1. Differential
Equations Pre-Test
or trafﬁc at the boarder
Vancouver, BC by ﬂickr user Tristen.Pelton
2. 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?
3. 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).
4. 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?
5. If y = 1 when x = 4 ﬁnd the solution to the differential equation
6. Find a function ƒ(x) which satisﬁes the equations ƒ(x)ƒ'(x)=x and
ƒ(0) = 1.
7. 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.