Complete following to the best of your ability. Hand in your results in stapled hardcopy at the beginning of class on Tuesday 9 January.

Part 1

Compute the following:

  1. \({\frac{\mathrm{d}^{}{t^3}}{\mathrm{d}{t}^{}}}\)

  2. \({\frac{\mathrm{d}^{2}{}}{\mathrm{d}{x}^{2}}}(x^5+2x+9)\)

  3. \({\frac{\mathrm{d}^{}{\cos(2\theta)}}{\mathrm{d}{\theta}^{}}}\)

  4. \({\frac{\partial^{}{}}{\partial{x}^{}}}(x^2+y^3+4c)\)

  5. \(\int\!x^3{\mathrm{d}{x}}\)

  6. \(\int\!\frac{{\mathrm{d}{x}}}{x}\)

  7. \(\int_1^5\!\frac{{\mathrm{d}{x}}}{x}\)

Part 2

During an algal bloom in Lake Erie, the biomass of algae grows at rate \(G(t)\) kg/da, where \(t\) is time (in da).

  1. What is the interpretation of \({\frac{\mathrm{d}^{}{G}}{\mathrm{d}{t}^{}}}\)?

  2. What is the interpretation of \(\int_0^{30}\!G(t)\,{\mathrm{d}{t}}\)?

  1. On the axes above, illustrate graphically the interpretation of \[\left.{\frac{\mathrm{d}^{}{G}}{\mathrm{d}{t}^{}}}\right\vert_{t=40}\quad\text{and}\quad\int_0^{30}\!G(t)\,{\mathrm{d}{t}}.\]

  2. On what day, approximately, did the bloom peak?

Part 3

  1. After the execution of the following pseudocode, what is the value of x?
x = 0
for j = 1 to 10
  x = x + j
end for
  1. After the execution of the following pseudocode, what is the value of x?
x = 1
j = 1
while j < 10
  if (j > 5) 
    x = x - j
  else 
    x = x + j
  end if
  j = j + 1
end while

Part 4

  1. Let \[f(x)=\frac{x^2}{x^2+1}.\] Draw the graph of \(f\).

  2. Let \[g(x)=\frac{x}{x^2-1}.\] Draw the graph of \(g\).

  3. Let \[h(x)=2\sin(2x)+1.\] Draw the graph of \(h\).

Part 5

Solve the following equations for \(y(x)\):

  1. \({\frac{\mathrm{d}^{}{y}}{\mathrm{d}{x}^{}}}=x^2\), \(y(0)=0\)

  2. \({\frac{\mathrm{d}^{}{y}}{\mathrm{d}{x}^{}}}=2\,y\), \(y(0)=5\)

  3. \({\frac{\mathrm{d}^{}{y}}{\mathrm{d}{x}^{}}}=H-y\), \(y(0)=a\)

Part 6

  1. When two dice are thrown, what is the chance of rolling either 7 or 11?

  2. You are dealt a 5-card poker hand. What is the chance you have two pairs?

Part 7

A population of ducks occupies two sites (A and B) on opposite sides of a pond. Individual ducks move randomly between A and B. Over the course of an hour, a duck at site A has a 20% chance of flying to site B and a duck at site B has a 50% chance of flying to site A.

Suppose that there are 100 ducks at A and 200 at B.

  1. What is the expected number of ducks present at each site after 2 hours?

  2. What is the probability that exactly 10 ducks will have flown from A to B in 1 hour?

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