Download MAT 3611 Calculus 1 Tutorial: Function Definition, Domains, Accumulation Points, Limits and more Quizzes Calculus in PDF only on Docsity!
University of Namibia Department of Computing, Mathematical and Statistical Sciences
MAT 3611 Calculus 1 Tutorial 1
February 19, 2022
- Decide wether the following relation define f as a function of x. In the positive case, give the domain of f.
a)
f (x) =
x^4 − 16 if x ≥ 0 , 6 − x if x^2 ≤ 4.
b) f (x) =
x^4 − 16 if x ≥ 0 , 2 − x if x^2 ≤ 4.
c) x^2 f (x) + x(f (x))^2 = 2 d) x^2 f (x) + x(f (x))^2 = 2 and f (x) + x 2 < 0
- Find the domain of the following function.
a) f (x) =
r log (^2) x 3 x −x^2 +9x+ b) f (x) = √ln(cos^ x) ex^2 − 2
c) f (x) = 4
r 2 −log (^1) 3 (3x−2) |x− 1 |− √ 2 x^2 − 1
d) f (x) = 6
q (
2 x)x^ − (2x)
√x
- Prove the following statements.
a) 1 is not an accumulation point of { 1 , 2 , 3 } b) 1 is an accumulation point of the open interval (1, 2) c) If A, B ⊆ R such that A ⊆ B, then every accumulation point of A is also an accumulation point of B
- Find the set of all accumulation points of the given set.
a) Q = the set of Rational numbers b) Z c) { (^) n^2 : n ∈ N} d) Each type of interval on the real number line. NB There is exactly 11 types of intervals on the real number line.
- Consider the function f (x) = x
(^4) − 10 x (^2) + x^2 − 2 x− 3. a) Find Df , the domain of f. b) Find the set A = {all accumulation points of Df , which are not in Df }. c) Compute lim x→a f (x), where a ∈ A.
- Prove the following statement using the ε, δ definiton of limits.
a) lim x→− 1. 5
x^2 x+3 = 1.^5 b) If a ̸= 0, then lim x→t (ax^2 + bx + c) = at^2 + bt + c
c) lim x→−√ 3
x^2 + 1 = 2
d) lim x→a
p |x − a| + b = b, where a, b ∈ R
e) lim x→ 1 +
√^1 x−1+2 =^
1 2
- Illustrate the precise definiton of limits by finding the values of δ that correspond to the given values of ε.
a) lim x→ 1
x + 1) = 2 for (i) ε = 2, (ii) ε = 10−^3
b) lim x→ 0
1 x+1 = 1 for (i)^ ε^ = 1,^ (ii)^ ε^ = 10
− 2
- Calculate the limits where they exist of the following functions, and clearly indicate why, when they do not exist. In each case, discuss the continuity of the function at the point where the limit is computed. If the function fail to be continuous, try to redefine the function, so that it will be continuous at that point.
a) lim x→ 5
x^2 − 6 x+ x− 5 b) lim x→ 3
x− 3 x^3 − 27
c) lim x→ 4
x−^1 − 4 −^1 x− 4
d) lim x→− 1
|x|− 1 x+
