Eric Yang 發問時間： 科學數學 · 5 年前

# 一題微分的應用問題

FINANCING A HOME

The Millers are planning to buy a home in the near future and estimate that they will need a 30-year-fixed-rate mortgage for \$240,000. Their monthly payment P (in dollars) can be computed by using the formula

where r is the interest rate per year.

(a)Find the differential of P.

(b)If the interest rate increases from the present rate of 5%/year to 5.2%/year between now and the time the Millers decide to secure the loan, approximately how much more will their monthly mortgage payment be? How much more will it be if the interest rate increases to 5.3%/year? To 5.4%/year? To 5.5%/year?

I have been stuck on this for almost an hour, any help is appreciated!

### 2 個解答

• 麻辣
Lv 7
5 年前
最佳解答

P=20000r/w(r)where r is interest rate per year & w(r)=1-(1+r/12)^(-360)

(a) Find the differential of P.w'=360(1+r/12)^(-361)/12=30(1+r/12)^(-361)P'=20000(w-r*w')/w^2=20000{w-30r(1+r)^(-361)}/w^2

(b-1) If the interest rate increases from the present rate of 5%/year to 5.2%/year between now and the time the Millers decide to secure the loan, approximately how much more will their monthly mortgage payment be? w(r)=1-(1+r/12)^(-360) w(0.05)=1-(1+0.05/12)^(-360)=0.776Δr=5.2%-5%=0.2%ΔP=20000{w-30r(1+r/12)^(-361)}Δr/w^2=20000{0.776-30*(1+0.05/12)^(-361)}*0.002/0.776^2=29.3356P=20000r/w=20000*0.05/0.776=1288P+ΔP=1288+29=1317(\$)

(b-2) How much more will it be if the interest rate increases to 5.3%/year? Δr=5.3%-5%=0.3%ΔP=20000{w-30r(1+r/12)^(-361)}Δr/w^2=20000{0.776-30*(1+0.05/12)^(-361)}*0.003/0.776^2=44(\$) (b-3) To 5.4%/year? To 5.5%/year? Δr=5.4%-5%=0.4%ΔP=20000{w-30r(1+r/12)^(-361)}Δr/w^2=20000{0.776-30*(1+0.05/12)^(-361)}*0.004/0.776^2=59(\$) (b-4) To 5.5%/year? Δr=5.5%-5%=0.5%ΔP=20000{w-30r(1+r/12)^(-361)}Δr/w^2=20000{0.776-30*(1+0.05/12)^(-361)}*0.005/0.776^2=73(\$)

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• 5 年前

P' = dP/dr 昆 derivative, 不是 differential. 後者是 dP = P' dr.

又: r 是 independent variable, 所以 dr = Δr;

但 dP 與 ΔP 不同, 只能說 ΔP approximate to dP.

第一小題當然是先算出 P', 但給的答案應是 P' dr.

第二小題是要計算 dP (當做 ΔP 私的近似).

根據所列各利率值得對應之 Δr, 設其為 dr, 而後代入前一小題

之 dP = P' dr 得 dP, 宣稱那是 ΔP 的近似值.

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