Assignment 2
MATH1061: Mathematics 1A
Semester 1, 2024
Please justify your answers. Correct answers without adequate justification will not receive full marks. A plot on its own is not considered adequate justification.
1. Consider f(x) = x3 + 3x2 − 9x − 6.
(a) Find and classify the critical point(s) for f(x) on [−4, 5].
(b) Find the global maximum and minimum values of f(x) on [−4, 5].
(c) Does f(x) have any point of inflection on [−4, 5]? If it does, locate this point of inflection; if not, explain why.
2. (a) For integer n ≥ 1, use upper and lower Riemann sums with n equal subdivisions to find an upper and lower bound for the value of
You may use the following fact without proof:
(b) Evaluate dx using the definition of Riemann integral.
3. Let f(x) = (9 + 5x + x2 )e−x.
(a) Find the 2nd order Taylor polynomial P2 (x) of f(x) centred at x = 0.
(b) Use the Lagrange form. of the remainder to obtain an upper bound for the remain- der R2 (x) when x = 1.
4. Let ℓ1 and ℓ2 be two lines in space defined by the parametric equations:
ℓ 1 : x = 3 − s, y = 4 + 5s, z = 3 + s (s ∈ R)
ℓ2 : x = 2 + t, y = −3 + t, z = −2 + 2t (t ∈ R)
Let P be the plane that contains ℓ1 and ℓ2 .
(a) Find the point of intersection of ℓ1 and ℓ2 .
(b) Find a general equation for P.
5. Let λ ∈ R, and consider the system of linear equations in the variables x,y,z given by
x − 5z = −4
x − λy − 2z = 2
x + 2y + λz = 2
(a) Row reduce the corresponding augmented matrix to row echelon form. (b) Find the values of the constant λ for which the system has
(i) no solutions
(ii) exactly one solution
(iii) infinitely many solutions
6. There are 2800 MATH1061 students, and they all do one of three things on a given night: they study linear algebra, they study calculus, or they watch netflix. We say a MATH1061 student
❼ is in State 1 if they study linear algebra;
❼ is in State 2 if they study calculus; and
❼ is in State 3 if they watch netflix.
MATH1061 students change their habits from night to night according to the following rules:
❼ If a student studies linear algebra one night, they have an 80% chance of studying linear algebra the next night; a 10% chance of studying calculus the next night; and a 10% chance of watching netflix the next night.
❼ If a student studies calculus one night, they have a 20% chance of studying linear algebra the next night; a 60% chance of studying calculus the next night; and a 20% chance of watching netflix the next night.
❼ If a student watches netflix one night, they have a 40% chance of studying linear algebra the next night; a 40% chance of studying calculus the next night; and a 20% chance of watching netflix the next night.
We encode the collection of probabilities of moving from one state to another in the matrix P = (pij )3×3, where
pij is the probability of moving from State j one night to State i the next night.
This means P is the matrix
where the middle column has been filled in for you.
(a) Finish writing down the matrix P.
(b) For night n we define the vector where
❼ x is the number of students in State 1;
❼ y is the number of students in State 2; and
❼ z is the number of students in State 3.
This means that for night n + 1 we have
xn+1 = Pxn.
Suppose initially we have 1000 students in State 1, 1000 students in State 2, and 800 students in State 3; or in other words,
Find the number of students in each state on night two, i.e. find x2 .
(c) By considering the system
(P − I3 )x = 0,
find all the vectors x that satisfy Px = x.
(d) Suppose that instead of the initial conditions in part (b), we initially have 1600 students studying linear algebra, 800 studying calculus, and 400 watching net-flix. Find the number of students studying linear algebra, the number of students studying calculus, and the number of students watching netflix on night n = 100.