The University of Sydney
School of Mathematics and Statistics
Assignment 2
MATH1021: Calculus of one variable Semester 1, 2023
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Copyright c© 2023 The University of Sydney 1
1. Using the definition of derivative, show that the function
f(x) =
{
xe
1
x , x < 0
x2, x ≥ 0
is differentiable at x = 0 and find f ′(0).
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2. Consider
f(x) = ax3 + x+ 1,
where a is a real-valued parameter. Find all possible values of a such that on the interval
[−1, 1], f has global maximum equal to 4/3 and global minimum equal to 2/3.
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3. Let
I =
∫ 1
0
arctan(x)
x
dx.
(a) Find the Taylor polynomial of order 2, P2(x), about x = 0 for the function
arctan(x).
(b) Use Lagrange’s formula for the remainder R2(x) = arctan(x)−P2(x) to show that∣∣∣∣∫ 1
0
arctan(x)
x
dx−
∫ 1
0
P2(x)
x
dx
∣∣∣∣ ≤ 19
(c) Hence calculate I with an error up to 1
9
.
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4. Let α > 0 be a (fixed) real number.
(a) Using lower and upper Riemann sums for
∫ n
0
xαdx, prove that for any integer
n ≥ 1, we have
1α + 2α + . . .+ (n− 1)α ≤
∫ n
0
xαdx ≤ 1α + 2α + . . .+ nα.
(b) Using part (a), or otherwise, prove that
lim
n→∞
1α + 2α + . . .+ nα
nα+1
=
1
α + 1
.
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