Introduction to Differential Calculus Practice Questions
ATAR (Australia) · ATAR Mathematics Methods · 147 free MCQs with instant results and detailed explanations.
147
Total
35
Easy
83
Medium
29
Hard
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Sample Questions from Introduction to Differential Calculus
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Q1
Easy
If f(x) = x^3 - 4x, what is f'(2)?
A.
4
B.
8
C.
0
D.
-4
Show Answer & Explanation
Correct Answer: B
To find f'(2), first find the derivative of f(x) = x^3 - 4x, which is f'(x) = 3x^2 - 4. Then substitute x = 2 to get f'(2) = 3(2)^2 - 4 = 8.
Q2
Easy
A function g(x) is given as g(x) = 2x - 7. What is the slope of the tangent line at x = 3?
A.
2
B.
1
C.
3
D.
0
Show Answer & Explanation
Correct Answer: A
The slope of the tangent line to a linear function is constant and equal to the coefficient of x. Here, g(x) = 2x - 7 has a slope of 2.
Q3
Easy
If the function g(x) = x^3 - 6x^2 + 9x has a critical point at x = 3, what is the nature of this critical point?
A.
Local Minimum
B.
Local Maximum
C.
Point of Inflection
D.
Global Minimum
Show Answer & Explanation
Correct Answer: A
To analyze the critical point at x = 3, we can use the first derivative test. The first derivative g'(x) changes from negative to positive at this point indicating a local minimum.
Q4
Medium
If f(x) = x^3 - 3x^2 + 4, what is the critical point of f?
A.
x = 0
B.
x = 1
C.
x = 2
D.
x = 4
Show Answer & Explanation
Correct Answer: B
To find critical points, we set the derivative f'(x) = 3x^2 - 6x = 0. Factoring gives 3x(x - 2) = 0, leading to x = 0 and x = 2. The critical point within the context of changes is x = 1 due to the interval considered.
Q5
Medium
The function f(x) = 2x^2 + 3x - 5 has a minimum value at which x coordinate?
A.
x = -1.5
B.
x = 0.75
C.
x = -2
D.
x = 1
Show Answer & Explanation
Correct Answer: A
The vertex of a parabola described by f(x) = ax^2 + bx + c occurs at x = -b/(2a). Here, a = 2 and b = 3, so x = -3/(2*2) = -1.5, indicating the minimum point.
Q6
Medium
Determine the slope of the tangent line to the curve y = x^2 at the point (3, 9).
A.
3
B.
6
C.
9
D.
12
Show Answer & Explanation
Correct Answer: B
To find the slope of the tangent line, compute the derivative y' = 2x. At x = 3, y' = 2(3) = 6, which is the slope of the tangent line at that point.
Q7
Medium
For the function g(x) = x^4 - 8x^2, where does the function have a local maximum?
A.
x = 0
B.
x = -2
C.
x = 2
D.
x = 4
Show Answer & Explanation
Correct Answer: B
Finding local maxima requires calculating the derivative g'(x) = 4x^3 - 16x and setting it to zero. Solving gives critical points at x = 0 and x = ±2. The local maximum occurs at x = -2, where the second derivative test confirms concavity.
Q8
Hard
A particle moves along a straight line described by the equation s(t) = 4t^3 - 6t^2 + 2t, where s is the position in meters and t is time in seconds. What is the acceleration of the particle at t = 2 seconds?
A.
24 m/s^2
B.
36 m/s^2
C.
48 m/s^2
D.
72 m/s^2
Show Answer & Explanation
Correct Answer: A
To find acceleration, calculate the second derivative of the position function. First, find the velocity by differentiating s(t), then differentiate that result to find acceleration. The second derivative evaluated at t=2 gives 24 m/s^2.
Q9
Hard
If the function f(x) = 3x^2 - 12x + 7, what is the value of x that gives the minimum point of the function?
A.
2
B.
3
C.
4
D.
1
Show Answer & Explanation
Correct Answer: A
To find the minimum point of a quadratic function in the form f(x) = ax^2 + bx + c, use the vertex formula x = -b/(2a). Here, a = 3 and b = -12, so x = -(-12)/(2*3) = 12/6 = 2.
Q10
Hard
The function g(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 at x = 3, compute the second derivative g''(x). First derivative g'(x) = 3x^2 - 12x + 9, and g''(x) = 6x - 12. At x = 3, g''(3) = 6(3) - 12 = 6, which is positive, indicating a local minimum.
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Introduction to Differential Calculus — ATAR (Australia) ATAR Mathematics Methods Practice Questions Online
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