Drill: Tell whether th e curve is concave up or down on the given interval

1. Drill: Tell whether the curve is concave up or down on the given interval • y = cos x on [-1, 1] • y = x4 – 12x-5 on [8, 17] • y = 4x3 – 3x2 + 6 on [-8, 0] 1. y’ = -sinx, y’’ = -cos x On [-1, 1], y” is negative, which indicates y is concave down. • y’ = 4x3 – 12, y” = 12x2 On [8, 17], y” is positive, which indicates y is concave up. • y’ = 12x2 – 6x; y” = 24x -6 On [-8, 0], the interval is negative, which indicates y is concave down.

2. Trapezoidal Rule Lesson 5.5

3. Objectives • Students will be able to • approximate the area under the graph of a nonnegative continuous function by using trapezoidal rule. • use Simpson’s Rule to approximate a definite integral.

4. The Trapezoidal Rule • To approximate use T = ( y0 + 2y1 + 2y2 + ……2yn-1 + yn ), where [a, b] is partitioned into n subintervals of equal length and h = (b-a)/n Equivalently, T = where LRAM and RRAM are the Riemann sums using the left and right endpoints, respectively, for f for the partition.

5. Example 1 Applying the Trapezoidal Rule Use the Trapezoidal Rule with n = 4 to estimate

6. Example 1 Applying the Trapezoidal Rule Use the Trapezoidal Rule with n = 4 to estimate

7. Example 2 Averaging Temperatures An observer measures the outside temperature every hour from noon until midnight, recording the temperatures in the following table. What was the average temperature for the 12-hour period?

8. Example 2 Averaging Temperatures An observer measures the outside temperature every hour from noon until midnight, recording the temperatures in the following table. What was the average temperature for the 12-hour period?

9. Example 2 Averaging Temperatures

10. Simpson’s Rule • To approximate use S = ( y0 + 4y1 + 2y2 + 4y3 ……2yn-2 +4yn-1 + yn ), where [a, b] is partitioned into an even number of n subintervals of equal length and h = (b-a)/n

11. Example 3 Applying Simpson’s Rule Use the Simpson’s Rule with n = 4 to estimate

12. Example 3 Applying Simpson’s Rule Use the Simpson’s Rule with n = 4 to estimate

13. Use the trapezoidal rule with n =4 to approximate the value of the integral. a = 0 b = 2 n = 4 h = (2-0)/4 = ½ h/2 = ½ ÷ 2 = ¼

14. Use the concavity of the function to predict whether the approximation is an overestimate or an underestimate. f(x) = x2 f’(x) = 2x f’’(x) = 2, which is positive, meaning f(x) is concave up. This indicates that the approximation is an overestimate. If f” <0, the approximation is an underestimate. If f” = 0, the approximation is exact.

15. c) Find the integrals exact value

16. Homework • Page 312/3: 3, 4, 9, 17, 18 • Page 315 1-3 (quick quiz)