AM10CO2020.dvi
AM10CO
1. Consider the function
f(x) =
x2
x3 + 4x
.
(a) Determine the domain and parity (if any) of the function.
(3 marks)
(b) Determine the critical points (maxima, minima and inflection points) of
f(x).
(6 marks)
(c) Determine regions of the domain where f(x) is increasing, decreasing,
concave and convex.
(6 marks)
(d) Determine the roots of f(x).
(3 marks)
(e) Plot the function f(x) using the provided graph paper.
(12 marks)
Total: 30 Marks
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AM10CO
2. Consider the following differential equation:
dy(x)
dx
+ 2xy(x) = f(x) , (1)
where
f(x) =



0 x < 0,
−2x3 0 < x < 1,
0 1 < x.
(2)
(a) Write and solve the associated homogeneous equation of (1).
(4 marks)
(b) Propose a solution to (1), for each of the regions where the non homoge-
neous term (2) has been defined, based on your results from (a) above.
(6 marks)
(c) Find a solution to the full problem.
(14 marks)
(d) Find a continuous solution to the full problem.
(6 marks)
(e) Adjust the free constant according to the condition y(1) = 1.
(5 marks)
Total: 35 Marks
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AM10CO
3. Consider the the following second-order differential equation:
d2y(x)
dx2
− 2
dy(x)
dx
+ 2y(x) = 2[sin(x/2)]2. (3)
(a) Write and solve the associated homogeneous equation of (3).
(10 marks)
(b) Propose a solution to (3), based on your results from (a) above.
(4 marks)
(c) Find the solution to the full problem.
Hint: Remember that 2[sin(x/2)]2 = 1− cos(x).
(16 marks)
(d) Adjust the free constants according to the conditions y(0) = 1 and y′(0) =
0.
(5 marks)
Total: 35 Marks
END OF ASSESSMENT
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AM10CO2019.dvi
AM10CO
1. Consider the function
f(x) =
{
1 x = 0
sin(x)
x
otherwise
.
(a) Determine the domain and parity (if any) of the function.
(3 marks)
(b) Determine the expressions satisfied by the critical points (maxima, min-
ima and inflection points), solving them for specific values when possible
(12 marks)
(c) Determine the roots of f(x).
(3 marks)
(d) Plot the function f(x) using the provided graph paper.
(12 marks)
Total: 30 Marks
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AM10CO
2. Find the continuous solution of the following equation:
3(x2 + 7)
dy(x)
dx
+ 2xy(x) = f(x) ,
where
f(x) =



0 x < 0
x 0 < x < 1
0 1 < x
,
satisfying the condtiton y(1) = 0.
Total: 35 Marks
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AM10CO
3. Solve the following equation:
d2y(x)
dx2
+
dy(x)
dx
− 2y(x) = x+ sin(x)
with y(0) = 1 and y′(0) = 0.
Total: 35 Marks
END OF EXAMINATION PAPER
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