Suppose saving $5000 a year from age 25 to 35 (10 deposits totaling $50,000) yields more at age 60 than saving $5000 a year from 35 to 60 (25 deposits totaling $125,000). Assuming an 8% (continuous) annual rate of return, use differential equations to verify or refute this claim.

Many people invest through a 401(k) or 403(b) plan where the contribution is a fixed percentage of salary, and salary changes every year. A starting professor is likely to make $55,000 and get a 3% raise every year. If a retirement plan has a 12% contribution, and a new professor works for 35 years, how much would their retirement plan be worth if it earns a good 8% annual return? (Ignore the usual 1-year eligibility delay for the sake of simplicity.) How does that compare to a very conservative portfolio that earns 3% instead?