MATH1021: Calculus of one variable Semester 1, 2023 Assignment 2
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Assignment 2
MATH1021: Calculus of one variable
Semester 1, 2023
1. Using the definition of derivative, show that the function
f (x) = ,x(x)e2
is differentiable at x = 0 and find f\ (0).
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.
3. Let
1 arctan(x)
+ x
(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 dx − +1 dx │ ≤
(c) Hence calculate I with an error up to .
4. Let α > 0 be a (fixed) real number.
n
(a) Using lower and upper Riemann sums for x dx, prove that for any integera
+
n ≥ 1, we have
n
1a + 2a + . . . + (n − 1)a ≤ xa dx ≤ 1a + 2a + . . . + na .
+
(b) Using part (a), or otherwise, prove that
1a + 2a + . . . + na 1
n人o naα1 α + 1 .
2023-05-12