3. If \( f(x, y) = \frac{xy}{x^2 + y^2} \), does \( \lim_{(x, y) \to (0, 0)} f(x, y) \) exists? Why? (10%)

4. A rectangular box without a lid is to be made from \( 12\mathrm{m}^2 \) of cardboard. Please use the method of Lagrange multiplier to find the maximum volume of such a box.(10%)

5. Find the radius of convergence and interval of convergence of the series \( \sum_{n=0}^{\infty} \frac{(-3)^n x^n}{\sqrt{n+1}} \). (15%)

一、 1、台北公司 20X0 年 10 月 1 日承租甲店面，租期 12 年，每年租金$360,000(每年期初 10 月 1 日支付)，租約不可撤銷，否則須支付違約金$3,000,000。20X8 年 9 月 30 日因捷運施工影響店面進出，台北公司改於其他地方租用乙店面；台北公司未撤銷甲店面租約且甲店面因地點不佳無法轉租而閒置。假設台北公司負債準備折現率為 6%。 試作：台北公司 20X8 年度財務報表上有關甲店面租約之淨損費(或淨利益)與負債金額各為何。(需詳列計算過程，否則不予計分)(10%)

2、三峽公司於 2018 年 1 月 1 日花費$60,000,000 完成興建一座垃圾掩埋場，預計可使用 10 年。垃圾掩埋場使用年限屆滿後，三峽公司必須進行場址復原與綠化工作，預計將支付$12,000,000 (以 2027 年底之物價估計)。 2019 年 12 月 31 日，三峽公司預期掩埋場場址復原與綠化成本應會下降至$10,000,000 (以 2027 年底之物價估計)。三峽公司採用直線法提列折舊，估計無殘值，折現率為 6%。 試作：三峽公司 2020 年度財務報表上有關該垃圾掩埋場之帳面金額、淨損費與負債金額各為何。(需詳列計算過程，否則不予計分)(15%)

二、 民生公司於 X5 年 12 月 31 日以$2,106,960(含交易成本)買入建國公司三年期公司債，面額$2,000,000，票面利率 8%，每年 12 月 31 日付息一次，原始有效利率 6%，該債券當日的 12 個月預期信用損失估計金額為$16,000。民生公司對該債券將採收取利息本金及出售的管理經營模式，並依 IFRS9 按公允價值衡量債券投資。 　X6 年 12 月 31 日，民生公司收到債券利息$160,000，該債券的信用風險已顯著增加，當日存續期間預期信用損失金額應為$208,000，公允價值為$1,900,000；民生公司當日稍後將該債券半數出售。 　X7 年 12 月 31 日，民生公司雖有收到債券利息$80,000；惟該債券不再於活絡市場中交易，已發生減損，當日存續期間預期信用損失金額應為$330,000，公允價值為$680,000。 　X8 年 12 月 31 日，民生公司只收到 X8 年的利息及本金共$730,200，其餘款項無法收回。 試作：民生公司 X7 年度財務報表上之債券投資帳面金額、處分投資損益、減損損失，與民生公司 X8 年度財務報表上之利息收入、減損損失各為何？(需詳列計算過程，否則不予計分)(25%)

1、Hunt Co. at the end of 2015, its first year of operations, prepared a reconciliation between pretax financial income and taxable income as follows: Estimated warranty expense of $800,000 will be deductible in 2016, $300,000 in 2017, and $100,000 in 2018. The use of the depreciable assets will result in taxable amounts of $450,000 in each of the next three years. What is income tax expense for 2015, assuming an income tax rate of 40% for all years.

2、Jack Company issued a two-year bond, face amount $10,000, 10% coupon rate on December 31, 20X0, interest payable annually for the proceeds of $10,000. Jack designated the bond measured at fair value through profit and loss, and considered that the recognition of change in the fair value of its own credit risk in other comprehensive income will not result in accounting mismatch. The original effective rate was 10% and the benchmark interest rate at the time of issuance was 6%. On December 31, 20X1, the fair value of the bond was $10,200 and the benchmark interest rate was 7%. What amount should be recognized in other comprehensive income (or loss) for year 20X1?

3、Rose Company granted 100 stock options for each of its 50 managers on January 1, 20X1. The vesting period was three years. The fair value of the stock option was $15 per unit. On December 31, 20X1, as the stock price of Rose Company had declined, the Company had reset the exercise price of the option downward, and the vesting period remained the same. The fair value of the option immediately before and after the reset was $6 and $8 respectively. On December 31, 20X1 and 20X2, the Company expected that 10 and 12 managers would leave the Company respectively. What compensation expense should the Company recognize for year 20X2?

(1) The company has decided to switch from the direct write-off method in accounting for bad debt expense to the percentage-of-sales approach. Assume that Haley Corporation has recognized bad debt expense as the receivables have actually become uncollectible in the following way: The controller estimates that an additional $65,400 will be charged off in 2017: $11,400 applicable to 2015 sales and $54,000 to 2016 sales.

(2) Inventory has been shipped on consignment. These transactions have been recorded as ordinary sales and billed as such on account. At December 31, 2016, inventory billed and in the hands of consignees amounted to $400,000. The percentage markup on selling price is 20%. Assume that consigned inventory is sold the following year. The company uses the perpetual inventory system. Instructions Assume that Haley Corporation erroneously reported net income of $1,000,000 for 2016. What should be the correct net income for 2016?

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