Calculus Practice Questions
IB (International Baccalaureate) · IB Math AI SL · 147 free MCQs with instant results and detailed explanations.
147
Total
37
Easy
73
Medium
37
Hard
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Sample Questions from Calculus
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
If the function f(x) = x^3 - 6x^2 + 9x has a critical point at x = 3, what is the nature of this critical point?
A.
Local maximum
B.
Local minimum
C.
Point of inflection
D.
Global maximum
Show Answer & Explanation
Correct Answer: B
To determine the nature of the critical point, we evaluate the second derivative at x = 3; f''(x) > 0 indicates a local minimum.
Q2
Easy
Evaluate the integral ∫ (2x + 3) dx from x = 0 to x = 2.
A.
9
B.
12
C.
8
D.
6
Show Answer & Explanation
Correct Answer: A
The integral ∫ (2x + 3) dx results in x^2 + 3x; evaluating from 0 to 2 gives 9.
Q3
Easy
The area under the curve y = 4x between x = 1 and x = 3 can be calculated using which integral?
A.
∫ from 1 to 3 of 4x dx
B.
∫ from 1 to 3 of 4x^2 dx
C.
∫ from 1 to 3 of 2x dx
D.
∫ from 1 to 3 of 4 dx
Show Answer & Explanation
Correct Answer: A
To find the area under the curve y = 4x from x = 1 to x = 3, we set up the integral ∫ from 1 to 3 of 4x dx. This will give the exact area under the curve between those limits.
Q4
Medium
If f(x) = x^3 - 3x, what is the x-coordinate of the local maximum?
A.
1
B.
-1
C.
0
D.
3
Show Answer & Explanation
Correct Answer: B
To find local maxima, set f'(x) = 0. f'(x) = 3x^2 - 3. Setting it to 0 gives x^2 = 1, so x = ±1. Test values to find local maximum at x = -1.
Q5
Medium
The function f(x) = 2sin(x) + cos(2x) has a maximum value. What is this maximum value?
A.
3
B.
2
C.
1
D.
0
Show Answer & Explanation
Correct Answer: A
The maximum value occurs when sin(x) = 1 (which gives 2), and cos(2x) = 1 (which gives 1). Therefore, the maximum value is 2 + 1 = 3.
Q6
Medium
What is the second derivative of the function f(x) = e^(2x) at x = 0?
A.
2
B.
1
C.
0
D.
4
Show Answer & Explanation
Correct Answer: D
The first derivative f'(x) = 2e^(2x). The second derivative f''(x) = 4e^(2x). Evaluating at x = 0 gives f''(0) = 4e^0 = 4.
Q7
Medium
The differential equation dy/dx = 3x^2 + 2y has what particular solution when y(0) = 1?
A.
y = x^3 + 1
B.
y = x^3 + 2
C.
y = 3x^3 + 1
D.
y = 3x^3 + 2
Show Answer & Explanation
Correct Answer: A
To solve the differential equation by integrating, we can separate variables and solve. The general solution comes out to be y = x^3 + C. Given y(0) = 1, we find C = 1. Therefore, the solution is y = x^3 + 1.
Q8
Hard
Given the function f(x) = x^3 - 3x^2 + 4, find the x-coordinate of the point where the function has a local maximum.
A.
1
B.
2
C.
0
D.
3
Show Answer & Explanation
Correct Answer: B
To find local maxima, we first find the derivative f'(x) = 3x^2 - 6x. Setting f'(x) = 0 gives x(3x - 6) = 0, leading to critical points at x = 0 and x = 2. Evaluating the second derivative f''(x) = 6x - 6 at x = 2 shows f''(2) = 6, indicating a local minimum, whereas f''(0) = -6 indicates a local maximum. Thus, the local maximum occurs at x = 2.
Q9
Hard
Evaluate the integral ∫ (2x^2 - 3x + 1) dx from x = 1 to x = 3.
A.
8
B.
10
C.
12
D.
14
Show Answer & Explanation
Correct Answer: B
To evaluate the definite integral, we first find the antiderivative F(x) = (2/3)x^3 - (3/2)x^2 + x. Evaluating this from 1 to 3 gives F(3) - F(1) = [(2/3)(3^3) - (3/2)(3^2) + 3] - [(2/3)(1^3) - (3/2)(1^2) + 1]. Calculating these values yields 10.
Q10
Hard
Consider the function f(x) = x^3 - 6x^2 + 9x. What is the x-coordinate of the point where the function has a local minimum?
A.
1
B.
3
C.
0
D.
2
Show Answer & Explanation
Correct Answer: D
To find local minima, we first calculate the first derivative f'(x) = 3x^2 - 12x + 9 and set it to zero. Solving 3(x^2 - 4x + 3) = 0 gives x = 1 and x = 3. To identify whether these points are minima or maxima, we use the second derivative test: f''(x) = 6x - 12. Evaluating f''(2) = 6(2) - 12 = 0, we check values around x = 2 and find it is a local minimum.
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Calculus — IB (International Baccalaureate) IB Math AI SL Practice Questions Online
This page contains 147 practice MCQs for the chapter Calculus in IB (International Baccalaureate) IB Math AI SL. The questions are organized by difficulty — 37 easy, 73 medium, 37 hard — so you can choose the right level for your preparation.
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