| Module code: MA4010NI Module title: Calculus and Linear Algebra Module leader: Mr. Indra Dhakal (Islington College) |
| Coursework Type: Individual Coursework Weight: This coursework accounts for 25% of your total module grades. Submission Date: Week 4, 8, 12, 20, 24 for each 5 problem sheets. When Coursework is Week 3 given out: Submission Submit the following in the MST portal of your classroom Instructions: before the due date: ● A report (document) in .pdf format in the Google Classroom or through any medium which the module leader specifies. Warning: London Metropolitan University and Islington College takes Plagiarism seriously. Offenders will be dealt with sternly. |
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Page 1 of 6
Problem Sheet 1 (5% of the module mark) [20 marks] Deadline: Week 4
Question 1
a) Define composite functions.
b) It is given that ��(��) = √�� + 1 , ��(��) =1��.
Write the formulas for �� ∘ �� and �� ∘ ��.
c) Find the domain of each.
d) Let ��(��) =��
��−2. Find a function �� = ��(��) so that (�� ∘ ��)(��) = ��.
Question 2
a) Consider the function
��(��) =
i) Find the domain of ��.
√3 + ��13 − 2 �� − 1.
ii) Use a graphing utility to graph the function. iii) Find lim ��→−27+��(��).
iv) Find lim��→1��(��).
Page 2 of 6
Problem Sheet 2 (5% of the module mark) [20 marks] Deadline: Week 8
Question 1
a) If ��(��) =��
��−1, find the derivative of �� from the first principle.
b) i) Determine the slope of the graph of 3(��2 + ��2)2 = 100���� at the point (3, 1). ii) Find ��′′ if ��2 + ��2 = 25.
Question 2
a) Find the tangent and normal line to the graph of ��2(��2 + ��2) = ��2 at the point (√22,√22).
b) Determine the points of inflection and discuss the concavity of the graph ��(��) = ��4 − 4��3.
Page 3 of 6
Problem Sheet 3 (5% of the module mark) [20 marks]
Deadline: Week 12
Question 1
a) Evaluate:
∫1
√�� + 1 + √�� − 1����
b) Using partial fraction method integrate ∫��2+��−1
��3+��2−6������.
Question 2
a) Show that
√��
∫ �� ln �� ���� =14
1
b) Find the area of the surface formed by revolving the graph of ��(��) = ��3 on the interval [0, 1] about �� −axis.
Page 4 of 6
Problem Sheet 4 (5% of the module mark) (20 marks) Deadline: Week 20
Question 1
a) Express the complex number 5−√3��
1−√3��in �� + ���� form. Also, find the modulus of
the number.
b) Find the six sixth roots if �� = −8 and graph these roots in the complex plane.
Question 2
a) If the position vector of the points ��, �� and �� are −2��⃗+ ��⃗ − ��⃗⃗, −4��⃗ + 2��⃗ + 2��⃗⃗ and 6��⃗ − 3��⃗ − 13��⃗⃗ respectively.
i) Find ����⃗⃗⃗⃗⃗⃗ and ����⃗⃗⃗⃗⃗⃗.
ii) Are ����⃗⃗⃗⃗⃗⃗ and ����⃗⃗⃗⃗⃗⃗ perpendicular?
iii) If ����⃗⃗⃗⃗⃗⃗ = ������⃗⃗⃗⃗⃗⃗, then find the value of ��.
2
b) Check whether the vectors ��⃗⃗ = (
2
1
independent.
) , ��⃗ = (
−4 6
5
) , ��⃗⃗ = (
−2 8
6
) are linearly
Page 5 of 6
Problem Sheet 5 (5% of the module mark) [20 marks]
Deadline: Week 24
Question 1
Consider the system of equations
x + 2y – 3z = 4
3x – y + 5z = 2
4x + y + (��2-14) z = ��+2
a) Reduce the system to echelon form.
b) Find the values of the constant �� that will give no solutions.
c) Find the values of the constant �� that will give an infinite number of solutions. d) Find the values of the constant �� that will give a unique solution.
Question 2
Determine a basis and dimension of the solution space of the homogeneous system ��1 − 4��2 − 3��3 − 7��4 = 0
2��1 − ��2 + ��3 + 7��4 = 0
��1 + 2��2 + 3��3 + 11��4 = 0
Express the general solution of each system as a span or using arbitrary constants. End of paper
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