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MATH1021: Calculus of One Variable
This individual assignment is due by 11:59pm Thursday 12 May 2022, via
Canvas. Late assignments will receive a penalty of 5% per day until the closing
date. Please make sure you review your submission carefully. What you see is
exactly how the marker will see your assignment. Submissions can be overwritten until
the due date. To ensure compliance with our anonymous marking obligations, please
do not under any circumstances include your name in any area of your assignment;
you may include your SID. The School of Mathematics and Statistics encourages some
collaboration between students when working on problems, but students must write
up and submit their own version of the solutions. If you have technical difficulties
with your submission, see the University of Sydney Canvas Guide, available from the
Help section of Canvas.
This assignment is worth 10% of your final assessment for this course. Your answers should be
well written, neat, thoughtful, mathematically concise, and a pleasure to read. Please cite any
resources used and show all working. Present your arguments clearly using words of explanation
and diagrams where relevant. After all, mathematics is about communicating your ideas. This
is a worthwhile skill which takes time and effort to master.
1. Evaluate the following limit using two different methods. Explain your working carefully
for each method.
lim
x→∞
3x4 + 7x2 + x− 3
2x4 − 4x3 + 2x
Which method do you prefer? (There is no right or wrong answer to this question. You
should just say which one you like better.) Explain why you prefer this method over the
other method.
2. (a) Explain why the function f(x) = ex
2
is not injective (one-to-one) on its natural
domain.
(b) Find the largest possible domain A, where all elements of A are non-negative and
f : A→ R, f(x) = ex2 is injective.
(c) Find a codomain B such that f : A→ B, f(x) = ex2 is surjective.
(d) Show that g : B → A, g(x) = √lnx is the inverse of f . Why is f−1(x) 6= −√lnx?
3. (a) Find the 5th order Taylor polynomial about a = pi for sin x.
(b) Use this Taylor polynomial to approximate sin 11pi
12
in terms of powers of pi
12
.
(c) The remainder term for a polynomial of order n expanded about x = a is
Rn(x) = f
(n+1)(c)
(x− a)n+1
(n+ 1)!
where c lies between a and x. Write down the remainder term for the polynomial
that you have found in part (a).
(d) Use the remainder term that you have found in part (c) to show that theTaylor
polynomial approximation to sin 11pi
12
that you found in part (b) is within 10−6 of
the actual value of sin 11pi
12
. Use your calculator to verify that the difference between
sin 11pi
12
and your approximation is, indeed, less than 10−6.
2
4. This question asks you to find an integral using Riemann sums. Recall from lectures,
that a definite integral
∫ b
a
f(x) dx can be defined as
lim
n→∞
Ln =
∫ b
a
f(x) dx = lim
n→∞
Un
where Un and Ln are upper and lower Riemann sums respectively.
(a) Sketch a diagram showing the rectangles corresponding to upper and lower Rie-
mann sums, Un and Ln on n equal subintervals for the integral∫ 1
0
x2 dx.
Include the ith subinterval in your diagram.
(b) Write down expressions for Un and Ln using sigma notation.
(c) The sum of the squares of the first n positive integers is given by the following
formula:
12 + 22 + 32 + 42 + 52 + . . .+ (n− 1)2 + n2 =
n∑
k=1
k2 =
n(n+ 1)(2n+ 1)
6
.
Use this fact to show that Un =
(n+ 1)(2n+ 1)
6n2
and (harder) that Ln =
(n− 1)(2n− 1)
6n2
.
(d) Find lim
n→∞
Un and lim
n→∞
Ln. Verify, using the Fundamental Theorem of Calcu-
lus (that is, conventional methods of integration), that these limits are equal to∫ 1
0
x2 dx.
