Differential Calculus Practice Questions
Matric (South Africa) · Matric Mathematics · 148 free MCQs with instant results and detailed explanations.
148
Total
32
Easy
76
Medium
40
Hard
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Sample Questions from Differential Calculus
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Q1
Easy
What is the derivative of f(x) = 3x^2 + 5x at x = 2?
A.
18
B.
12
C.
24
D.
22
Show Answer & Explanation
Correct Answer: A
To find the derivative, we differentiate f(x) to get f'(x) = 6x + 5. At x = 2, f'(2) = 6(2) + 5 = 12 + 5 = 18.
Q2
Easy
The slope of the tangent to the curve y = 2x^2 + 3 at x = -1 is what?
A.
-4
B.
0
C.
1
D.
4
Show Answer & Explanation
Correct Answer: A
To find the slope of the tangent, we first differentiate y to get dy/dx = 4x. Then, substituting x = -1 gives dy/dx = 4(-1) = -4.
Q3
Easy
What is the slope of the tangent line to the curve y = 2x² + 3 at x = 1?
A.
3
B.
4
C.
5
D.
6
Show Answer & Explanation
Correct Answer: B
The slope of the tangent line is given by the derivative of the function. For y = 2x² + 3, the derivative is y' = 4x. Thus, at x = 1, y'(1) = 4(1) = 4.
Q4
Medium
If the function f(x) = x^2 - 4x + 6 has a minimum point, what is the x-coordinate of this minimum point?
A.
2
B.
4
C.
0
D.
-2
Show Answer & Explanation
Correct Answer: A
The vertex form of a parabola gives the x-coordinate of the minimum point as -b/(2a). Here, a = 1 and b = -4, so x = 2.
Q5
Medium
What is the second derivative of the function f(x) = 2x^3 - 6x^2 + 3?
A.
12x - 12
B.
6x - 6
C.
6x + 6
D.
12x + 12
Show Answer & Explanation
Correct Answer: A
First, the first derivative f'(x) is 6x^2 - 12x, and the second derivative f''(x) is 12x - 12.
Q6
Medium
At what point does the function f(x) = x^3 - 3x^2 + 4 have a stationary point?
A.
x = 1
B.
x = 2
C.
x = 0
D.
x = 3
Show Answer & Explanation
Correct Answer: B
To find stationary points, set the first derivative f'(x) = 3x^2 - 6x to zero. Factoring gives x(x - 2) = 0, yielding x = 0 and x = 2; thus, the stationary point is at x = 2.
Q7
Medium
What is the derivative of the function f(x) = 3x^2 + 5x - 7?
A.
6x + 5
B.
3x + 5
C.
3x^2
D.
5x - 7
Show Answer & Explanation
Correct Answer: A
The derivative of f(x) with respect to x is calculated using the power rule. The derivative of 3x^2 is 6x, and the derivative of 5x is 5. Thus, the derivative is 6x + 5.
Q8
Hard
A function f(x) = x^3 - 6x^2 + 9x + 15 is given. What are the critical points of this function?
A.
(0, 15)
B.
(2, 3)
C.
(3, 0)
D.
(1, 19)
Show Answer & Explanation
Correct Answer: B
To find the critical points, we first need to compute the derivative f'(x) = 3x^2 - 12x + 9 and set it to zero. Solving 3x^2 - 12x + 9 = 0 gives us the critical points at x = 2 and x = 3, thus the point (2, 3) is a critical point based on evaluating f(2).
Q9
Hard
Given the function g(t) = 4t^4 - 16t^3 + 24t^2 - 12, determine the nature of the critical point at t=1.
A.
Local minimum
B.
Local maximum
C.
Point of inflection
D.
Neither maximum nor minimum
Show Answer & Explanation
Correct Answer: A
To determine the nature of the critical point at t=1, compute the second derivative g''(t) = 48t^2 - 96t + 48. Evaluating g''(1) gives 0, indicating a possible inflection point. However, checking the first derivative sign changes around t=1 shows it is a local minimum.
Q10
Hard
A curve is defined by the equation y = x^4 - 4x^3 + 6x^2 - 4. Which of the following statements is true about the critical points of this curve?
A.
It has two critical points, both of which are maxima.
B.
It has one critical point which is an inflection point.
C.
It has three critical points, with one being a minimum.
D.
It has four critical points, all of which are maxima.
Show Answer & Explanation
Correct Answer: C
To find critical points, we differentiate y to get y' = 4x^3 - 12x^2 + 12x. Setting y' to 0 yields three critical points. Analyzing the second derivative confirms one is a minimum. Thus, option C is correct.
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Differential Calculus — Matric (South Africa) Matric Mathematics Practice Questions Online
This page contains 148 practice MCQs for the chapter Differential Calculus in Matric (South Africa) Matric Mathematics. The questions are organized by difficulty — 32 easy, 76 medium, 40 hard — so you can choose the right level for your preparation.
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