| 1. |
Find the distance between the points with coordinates  and  .
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| 2. |
What is the formula of the sphere centered at (1,2,3) and of radius equal to 5?
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| 3. |
What is the graph of the equation  ?
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| 4. |
Let  . Compute  .
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| 5. |
Find the domain of  .
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| 6. |
Find the domain of  .
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| 7. |
Find the domain of  .
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| 8. |
Compute  for the following functions:
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| 9. |
Classify the following equations as to their order, linearity/non-linearity, and as to whether they have constant coefficients. In all the problems below assume that  is a function of independent variable  .
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| 10. |
Solve the following differential equations:
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| 11. |
A certain population of bacteria, as a function of time measured in hours, is growing at the uniform rate equal to 2% of the population size. Suppose that the initial population contains 1000 cells. What will the population be at time  hours?
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| 12. |
A population of rabbits on an island triples every 5 years. How long does it take this population to double if the growth rate is always proportional to the population size?
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| 13. |
If a radiation dose of 1 rad kills 3% of cancer cells, how much radiation would kill 99% of the cells? (Assume that cancer cell death rate is always proportional to the number of cells.)
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| 14. |
Suppose a certain radioactive element has a half-life of 1000 years. How long before 99 % of the original amount is lost?
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| 15. |
Suppose a certain population of animals has a uniform birth rate of 20% with 2 young surviving per birth on the average and a uniform death rate of 10% when time is measured in years. What is the population size as a function of time, if the initial population was 200 animals?
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| 16. |
Solve the following equations.
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| 17. |
Suppose a certain population of animals has a uniform birth rate of 20% with 2 young surviving per birth on the average and a uniform death rate of 10% when time is measure in years. Assume also that each year 100 animals are moving into the area. What is the population size as a function of time, if the original population was 200 animals?
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| 18. |
Solve the following equations.
| (a) |
 ,  .
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| (b) |
,  .
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| (c) |
 , 
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| (d) |
 ,
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| (e) |
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| (f) |
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| 19. |
Suppose a population of mice in a house follows a logistic model with the maximum population equal to 300 mice. Initially the house had 10 mice. After 1 year the house had 50 mice. How many mice will live in the house after 5 years?
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