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Gunadarma University
Subject : Basic Mathematics
Faculty : Industrial Technology
Program of Study : Informatics Engineering Major
Years : 2024
Calculus Practice Quiz
1. Evaluate the limit : lim
x→ 2
𝑥
2
x - 2
A) 2
B) 4
C) 0
D) 8
Explanation:
lim
x→ 2
𝑥
2
x - 2
= lim
x→ 2
(x - 2 )(x + 2 )
x - 2
= lim
x→ 2
(x + 2 ) = 2 + 2 = 4
2. Determine if the following limit exists: lim
x→ 0
sin(x)
x
A) 0
B) 1
C) Does not exist
D) − 1
Explanation: This is a standard limit. By applying L'Hôpital's Rule or knowing the special limit:
lim
x→ 0
sin(x)
x
3. Find the limit: lim
x→∞
3 𝑥
2
𝑥
2
−𝑥+ 4
A) 1
B) 2
C) 3
D) 0
Explanation: Divide the numerator and denominator by x
2
lim
x→∞
3 +
2
x
1
𝑥
2
1 -
1
x
4
𝑥
2
3 + 0 + 0
1 - 0 + 0
- Evaluate the limit: lim
𝑥→ 0
A) −∞
B) 0
C) ∞
D) Does not exist
Explanation: As xxx approaches 000 from the positive side, the natural logarithm function goes
to −∞:
lim
𝑥→ 0
- Find the derivative of the function: f(x) = 3x
4
− 5x
3
A) 12x
3
− 15x
2
B) 12x
3
− 10x
2
C) 12x
4
− 15x
3
D) 12x
3
− 15x + 2
Explanation: Using logarithmic differentiation:
y = x
2
⟹ ln(y) = x ln(x) ⟹
1
y
dy
dx
= ln(x) + 1 ⟹
dy
dx
= y(ln(x) + 1) = x
2
- Find the critical points of the function f(x) = x
3
− 3x
2
A) x = 0
B) x = 2
C) x = 1
D) x = − 1
Explanation: Set the derivative equal to zero:
f′(x) = 3x
2
− 6x = 3x(x − 2) = 0 ⟹ x = 0 or x = 2
The critical point in the given options is x=1x = 1x= 1.
- Determine the intervals on which f(x) =
1
𝑥
2
is increasing or decreasing.
A) Increasing on (−∞, −1) ∪ (1, ∞), decreasing on (−1, 1)
B) Increasing on (−1, 1), decreasing on (−∞, −1) ∪ (1, ∞)
C) Increasing on (−∞, ∞)
D) Decreasing on (−∞, ∞)
Explanation: Compute the derivative:
f′(x) = -
2x
(𝑥
2
2
Analyze the sign of f′(x):
o f′(x) > 0 when − 1 < x < 1
o f′(x) < 0 when x < −1x or x > 1
