Q. The function f(x) = { x^2, x < 0; 2, x = 0; x + 1, x > 0 } is continuous at x = 0?
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A.
Yes
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B.
No
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C.
Only left continuous
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D.
Only right continuous
Solution
At x = 0, lim x→0- f(x) = 0 and lim x→0+ f(x) = 1, hence it is discontinuous at x = 0.
Correct Answer: B — No
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Q. What is the critical point of f(x) = x^2 - 4x + 4? (2022)
Solution
Set f'(x) = 2x - 4 = 0; solving gives x = 2.
Correct Answer: B — 2
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Q. What is the critical point of the function f(x) = x^2 - 4x + 4? (2022)
Solution
Find f'(x) = 2x - 4. Set f'(x) = 0, giving 2x - 4 = 0, hence x = 2.
Correct Answer: B — 2
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Q. What is the derivative of f(x) = 3x^4 - 5x^2 + 2? (2021)
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A.
12x^3 - 10x
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B.
12x^3 - 5
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C.
6x^3 - 5x
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D.
3x^3 - 5
Solution
Using the power rule, f'(x) = 12x^3 - 10x.
Correct Answer: A — 12x^3 - 10x
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Q. What is the derivative of f(x) = 5x^5 - 3x + 7? (2020)
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A.
25x^4 - 3
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B.
15x^4 - 3
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C.
5x^4 - 3
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D.
20x^4 - 3
Solution
Using the power rule, f'(x) = 25x^4 - 3.
Correct Answer: A — 25x^4 - 3
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Q. What is the derivative of f(x) = ln(x)? (2019)
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A.
1/x
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B.
x
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C.
e^x
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D.
x^2
Solution
The derivative f'(x) = d/dx(ln(x)) = 1/x.
Correct Answer: A — 1/x
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Q. What is the derivative of f(x) = x^2 * sin(x)? (2023)
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A.
2x * sin(x) + x^2 * cos(x)
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B.
2x * cos(x) + x^2 * sin(x)
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C.
2x * sin(x) - x^2 * cos(x)
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D.
x^2 * sin(x) + 2x * cos(x)
Solution
Using the product rule, f'(x) = 2x * sin(x) + x^2 * cos(x).
Correct Answer: A — 2x * sin(x) + x^2 * cos(x)
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Q. What is the derivative of f(x) = x^3 * ln(x)? (2023)
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A.
3x^2 * ln(x) + x^2
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B.
3x^2 * ln(x) + x^3/x
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C.
3x^2 * ln(x) + 3x^2
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D.
3x^2 * ln(x) + 1
Solution
Using the product rule, f'(x) = 3x^2 * ln(x) + x^2.
Correct Answer: A — 3x^2 * ln(x) + x^2
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Q. What is the derivative of the function f(x) = 3x^2 + 5x - 7? (2021)
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A.
3x + 5
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B.
6x + 5
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C.
6x - 5
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D.
3x^2 + 5
Solution
The derivative f'(x) = d/dx(3x^2 + 5x - 7) = 6x + 5.
Correct Answer: B — 6x + 5
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Q. What is the limit: lim (x -> 0) (cos(x) - 1)/x^2? (2019)
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A.
0
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B.
-1/2
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C.
1
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D.
Undefined
Solution
Using the Taylor series expansion for cos(x), we find that lim (x -> 0) (cos(x) - 1)/x^2 = -1/2.
Correct Answer: B — -1/2
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Q. What is the limit: lim (x -> 0) (e^x - 1)/x? (2022)
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A.
1
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B.
0
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C.
e
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D.
Undefined
Solution
Using the derivative of e^x at x = 0, we find that lim (x -> 0) (e^x - 1)/x = 1.
Correct Answer: A — 1
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Q. What is the slope of the tangent line to the curve y = x^2 at the point (3, 9)? (2020)
Solution
The derivative y' = 2x. At x = 3, y' = 2(3) = 6.
Correct Answer: B — 6
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Q. Which of the following statements is true about the function f(x) = 1/(x-3)?
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A.
Continuous at x = 3
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B.
Continuous everywhere
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C.
Not continuous at x = 3
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D.
Continuous at x = 0
Solution
The function f(x) = 1/(x-3) is not defined at x = 3, hence it is not continuous at that point.
Correct Answer: C — Not continuous at x = 3
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Q. Which of the following statements is true regarding the function f(x) = |x|?
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A.
Continuous everywhere
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B.
Discontinuous at x = 0
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C.
Continuous only for x > 0
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D.
Discontinuous for x < 0
Solution
The function f(x) = |x| is continuous everywhere, including at x = 0.
Correct Answer: A — Continuous everywhere
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