Presentation on theme: "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."— Presentation transcript:
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 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
Rates Model : Mike earns $120 in an 8 hour shift. (a) What is his rate of pay? (b) How much would he earn in 7 hours? (a) Rate of pay = 120/8 $/hr = $15/hr (b) In 7 hours, he earns 7 x 15 = $105
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
EXAMPLE 3: page 187 Volume (L)512152025 Mass (Kg)4.19.31215.719.75 Kg/L
EXAMPLE 3: page 187 Volume (L)512152025 Mass (Kg)4.19.31215.719.75 Kg/L0.820.780.800.79 Within experimental error, these variables are related by a fixed rate ( ≈ 0.79 Kg/L)
Calculator Steps for Linear Regression TI – 83 (Enter data via Stat – Edit) 2 nd 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: C1y: C2 Frequency: no Enter to save You’ll return to this screen (ESC) Set window
Add a Regression Line TI – 83 Turn on DiagnosticOn (via catalog) Stat – Calc 4: LinReg LinReg L1, L2, Y 1 Enter (examine stats) Graph TI – 89 F5: calc Calc type → 5: LinReg x: C1y: C2 Store regEQ → y 1 Freq → no Enter to save Graph
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.
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
Consider the following situation: A car travels from Bundaberg to Miriamvale (100 km) at 80 km/h. How fast must he travel coming home to average 100 km/h for the entire trip?
Finding Tangents An algebraic approach Differentiation by First Principles Differentiation 1: (11B) -Tangent appletTangent applet
Let P [ x, f(x) ] be a point on the curve y = f(x) P [ x, f(x) ] and let Q be a neighbouring point a distance of h further along the x-axis from point P. x+h - x f(x+h) – f(x) Q [ x+h, f(x+h) ] Q [ x+h, f(x+h) ] h
P(x,f(x)) Q[x+h,f(x+h)] f(x+h) – f(x) x+h - x Gradient of tangent = lim f(x+h) – f(x) h 0 h Let P [ x, f(x) ] and let Q [ x+h, f(x+h) ]