UNIVERSITY OF SOUTH AFRICA (UNISA)
College of Science, Engineering and Technology
⋄
ASSIGNMENT 1
Year Module — 2026
⋄
Module Code: MAT1613
Module Name: Calculus B
Assignment No.: Assignment 1
Due Date: 15 May 2026
Year: 2026
Submitted in partial fulfilment of the requirements for MAT1613 — Calculus B
at the University of South Africa.
,UNISA | MAT1613 Calculus B — Assignment 1
Question 1 — Asymptotes, Curve Analysis (21 Marks)
Question 1.1 — Horizontal and Vertical Asymptotes (5 marks)
Question: Find the horizontal and vertical asymptote of
√
2x2 + 1
f (x) =
3x − 5
Step 1: Vertical Asymptote
The vertical asymptote occurs where the denominator equals zero, provided the numerator is
non-zero at that point. Setting the denominator equal to zero:
5
3x − 5 = 0 =⇒ x =
3
5
q
5 2
At x = , the numerator 2 3 + 1 ̸= 0, so a vertical asymptote exists.
3
Step 2: Horizontal Asymptote as x → +∞
Factor x2 from inside the square root:
q q
1 1
x2 2 + x2
|x| 2 + x2
f (x) = =
3x − 5 3x − 5
For x → +∞, |x| = x. Dividing numerator and denominator by x:
q
1
2+ x2
f (x) = 5
3− x
1 5
As x → +∞, 2
→ 0 and → 0, therefore:
x x
√
2
lim f (x) =
x→+∞ 3
Page 1 of 15
, UNISA | MAT1613 Calculus B — Assignment 1
Step 3: Horizontal Asymptote as x → −∞
For x → −∞, |x| = −x, so:
q q
1 1
−x 2 + x2
− 2+ x2
f (x) = = 5
3x − 5 3− x
As x → −∞: √
− 2
lim f (x) =
x→−∞ 3
Key Result
5
Vertical asymptote: x =
3 √ √
2 2
Horizontal asymptotes: y = as x → +∞ and y = − as x → −∞
3 3
Question 1.2 — Analysis of f (x) = 2x3 − 9x2 + 12x − 3
(a) Increasing and Decreasing Intervals using Sign Pattern (5 marks)
Question: Use the sign pattern to find the intervals on which f is increasing or decreasing for
f (x) = 2x3 − 9x2 + 12x − 3.
Step 1: Find the first derivative.
f ′ (x) = 6x2 − 18x + 12
Step 2: Factor f ′ (x).
f ′ (x) = 6(x2 − 3x + 2) = 6(x − 1)(x − 2)
Step 3: Identify critical points.
Setting f ′ (x) = 0:
6(x − 1)(x − 2) = 0 =⇒ x = 1 and x = 2
Step 4: Sign table for f ′ (x).
Page 2 of 15
College of Science, Engineering and Technology
⋄
ASSIGNMENT 1
Year Module — 2026
⋄
Module Code: MAT1613
Module Name: Calculus B
Assignment No.: Assignment 1
Due Date: 15 May 2026
Year: 2026
Submitted in partial fulfilment of the requirements for MAT1613 — Calculus B
at the University of South Africa.
,UNISA | MAT1613 Calculus B — Assignment 1
Question 1 — Asymptotes, Curve Analysis (21 Marks)
Question 1.1 — Horizontal and Vertical Asymptotes (5 marks)
Question: Find the horizontal and vertical asymptote of
√
2x2 + 1
f (x) =
3x − 5
Step 1: Vertical Asymptote
The vertical asymptote occurs where the denominator equals zero, provided the numerator is
non-zero at that point. Setting the denominator equal to zero:
5
3x − 5 = 0 =⇒ x =
3
5
q
5 2
At x = , the numerator 2 3 + 1 ̸= 0, so a vertical asymptote exists.
3
Step 2: Horizontal Asymptote as x → +∞
Factor x2 from inside the square root:
q q
1 1
x2 2 + x2
|x| 2 + x2
f (x) = =
3x − 5 3x − 5
For x → +∞, |x| = x. Dividing numerator and denominator by x:
q
1
2+ x2
f (x) = 5
3− x
1 5
As x → +∞, 2
→ 0 and → 0, therefore:
x x
√
2
lim f (x) =
x→+∞ 3
Page 1 of 15
, UNISA | MAT1613 Calculus B — Assignment 1
Step 3: Horizontal Asymptote as x → −∞
For x → −∞, |x| = −x, so:
q q
1 1
−x 2 + x2
− 2+ x2
f (x) = = 5
3x − 5 3− x
As x → −∞: √
− 2
lim f (x) =
x→−∞ 3
Key Result
5
Vertical asymptote: x =
3 √ √
2 2
Horizontal asymptotes: y = as x → +∞ and y = − as x → −∞
3 3
Question 1.2 — Analysis of f (x) = 2x3 − 9x2 + 12x − 3
(a) Increasing and Decreasing Intervals using Sign Pattern (5 marks)
Question: Use the sign pattern to find the intervals on which f is increasing or decreasing for
f (x) = 2x3 − 9x2 + 12x − 3.
Step 1: Find the first derivative.
f ′ (x) = 6x2 − 18x + 12
Step 2: Factor f ′ (x).
f ′ (x) = 6(x2 − 3x + 2) = 6(x − 1)(x − 2)
Step 3: Identify critical points.
Setting f ′ (x) = 0:
6(x − 1)(x − 2) = 0 =⇒ x = 1 and x = 2
Step 4: Sign table for f ′ (x).
Page 2 of 15
