differentiation

average rate of change §

• straight line graphs have constant gradients.
• non linear functions have gradients that vary, and the gradient is often estimated from chords or secants.
  • chord: a line between two points on a curve.
  • secant: a line through two points on a curve.
• for any function f(x), the gradient of the chord or secant from points (a, f(a)) to (b, f(b)) is the average rate of change over the interval of [a, b].

difference quotient §

• h is the small change in x, corresponding to b-a or x2 - x1, hence x+h corresponds to b or x2 $\frac{f(x+h)-f(x)}{h}$

lebiniz notation §

• $\delta$ means “small change in or “increment in”$\frac{\delta y}{\delta x}$

example questions (done in class) §

• f(x) = x^2 - 2x + 5
  • x at 1 = 1 - 2 + 5 = 4
  • x at 5 = 25 - 10 + 5 = 20
  • change of 16/4 = 4
• T = 600/(t^2+2t+30) where t is time after 6pm
  • temp @ 6pm: 600/30 = 20degC
  • temp @ midnight: 600/(36+12+30) = 7.7 deg C
  • rate of change: (20-7.7) / (6-12) = -2.05 deg C / hour

instantaneous rate of change §

• instantaneous rate of change occurs as there is smaller and smaller intervals to create a gradient.
• expressed as $rat{e}_{inst}=\frac{\delta p}{\delta q}$ (i dont know if this is right)