The Language of Differential Equations
Answers

1. Let N(t) denote the location measured in miles north of the start of the highway of the car at time t measured in hours. Notice that N(0) = 23 and N'(t) = 60.
   

2. Let p(x) denote the total cost (including shipping) of x pounds of coffee. Notice p(0) = $12.00 for shipping and

   
            10.00, dollars per pound if x <= 5;
   p'(x) =
             8.00, dollars per pound if 5 < x.
   

   

3. Let T(s) denote the total tax for a single taxpayer whose income is s dollars. Notice T(8000) = 0 and if s > 8000 then T'(s) = 0.17.
   

4. Let p(t) denote the population of the city at time t measured in years with t = 0 being now. Notice p(0) = 50,000 and p'(t) = 1,200.