on the curve y = f(x) and let Q be a neighbouring point a distance of h further along the x-axis from point P. Q [ x+h, f(x+h) ] Q [ f(x+h) ] x+h, f(x+h) – f(x) P[x, f(x)] h x+h - x

  21. Let P[ x, f(x) ]and let Q[ x+h, f(x+h) ] Gradient of tangent = limf(x+h) – f(x) h0 h Q[x+h,f(x+h)] f(x+h) – f(x) P(x,f(x)) x+h - x

  22. First Principles

  23. Model Find the gradient of the tangent to f(x)= x2 at the point where x = 2 Let P be the point (2, 4) and let Q be the point [(2+h), f(2+h)] Q P

  24. Scootle: First Principles • Models • Use first principles to find the gradient of the curve • (a) y = 2x2 – 13x + 15 at x=5 • (b) y = x2 + 3x - 8 at any point Differential Graphing Tool

  25. Exercise NewQ P 262 Exercise 8.2 2-5 Differentiation 1: (11B) • Tangent applet • 3 derivative puzzles Q: Differential Functions (11B) Q: Differentiate Polynomials (11B)

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