- Talk about derivative extra credit from Newton lab on Monday
- Exam 2
- Next Friday (April 8)
- Material since 1st hour exam (e.g., if, while, for, function handles, Newton, Riemann sums & trapezoid rule, etc.)
- Rules and format similar to 1st exam, especially open references, open computer
- Sample exam on our “Exercises” page
- Riemann sum and trapezoid rule videos
- Estimates of area under curve
- More rectangles or trapezoids give more accurate estimates
- With rectangles can take height at left edge, midpoint, or right
- Formulas
- Example
- Use trapezoid rule
- Inputs: f(x) = 2x + 1, a = 3, b = 15, n = 5
- x1 = 3, x2 = 5.4, x3 = 7.8, x4 = 10.2, x5 = 12.6, x6 = 15
- Areas
- A1 = ( f(3)+f(5.4))/2 * 2.4 = 22.56
- A2 = (f(5.4)+f(7.8)/2 * 2.4 = 34.08
- A3 = (f(7.8)+f(10.2))/2 * 2.4 = 45.6
- A4 = (f(10.2)+f(12.6))/2 * 2.4 = 57.12
- A5 = (f(12.6)+f(15))/2 * 2.4 = 68.64
- Total area = 228
- Trapezoid algorithm, based in large part on capturing in pseudocode what you
did/described in English
trapezoid( f, a, b, n ) x = linspace( a, b, n+1 ) deltaX = (b-a)/n for i = 1:n % i is number of current trapezoid A(i) = ( f(x(i)) + f(x(i+1)) ) / 2 * deltaX return sum(A)
- Lab day for integration lab
- Including demo of intended output
- No reading