Calculus Practice Questions
IB (International Baccalaureate) · IB Math AA HL · 141 free MCQs with instant results and detailed explanations.
141
Total
31
Easy
76
Medium
34
Hard
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Sample Questions from Calculus
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Q1
Easy
What is the derivative of the function f(x) = 3x^4 - 5x^2 + 6?
A.
12x^3 - 10x
B.
12x^4 - 10x^2
C.
3x^3 - 5x
D.
6x^3 - 10
Show Answer & Explanation
Correct Answer: A
The derivative of f(x) is found by applying the power rule. For each term, multiply the coefficient by the exponent and decrease the exponent by one. Thus, the derivative is 12x^3 - 10x.
Q2
Easy
Which of the following represents the local maximum of the function f(x) = -x^2 + 4x?
A.
At x = 2
B.
At x = 0
C.
At x = 4
D.
At x = 1
Show Answer & Explanation
Correct Answer: A
To find the local maximum, we first take the derivative f'(x) = -2x + 4 and set it to zero. Solving gives x = 2, which is a maximum for this downward-opening parabola.
Q3
Easy
If the area under the curve f(x) = x^3 from x = 1 to x = 3 is calculated, what is the value of this integral?
A.
20
B.
32
C.
24
D.
18
Show Answer & Explanation
Correct Answer: C
The integral of f(x) = x^3 from 1 to 3 is calculated as [1/4 * x^4] evaluated from 1 to 3, which gives (1/4 * 3^4) - (1/4 * 1^4) = 20 - 0.25 = 24.
Q4
Medium
Evaluate the limit: lim (x → 3) (x^2 - 9)/(x - 3).
A.
6
B.
9
C.
0
D.
Undefined
Show Answer & Explanation
Correct Answer: A
Factoring the numerator gives (x-3)(x+3)/(x-3), which simplifies to x+3. Plugging in x=3 results in 6.
Q5
Medium
Evaluate the integral ∫ (2x^3 - 4x^2 + 6) dx from x = 1 to x = 3.
A.
12
B.
20
C.
16
D.
18
Show Answer & Explanation
Correct Answer: B
The definite integral evaluates the area under the curve from x=1 to x=3. First, find the antiderivative: (1/2)x^4 - (4/3)x^3 + 6x. Plugging in the limits gives (81/2 - 8/3 + 18) = 20.
Q6
Medium
A particle moves along a line, and its position is given by s(t) = t^3 - 6t^2 + 9t. What is the acceleration of the particle at t = 2?
A.
6
B.
0
C.
12
D.
9
Show Answer & Explanation
Correct Answer: B
Acceleration is the second derivative of the position function. First, find the first derivative (velocity): s'(t) = 3t^2 - 12t + 9. Then, differentiate again to get acceleration: s''(t) = 6t - 12. At t = 2, s''(2) = 0.
Q7
Medium
Consider the function f(x) = x^2ln(x). What is the value of f'(1)?
A.
1
B.
2
C.
0
D.
ln(1) + 1
Show Answer & Explanation
Correct Answer: B
Using the product rule, f'(x) = 2xln(x) + x^2(1/x) = 2xln(x) + x. Evaluating at x = 1 gives f'(1) = 2(1)(0) + 1 = 2.
Q8
Hard
Consider the function f(x) = x^3 - 3x^2 + 4. What is the x-coordinate of the local minimum of this function?
A.
1
B.
2
C.
0
D.
3
Show Answer & Explanation
Correct Answer: B
To find the local minimum, we first take the derivative f'(x) = 3x^2 - 6x. Setting this to zero gives x(3x - 6) = 0, leading to x = 0 or x = 2. Evaluating the second derivative f''(x) = 6x - 6 shows that f''(2) = 6(2) - 6 = 6 > 0, indicating a local minimum at x = 2.
Q9
Hard
A particle moves along a line such that its position is given by s(t) = 4t^3 - 12t^2 + 9t. What is the maximum displacement from the origin?
A.
3
B.
4
C.
5
D.
6
Show Answer & Explanation
Correct Answer: A
To find the maximum displacement, we first take the derivative of the position function, s'(t) = 12t^2 - 24t + 9, and set it to zero. Solving gives t = 1 and t = 0.75. Evaluating s(t) at these points reveals that the maximum displacement occurs at s(1) = 3 and is the correct answer.
Q10
Hard
Given the function f(x) = 3x^4 - 12x^3 + 9x^2, what is the x-coordinate of the local maximum?
A.
1
B.
2
C.
3
D.
0
Show Answer & Explanation
Correct Answer: B
To find the local maximum, we first need to find the critical points by setting the first derivative f'(x) = 12x^3 - 36x^2 + 18x to zero. Factoring gives us x(12x^2 - 36x + 18) = 0, leading to x = 0 or solving the quadratic gives us x = 2. Evaluating the second derivative confirms that x = 2 is a local maximum.
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Calculus — IB (International Baccalaureate) IB Math AA HL Practice Questions Online
This page contains 141 practice MCQs for the chapter Calculus in IB (International Baccalaureate) IB Math AA HL. The questions are organized by difficulty — 31 easy, 76 medium, 34 hard — so you can choose the right level for your preparation.
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