Introduction to Calculus Worksheet 2, Section 4.2 (Sigma Notation)

Worksheet 2, Section 4.2 (Sigma Notation) Name: Date: Directions: You are encouraged to work in groups, but you don’t need to do so. We’ll turn in what we could finish at the end of class. The exercises below are 4, 5, and 6 from Section 4.2. 4. Consider the function f (x) = 3x + 4. (a) Compute M4 for f (x) on the interval [2, 5]. Be sure to clearly identify the value of ∆x as well as the locations of x0 , x1 , . . . x4 . Include a careful sketch of the function and the corresponding rectangles being used in the sum. Starting out, ∆x = (5 − 2)/4 = 3/4. Therefore, it’s probably easiest to build a table of values: x0 xi 2 x1 2.75 x2 3.5 x3 4.25 x4 5 Midpts y 2.375 11.125 3.125 13.375 3.875 15.625 4.625 17.875 The sum of the heights is 58, multiply by the width of 3/4 to get that M4 = 43.5. (b) Use a familiar geometric formula to determine the exact value of the area of the region bounded by y = f (x) and the x-axis on [2, 5]. SOLUTION: THe shape is a trapezoid- If you make it into a rectangle at the bottom with a triangle on top, then the area is 1 3 · 10 + · 3 · 9 = 43.5 2 (c) Explain why the values you computed in (a) and (b) turn out to be the same. Will this be true if we use a number different than n = 4 and compute Mn ? Will L4 or R4 have the same value as the exact area of the region found in (b)? SOLUTION: These are questions to think about, but we won’t go in depth on them. Parts (a) and (b) were the same because we had a linear function, but L4 and R4 will not give the same answer. 5. Let S be the sum given by S = ((1.4)2 + 1) · 0.4 + ((1.8)2 + 1) · 0.4 + ((2.2)2 + 1) · 0.4 + ((2.6)2 + 1) · 0.4 + ((3.0)2 + 1) · 0.4 (a) Assume that S is a right Riemann sum. For what function F and what interval [a, b]is S this function’s Riemann sum? SOLUTION: We see that f (x) = x2 + 1 and ∆x = 0.4. Since there are 5 rectangle, we know that b−a = 0.4 5 ⇒ b−a=2 so the length of the interval is 2. If we’re using right endpoints, then b = 3, and so a = 1. (b) How does your answer to (a) change if is a left Riemann sum? a middle Riemann sum? SOLUTION: If the sum is a left Riemann sum, then a = 1.4 and b = 1.4 + 2 = 3.4. If we’re using middle points, a = 1.2 and b = 3.2 (c) Suppose that S really is a right Riemann sum. What is geometric quantity does S approximate? SOLUTION: S is the area under the graph of x2 + 1 on [1, 3]. (d) Use sigma notation to write a new sum R that is the right Riemann sum for the same function, but that uses twice as many subintervals as S. 2 = 0.2. The first right endpoint is SOLUTION: For twice as many rectangles, n = 10, so ∆x = 10 1 + 0.2, and the last endpoint is 3. 10 X ((1 + k · 0.2)2 + 1) · 0.2 k=1 1. A car traveling along a straight road is braking and its velocity is measured at several different points in time, as given in the following table, where t is measured in seconds and v(t) is measured in feet/sec. 0 0.3 t v(t) 100 88 0.6 74 0.9 59 1.2 40 1.5 19 1.8 0 (a) Plot the given data on a set of axes with time on the horizontal axis and the velocity on the vertical axis. SOLUTION: This is straightforward. (b) Estimate the total distance traveled during the car the time brakes using a middle Riemann sum with 3 subintervals. SOLUTION: The width of each rectangle is 0.6, and use midpoints for heights: 88 · 0.6 + 59 · 0.6 + 19 · 0.6 = 99.6 (c) Estimate the total distance traveled on [0, 1.8] by computing L6 , R6 , and 12 (L6 + R6 ). SOLUTION: The width of each rectangle is 0.3, and L6 = 0.3(100 + 88 + 74 + 59 + 40 + 19) R6 = 0.3(88 + 74 + 59 + 40 + 19 + 0) The last part is the average of the two numbers we computed. (d) Assuming that v(t) is always decreasing on [0, 1.8], what is the maximum possible distance the car traveled before it stopped? Why? The value of L6 will be an overestimate.