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MAT-2510 Midterm: MATH 2510 App State Spring2015 Test2 answer key

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turquoisegnu128

15 Feb 2019

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Be sure to show your work: (30 points) converging questions (a) prove that (cid:28) 2n3 + n n3 + 3 (cid:29) converges. Notice that for n n we have n 6 so |n 6| = n 6 < n. for all n n , 2n3 + n n3 + 3 2 n3 + 3 n3 + 3(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) n 6 n3 + 3(cid:12)(cid:12)(cid:12)(cid:12) n 6 n3 + 3 n n3 + 3 n n3 = Hence, by the de nition of convergence, we have that (cid:28) 2n3 + n n3 + 3 (cid:29) converges to 2. 2n3 + n (cid:12)(cid:12)(cid:12)(cid:12) n2 + 1 (cid:29) converges. (b) prove that (cid:28) ( 1)nn. Notice that for n n we have n2 + 1 (cid:12)(cid:12)(cid:12)(cid:12) n2 + 1 0(cid:12)(cid:12)(cid:12)(cid:12) Hence, by the de nition of convergence, we have that (cid:28) ( 1)nn n2 + 1 (cid:29) converges to 0. (c) prove that (cid:28) n4 n 1(cid:29) n=2 diverges.

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ey table is seen below xeecions tor Chapter 1 12, Fori, b â¬ Î.el * b-aa (0 (D 14 ForVEQ r the operation defined in Example 1 on the wr John Sue Henry. P iDFor 15ine if is ass ociative by determining if the final six engu defined ang property (ii)of Theorem 1 . So no element i if the operntion debined in Example 1 of Emle if each element has an inverse Section 1.3 Groups iw comutatityn ild Chapter 3 where the Funda the exercises that follow etermining why they are not Write out the Cayley table for the gru(o)nud lensify thee element the 10 elements of the group Z 18 its operation uses ts in the first coordinnte and +2 in the secotu Ldessity the Zn and write out the Cayley table. Reeall th of each element does Theorem 1.14 tell us aont the entries in the Cayley table ol s xrewp that is set with exactly one element. A {n), will always form group since there 19. What nly one way to define an operation on the set. Be sure to shhow that the properi combination of the usual notice what set the rule is be of a group all hold. 21. We can create even create groupe with games! Consider four cups placed in a ssare patter table. If we have a penny in one of the cups there are four ways we can move it to t cup: Horizontally, Vertically, Diagonally, or Stay nehere it is. We will p, S. To define nn operation, consider two movements in a row, v.e.,m ove the penny as r tells us to, then after that move the penny as y instruets. the set of integers the set of real numbers n the set of positive integers. e set of positive integers. n the set of positive integers. on the set of integers e set of negative integers. we ple, HV D since if we first move it horizontally and then vertically nltogether e moved it diagonally. Create the Cayley table for this group, identify the identity and the inverse of cach element 22. Prove that the set A 0 under matrix multiplication. Is it abelian? 23. Verify that 3(R) is a group using the addition of funetions as defined in Example 13 e set of nonzero rational n he set of nonzero real num n on the set R (-1 Section 1.4 Basic Algebra in Groups 24. Prove the Cancellation Law (Theorem 1.20). Do not use any theorems that occur after it in the text ssociative, commutative, has roof or counterexample fo 25. Assume G is a group and a e G. Consider a fixed integer k and use PMI (Theorem 0.3) to prove that for all nEN, a aYou will need to consider cases here since k can be positive, negative, or o 26. Prove Theorem 1.23 (ii) for the case of n > 0 and m

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Complete this assignment in Maple. When you are finished, save it to your hard drive and email a copy of your Maple worksheet to me at Write "Math 1204 Lab 4: Name1 and Name2" exactly like this as the subject of your email, where Name1 and Name2 are your names (if you work on this alone, put one name). This lab will get you to construct an interpolating polynomial from given data. Open Maple and load the Linear Algebra package as usual. A chemical company is testing the exothermic properties of a new molecule being combined with 100 ml Hydrochloric Acid at 21 degrees C. Experimentally, they have recovered the following data: Look in your notes for Section 2.7 and remind yourself how polynomial interpolation works. Using MAPLE and our techniques from Section 2.7, find an interpolating polynomial that passes through all six data points. To do this, it might be easier to set each amount of molecule as a variable, like this: and so on. Using this polynomial estimate the temperature of the HCl when 400 mg of the molecule is added 125 mg is added. 50 mg is added. We will now graph the data points and your polynomial on the same graph. First, we need to include the 'plots' package from Maple with the following command: > with (plots); Make a list of your ordered pairs of data and call it K. Example: if I wanted to store the ordered pairs (1, 2) and (6, -1) in a list named K, I would input the following command: > K: [[1, 1], [6, -1]]; Note the MAPLE uses square brackets for ordered pairs, and lists as well. Plot your points with the following command: > pointplot(K); Input your interpolating polynomial into MAPLE and call it f. Plot your interpolating polynomial with the following command: > plot(f, x = 1..6); Note that the second part of the plot function tells MAPLE the bounds for x in our graph. Plot these together on the plot function tells MAPLE the bounds for x in our graph. Plot these together on the same graph in the following way: > display ({pointplote(K), plot(f,x=-1..6)}); What do you notice about your data points in relation to your function? An intern is asked to run the experiment again. They recover the following data. Attempt to use polynomial interpolation for this set of data. Why doesn't work? Explain why there is no function that can model this data.