8.Find the eigenfunction of the following equation: y"+ ky =0, y(0)-y(L)-0.(15%)

9. Find the Fourier series in complex form for the periodic function f(t) with period 2, .(18%) 

7. Bigly or Big League? "Did he say bigly?" This question had been asked repeatedly on social media soon after the then-candidate Donald Trump was reported to use the word during the first US presidential debate. Many wondered whether bigly was really an English word or a "Trumpism" presumably derived from big and the adverb suffix -ly. However, based on acoustic analyses, some phoneticians argued that what Mr. Trump actually said was big league but the final /g/ in his league was just difficult for the untrained ear to detect. Figure 2 shows the waveform and spectrogram of Trump's production of what appears to be Figure 1 Donald Trump, the to-be ambiguous between bigly and big league. Look at the figure. president of the most powerful What are the acoustic cues to final /g in league? Name at least  TWO of them.

Question 2-1. Use the substitution test described above to identify all constituents in the following sentence. Explain how you discovered the constituents. (20%) The tall man might have taken the pen away.

Question 2-2. Sometimes a test may give false negative results. That is, a word string is a constituent, but the results suggest otherwise. Therefore, syntacticians usually apply multiple diagnoses. In addition to substituting pronouns, we can use one. Consider this sentence: I met the guy with brown hair from the US, but not the one from the UK. Here one is substituted for guy with brown hair, indicating that guy with brown hair is a constituent. Apply this one substitution test to the sentence in the above question. Does this test reveal any more constituents than what you have just identified? Why or why not? (5%)

Question 3-1. Describe the trends and features in the graph. (10%)

Question 3-2. Why did Labov reply "Excuse me?" to get a repetition of fourth floor from the employee? What do you think was the purpose of doing this? According to the graph, do the two pronunciations of the phrase differ? If yes, how? (8%)

Question 3-3. What could Labov's results tell us about the social meaning and function of postvocalic /x/ in New York? (7%)

Part 4.(25%) English has a more complex syllable structure than Mandarin Chinese does in the sense that English allows more consonants in the onset and coda. When an English word is borrowed into Mandarin Chinese, sound adaptation is often necessary. Examine the following data set (cited from Lin 2007: 241)  Question 4. Generalize how the onset and coda consonant clusters in English are modified by nucleus insertion when its words are borrowed into Mandarin Chinese. Explicitly state the conditions in which a certain vowel is chosen to be inserted. (25%) Pinyin [d3um]

(a) Write down the consumer's current period budget constraint and its future-period budget constraint. Also, derive the (present-value) lifetime budget constraint of the consumer. [10 pts]

(b) Use the graphical approach to show the optimal consumption/saving bundle for the consumer to be a lender. The figure should have C' on the vertical axis and C on the horizontal axis. It should also contain the lifetime budget of the consumers, and the indifference curve (IC). Make sure you mark the slope of the budget line, its endpoints, the endowment point, the equilibrium current and future consumption (C* and C'*), and the equilibrium saving (s*). [10 pts]

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