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MATH1062: Mathematics
Part A: Calculus Questions
Please justify your answers by showing all relevant works. Correct answers without adequate
justification will not receive full marks.
1. Write the following differential equations for the unknown function, y(x) with the variable
x, in the standard form and illustrate the direction field by drawing the tangent intervals
at the four points: (1, 0), (0, 1), (−1, 0) and (0,−1) in the (x, y)-plane.
For each differential equation, find the general solution, then find the particular solution
satisfying y(0) = 1, finally sketch the solution curve in the above illustration of direction
field. All drawing and sketch must be done manually by hand.
(a)
x
dy
dx
− 2x2ex2 = 0.
(b)
y
dy
dx
+ x = 0.
2. Consider a cubic water tank with the side length 4, placed horizontally. There is a hole
at the bottom. Due to a pump, the water will leave the tank according to the following
differential equation:
dV
dt
= −(1 + sin t)
√
V
where V (t) is the volume of water remaining in the tank at the time t.
(a) Find the general solution of this differential equation.
(b) Find the particular solution if the tank is full to start with, that is, V (0) = 43 = 64.
(c) For the solution from (b), let T be the time when the tank becomes empty, that
is V (T ) = 0. Is T > 14 or T < 14? Explain your choice.
2
Part B: Statistics Questions
Part B questions are provided in the R Markdown worksheet at https://canvas.sydney.edu.
au/courses/59770/files/38709347. You should answer these questions using the worksheet
and use the Knit button in R Studio to generate the html file. We can only mark the html
file.