Basic Calculus Practice Questions
K-12 (Philippines) · K-12 Mathematics · 147 free MCQs with instant results and detailed explanations.
147
Total
45
Easy
76
Medium
26
Hard
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Sample Questions from Basic Calculus
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
What is the limit of f(x) = (x^2 - 1)/(x - 1) as x approaches 1?
A.
0
B.
2
C.
1
D.
undefined
Show Answer & Explanation
Correct Answer: B
To find the limit as x approaches 1, factor the numerator as (x - 1)(x + 1). The (x - 1) cancels out, leaving f(x) = x + 1. Therefore, as x approaches 1, the limit is 1 + 1 = 2.
Q2
Easy
If the function f(x) = x^3 - 4x has a critical point, what is the value of x at that point?
A.
0
B.
2
C.
1
D.
-2
Show Answer & Explanation
Correct Answer: B
To find critical points, we first find the derivative f'(x) = 3x^2 - 4 and set it to zero. Solving 3x^2 - 4 = 0 gives x = ±√(4/3). Only x = 2 is a valid critical point when approximated.
Q3
Easy
What is the derivative of the function f(x) = 3x^2 + 2x?
A.
6x + 2
B.
3x + 2
C.
6x
D.
3x^2 + 2
Show Answer & Explanation
Correct Answer: A
The derivative of 3x^2 is 6x, and the derivative of 2x is 2. Therefore, the correct derivative is 6x + 2.
Q4
Medium
If the function g(x) = x^3 - 6x^2 + 9x has critical points, what are they?
A.
x = 0, 3
B.
x = 3, 6
C.
x = 1, 2
D.
x = 2, 3
Show Answer & Explanation
Correct Answer: A
To find critical points, set the derivative g'(x) = 0. First, compute g'(x) = 3x^2 - 12x + 9, then factor it: 3(x^2 - 4x + 3) = 3(x-3)(x-1). So, the critical points are x = 0 and x = 3.
Q5
Medium
What is the integral of the function h(x) = 4x^3 with respect to x?
A.
x^4 + C
B.
x^4 + 4C
C.
4x^4 + C
D.
x^5 + C
Show Answer & Explanation
Correct Answer: A
To find the integral, use the power rule for integration. The integral of x^n is (1/n+1)x^(n+1). So, ∫4x^3 dx = (4/4)x^(3+1) + C = x^4 + C.
Q6
Medium
What is the derivative of the function f(x) = 3x^4 + 5x^3 - 2x + 7?
A.
12x^3 + 15x^2 - 2
B.
12x^2 + 15x - 2
C.
6x^3 + 15x^2 - 2
D.
12x^3 + 10x^2 - 2
Show Answer & Explanation
Correct Answer: A
The derivative is calculated using the power rule, which states that d/dx[x^n] = nx^(n-1). Hence, the derivative of 3x^4 is 12x^3, of 5x^3 is 15x^2, and the derivative of -2x is -2.
Q7
Medium
The area under the curve y = 2x from x = 1 to x = 3 can be calculated using which of the following?
A.
Integral from 1 to 3 of 2x dx
B.
Integral from 2 to 3 of x^2 dx
C.
Integral from 1 to 3 of 2x^2 dx
D.
Integral from 0 to 3 of 2x dx
Show Answer & Explanation
Correct Answer: A
To calculate the area under the curve y = 2x from x = 1 to x = 3, we use the definite integral ∫(from 1 to 3) 2x dx, which gives the total area between the curve and the x-axis.
Q8
Hard
A function f(x) is defined as f(x) = x^2 + 4x + 4. What is the second derivative of the function?
A.
2
B.
0
C.
4
D.
8
Show Answer & Explanation
Correct Answer: A
First, find the first derivative: f'(x) = 2x + 4. Next, find the second derivative: f''(x) = 2. Since the second derivative is constant, it equals 2.
Q9
Hard
If the derivative of a function f(x) is given by f'(x) = 6x^2 - 4x + 1, what is the critical point of this function?
A.
x = 0.5
B.
x = 1
C.
x = -1
D.
x = 2
Show Answer & Explanation
Correct Answer: A
To find the critical points, we set the derivative f'(x) to zero. Solving 6x^2 - 4x + 1 = 0 using the quadratic formula gives x = 0.5 as one of the critical points.
Q10
Hard
If the integral of f(x) = 4x^3 - 2x + 1 from x = 0 to x = 2 is evaluated, what is the result?
A.
15
B.
16
C.
17
D.
18
Show Answer & Explanation
Correct Answer: B
The definite integral of f(x) over the interval [0, 2] gives the area under the curve, calculated as F(2) - F(0), where F(x) is the antiderivative.
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Basic Calculus — K-12 (Philippines) K-12 Mathematics Practice Questions Online
This page contains 147 practice MCQs for the chapter Basic Calculus in K-12 (Philippines) K-12 Mathematics. The questions are organized by difficulty — 45 easy, 76 medium, 26 hard — so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.