So I'm in week 4 of my Calc 2 class and there's two questions that I'm not sure where to even start. I've looked through everything that we've done and we touched on Reimann sums, but the professor didn't explain them that well. I've been over on Khan Academy trying to learn, but I'm not sure how their lessons pertain to my question.
Question 1:
let f be a continuous positive valued function defined on an interval [a, b]. Take a’partition’ of the interval consisting of only the interval [a, b] itself. Let U[a,b] be theupper Riemann sum of f on that partition. Create a refinement of the partitionby adding another point c so that the partition is now [a, c, b] and let S[a,c,b] be anyRiemann sum on the finer partition. Show that S[a,c,b] ≤ U[a,b]. Show by inductionthat this holds for any refinement of the partition by adding any number of pointsto the partition.
For question 2, I can't find anything like it in our textbook. The only example that I could find even halfway similar would be one that requires the radius of the cup. Since that is not given, I'm not sure what to do.
Question 2:
A cup of soy milk weighs 243 grams=0.243 kg and contains 131 (nutritional) calories.If one drinks the cup is three sips each time lifting the cup 40 cm =0.4meters and drinking a third of it. How much work is expanded on the act of drinking.Considering the work expanded in the drinking what is the net caloric intake resulting from drinking this cup ? Neglect teh weight of the cup itself.[hint: it is important to keep track of the units of work (Newtons) and convert them to energy units of nutritional calories. To calculate Newtons the mass must be in Kg and the distance in meters]
I'm not really looking for anyone to do the work for me, instead, I'd rather have someone explain what to do to these problems. Even better would be a link to something that converts the first problem to English and a lesson on the 2nd one. Thanks ahead of time!
Question 1:
let f be a continuous positive valued function defined on an interval [a, b]. Take a’partition’ of the interval consisting of only the interval [a, b] itself. Let U[a,b] be theupper Riemann sum of f on that partition. Create a refinement of the partitionby adding another point c so that the partition is now [a, c, b] and let S[a,c,b] be anyRiemann sum on the finer partition. Show that S[a,c,b] ≤ U[a,b]. Show by inductionthat this holds for any refinement of the partition by adding any number of pointsto the partition.
For question 2, I can't find anything like it in our textbook. The only example that I could find even halfway similar would be one that requires the radius of the cup. Since that is not given, I'm not sure what to do.
Question 2:
A cup of soy milk weighs 243 grams=0.243 kg and contains 131 (nutritional) calories.If one drinks the cup is three sips each time lifting the cup 40 cm =0.4meters and drinking a third of it. How much work is expanded on the act of drinking.Considering the work expanded in the drinking what is the net caloric intake resulting from drinking this cup ? Neglect teh weight of the cup itself.[hint: it is important to keep track of the units of work (Newtons) and convert them to energy units of nutritional calories. To calculate Newtons the mass must be in Kg and the distance in meters]
I'm not really looking for anyone to do the work for me, instead, I'd rather have someone explain what to do to these problems. Even better would be a link to something that converts the first problem to English and a lesson on the 2nd one. Thanks ahead of time!