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# AP Calculus AB April 14, 2009

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Differential Equations pre-test.

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### AP Calculus AB April 14, 2009

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