Day 30 in Mat129

Last time: More Applied Optimization

Next time: Integration

Announcements
- A comment about using your calculator... Please do!
- Hand in 63,65, and 69 on page 218.
- Thu 4/1 Section 4.8 Anti-derivatives
  Begin On-line homework for 4.8 (due 4/8).
Last time: More Optimization
- Redo #48, p. 228! I pulled a fast one last time....
Today: Section 4.8 (Anti-derivatives)
- Definition:
- All antiderivatives are the same up to a constant:
- Differential Equations:
  - A differential equation is an equation linking a function and its derivative(s).
  - The simplest differential equations look like this:
    
    ${\frac{dy}{dx}=g(x)}$
  - Usually an initial condition (or conditions) will also be provided -- so that the solution (y(x)) has value y=y₀ at x=x₀.
  - Our job then is to find the function y(x) which satisfies both of those conditions.
  - Examples:
    - #81, p. 247
    - A spring satisfies Hooke's law:
      
      ${F=-kx}$
      Newton said that
      
      ${F=ma}$
      If we put those two laws together, we get
      
      ${\frac{d^2x}{dt^2}=-kx}$
      For initial conditions, we should give initial position and initial velocity.
    - Gravity is essentially constant at the Earth's surface: we can use a differential equation to deduce the formula for the height h of an object as a function of time:
      
      ${a=9.81m/s^2}$
      Newton said that
      
      ${F=ma}$
      If we put those two laws together, we get
      
      ${\frac{d^2h}{dt^2}=-a}$
      Once again, we should give initial position and initial velocity.
- Notation:
- The integral is an operator that also is linear:
- A couple of important special cases:
  - Power rule:
  - Trig rules:
- Examples:
  - #3, p. 242
  - #29, p. 243
  - #48, p. 243
  - #63, p. 244

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