1. Problem of the Day The graph of a function f is shown above.Which of the following statements about f is false? A) f is continuous at x = a B) f has a relative maximum at x = a C) x = a is in the domain of f D) lim f(x) is equal to lim f(x) E) lim f(x) exists x a + x a - x a a

2. Problem of the Day The graph of a function f is shown above.Which of the following statements about f is false? A) f is continuous at x = a B) f has a relative maximum at x = a C) x = a is in the domain of f D) lim f(x) is equal to lim f(x) E) lim f(x) exists x a + x a - x a a

3. You have 1 minute to sum all the integers from 1 to 100.

4. Mathematicians needed a concise way for writing sums Sigma notation does this It uses the uppercase Greek letter sigma Σ

5. Sigma notation does this The sum of n terms a 1, a 2, a 3,..., a n equals Σ a i and means a 1 + a 2 +... + a n i = 1 n index of summation (i, j, k most of the time) upper bound of summation lower bound of summation (any interger < upper bound) Σ j 3 = 2 3 + 3 3 + 4 3 + 5 3 j = 2 5

6. Properties of Summation Σ Ka i = i = 1 n K Σ a i i = 1 n Σ (a i ± b i ) = i = 1 n Σ a i + i = 1 n Σ b i i = 1 n

7. Summation Formulas (page 260) Σ c = i = 1 n cn Σ i = i = 1 n n(n + 1) 2 Σ i 2 = i = 1 n n(n + 1)(2n + 1) 6 Σ i 3 = i = 1 n n 2 (n + 1) 2 4

8. Evaluating a Sum Σ i + 1 n 2 i = 1 n Evaluate for n = 10, 100, 1000 Σ i + 1 i = 1 n 1 n 2 Factor out constant (property 1)

9. Evaluating a Sum Σ i + 1 n 2 i = 1 n Evaluate for n = 10, 100, 1000 Σ i + 1 i = 1 n 1 n 2 Factor out constant (property 1) Split apart the sum (property 2) Σ i i = 1 n 1 n 2 Σ 1 i = 1 n + ( (

10. Evaluating a Sum Σ i + 1 n 2 i = 1 n Evaluate for n = 10, 100, 1000 Σ i + 1 i = 1 n 1 n 2 1 n 2 n(n + 1) + n 2 ( ( Factor out constant (property 1) Split apart the sum (property 2) Summation formulas n + 3 2n Simplify Σ i i = 1 n 1 n 2 Σ 1 i = 1 n + ( (

11. n + 3 2n as n approaches ∞ what does the sum approach? what do you get when n = 10, 100, 1000?

12. n + 3 2n as n approaches ∞ what does the sum approach? lim n ∞ n + 3 = 1 2n 2 what do you get when n = 10, 100, 1000?

13. You have 1 minute to sum all the integers from 1 to 100. 1 2 3 4... 100 100 99 98 97 1 101 101 101 101 101 + 100 x 101 = 5050 2 (because you did each number twice) Σ i = (100)(101) = 5050 2 i = 1 n

14. In truth, Carl Friedrich Gauss (1777 - 1855) was asked by his teacher to sum the integers from 1 to 100. When he had the correct answer in only a few moments, the teacher was astonished. He developed the method mentioned on the previous screen.

15. Problems 1& 3 on your homework are to be written out (do not use properties and formulas)

16. Attachments