# X y y 1 2 x 1 3 q4 let f x be a function that is

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x y y = 1 2 x 1 3 Q[4]: Let f ( x ) be a function that is decreasing from x = 0 to x = 5. Which Riemann sum approximation of ż 5 0 f ( x ) d x is the largest–left, right, or midpoint? Q[5]: Give an example of a function f ( x ) , an interval [ a , b ] , and a number n such that the midpoint Riemann sum of f ( x ) over [ a , b ] using n intervals is larger than both the left and right Riemann sums of f ( x ) over [ a , b ] using n intervals. In Questions 6 through 10 , we practice using sigma notation. There are many ways to write a given sum in sigma notation. You can practice finding several, and deciding which looks the clearest. Q[6]: Express the following sums in sigma notation: (a) 3 + 4 + 5 + 6 + 7 (b) 6 + 8 + 10 + 12 + 14 (c) 7 + 9 + 11 + 13 + 15 (d) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 3
I NTEGRATION 1.1 D EFINITION OF THE I NTEGRAL Q[7]: Express the following sums in sigma notation: (a) 1 3 + 1 9 + 1 27 + 1 81 (b) 2 3 + 2 9 + 2 27 + 2 81 (c) ´ 2 3 + 2 9 ´ 2 27 + 2 81 (d) 2 3 ´ 2 9 + 2 27 ´ 2 81 Q[8]: Express the following sums in sigma notation: (a) 1 3 + 1 3 + 5 27 + 7 81 + 9 243 (b) 1 5 + 1 11 + 1 29 + 1 83 + 1 245 (c) 1000 + 200 + 30 + 4 + 1 2 + 3 50 + 7 1000 Q[9]: Evaluate the following sums. You might want to use the formulas from Theorems 1.1.5 and 1.1.6 in the CLP–II text. (a) 100 ÿ i = 0 3 5 i (b) 100 ÿ i = 50 3 5 i (c) 10 ÿ i = 1 i 2 ´ 3 i + 5 (d) b ÿ n = 1 1 e n + en 3 , where b is some integer greater than 1. Q[10]: Evaluate the following sums. You might want to use the formulas from Theorem 1.1.6 in the CLP–II text. (a) 100 ÿ i = 50 ( i ´ 50 ) + 50 ÿ i = 0 i (b) 100 ÿ i = 10 ( i ´ 5 ) 3 (c) 11 ÿ n = 1 ( ´ 1 ) n (d) 11 ÿ n = 2 ( ´ 1 ) 2 n + 1 Questions 11 through 15 are meant to give you practice interpreting the formulas in Definition 1.1.11 of the CLP–II text. The formulas might look complicated at first, but if you understand what each piece means, they are easy to learn. 4
I NTEGRATION 1.1 D EFINITION OF THE I NTEGRAL Q[11]: In the picture below, draw in the rectangles whose (signed) area is being computed by the midpoint Riemann sum 4 ÿ i = 1 b ´ a 4 ¨ f a + i ´ 1 2 b ´ a 4 . x y b a y = f ( x ) Q[12]( ˚ ): 4 ÿ k = 1 f ( 1 + k ) ¨ 1 is a left Riemann sum for a function f ( x ) on the interval [ a , b ] with n subintervals. Find the values of a , b and n . Q[13]: Draw a picture illustrating the area given by the following Riemann sum. 3 ÿ i = 1 2 ¨ ( 5 + 2 i ) 2 Q[14]: Draw a picture illustrating the area given by the following Riemann sum. 5 ÿ i = 1 π 20 ¨ tan π ( i ´ 1 ) 20 Q[15]( ˚ ): Fill in the blanks with right, left, or midpoint; an interval; and a value of n. 3 ř k = 0 f ( 1.5 + k ) ¨ 1 is a Riemann sum for f on the interval [ , ] with n = . Q[16]: Evaluate the following integral by interpreting it as a signed area, and using geometry: ż 5 0 x d x 5
I NTEGRATION 1.1 D EFINITION OF THE I NTEGRAL Q[17]: Evaluate the following integral by interpreting it as a signed area, and using geometry: ż 5 ´ 2 x d x §§ Stage 2 Q[18]( ˚ ): Use sigma notation to write the midpoint Riemann sum for f ( x ) = x 8 on [ 5, 15 ] with n = 50. Do not evaluate the Riemann sum. Q[19]( ˚ ): Estimate ż 5 ´ 1 x 3 d x using three approximating rectangles and left hand end points.