The document contains several problems related to differential equations: 1) It describes the location of a car over time based on its initial location and velocity formula. It asks to find the car's velocity at t=1 and location at t=5. 2) It asks to find a function that satisfies the differential equation f(x)f'(x)=x with the initial condition f(0)=1. 3) It asks to model the decay of radium over time based on its initial amount and decay rate, and express the amount remaining after t years.

1. Differential
Equations Pre-Test
or traffic at the boarder
Vancouver, BC by flickr user Tristen.Pelton

6. The location of a slow-moving automobile in miles north of the Canada/USA
border on highway 75 is given by a function y = ƒ(t), where t represents
time in hours since noon yesterday. Suppose that ƒ(1) = 40, that is, the car
is 40 miles north of the border at 1:00 pm. The velocity of the car (in miles
per hour) depends on both time and location, and is given by the formula:
(a) Find the car’s velocity at t = 1.
(b) Determine an explicit formula for the location y = ƒ(t).
(c) Where is the car at 5:00 pm?

7. The location of a slow-moving automobile in miles north of the Canada/USA
border on highway 75 is given by a function y = ƒ(t), where t represents
time in hours since noon yesterday. Suppose that ƒ(1) = 40, that is, the car
is 40 miles north of the border at 1:00 pm. The velocity of the car (in miles
per hour) depends on both time and location, and is given by the formula:
(b) Determine an explicit formula for the location y = ƒ(t).

8. The location of a slow-moving automobile in miles north of the Canada/USA
border on highway 75 is given by a function y = ƒ(t), where t represents
time in hours since noon yesterday. Suppose that ƒ(1) = 40, that is, the car
is 40 miles north of the border at 1:00 pm. The velocity of the car (in miles
per hour) depends on both time and location, and is given by the formula:
(c) Where is the car at 5:00 pm?

9. If y = 1 when x = 4 find the solution to the differential equation

10. Find a function ƒ(x) which satisfies the equations ƒ(x)ƒ'(x)=x and
ƒ(0) = 1.

11. Radium decomposes at a rate proportional to the amount present.
Find an expression for the amount R left after t years, if R0 is present
initially and c is the negative constant of proportionality.