2. In this question, we will explore the notion of a sample point in more depth. The multiple parts are there to gradually guide you through the analysis of a somewhat "pathological function and whether or not it is integrable. Consider the function (a) Graph this function. What is does your gut instinct tell you about the integrability of this function on (0,8)? Describe in a few sentences what makes you think this function is for is not) integrable. Don't use any formal mathematics at this point, we are just exploring (b) Split the interval (0.8) into 4 equal subintervals. For each subinterval, choce a sample point and calculate the associated finite Riemann sun, being sure to represent your sum with a picture. Do the same thing, but with a different set of sample points. Do your answers match or not? Why does this make sense? (e) Split the interval (0.8) into 5 equal subintervals. For each subinterval, choose a sample point and calculate the associated finite Riemann sum. being sure to represent your sum with a picture. Do the same thing, but with a different set of sample points. How was the choice of sample points different from the previous problem where we used 4 subintervals? (d) Let's make this more formal now. In order to show that a function f(t) is integrable on (1,6], we need to show that lim (+7) Ar exists and is equal for all choices of sample points, 1;. Draw out this Riemann sum as a picture for our function y(1) defined above. How many choices" are there for a sample point in a given subinterval? What are the possibilities for the output of the function for that sample point? Are there any choices of specific sample points that complicate matters? Why or why not? (e) Distil your observations in part (d) to show that there are exactly two cases for sets of sample points and their associated outputs. (1) Evaluate lim (4) Ar for the two cases of sample points you found in part (e). (8) What does your work in part (d) tell you about the integrability of g(x) on 10,8)? (h) Suppose that instead, we had h(2) - {% +2,46 Do you think h(x) would be integrable? Why or why not? Describe in a few sentences, tsing your work in the above parts to inform your answer. Again, you don't need to formally prove your claim. 2. In this question, we will explore the notion of a sample point in more depth. The multiple parts are there to gradually guide you through the analysis of a somewhat "pathological function and whether or not it is integrable. Consider the function (a) Graph this function. What is does your gut instinct tell you about the integrability of this function on (0,8)? Describe in a few sentences what makes you think this function is for is not) integrable. Don't use any formal mathematics at this point, we are just exploring (b) Split the interval (0.8) into 4 equal subintervals. For each subinterval, choce a sample point and calculate the associated finite Riemann sun, being sure to represent your sum with a picture. Do the same thing, but with a different set of sample points. Do your answers match or not? Why does this make sense? (e) Split the interval (0.8) into 5 equal subintervals. For each subinterval, choose a sample point and calculate the associated finite Riemann sum. being sure to represent your sum with a picture. Do the same thing, but with a different set of sample points. How was the choice of sample points different from the previous problem where we used 4 subintervals? (d) Let's make this more formal now. In order to show that a function f(t) is integrable on (1,6], we need to show that lim (+7) Ar exists and is equal for all choices of sample points, 1;. Draw out this Riemann sum as a picture for our function y(1) defined above. How many choices" are there for a sample point in a given subinterval? What are the possibilities for the output of the function for that sample point? Are there any choices of specific sample points that complicate matters? Why or why not? (e) Distil your observations in part (d) to show that there are exactly two cases for sets of sample points and their associated outputs. (1) Evaluate lim (4) Ar for the two cases of sample points you found in part (e). (8) What does your work in part (d) tell you about the integrability of g(x) on 10,8)? (h) Suppose that instead, we had h(2) - {% +2,46 Do you think h(x) would be integrable? Why or why not? Describe in a few sentences, tsing your work in the above parts to inform your answer. Again, you don't need to formally prove your claim