Integrals and Functions of Time

Question

1. There are two rival companies, Winner's Inc. and Loser's Co.. Winner's Inc.'s net worth is growing at a rate of 1/2 * t² thousand dollars per month (i.e. t is measured in months). Loser's Co. is operating at a loss, and their net worth is decreasing at a rate of t/3 thousand dollars per month (that is, it is given by the function −t/3). Winner's Inc. and Loser's Co. both start the year with $10,000 in the bank.
2. What does the function f(x)=∫ 1/2 * t² dt from 0 to x compute? What does g(x) = ∫ -t/3 dt from 0 to x compute? Construct a function that computes the difference of net worth between Winner's Inc. and Loser's Co. as a function of time.

Answer 1

The integrals both tell you the change in value of the company from t = 0 to a time of x. If you want to know the value of a company, it will be V₀ + integral. Note that the second company that is losing money will have a negative growth, so the integral is still just the change in value (it happens to be negative) so the overall value will be less than V₀