Q. What is the limit: lim (x -> 0) (ln(1 + x)/x)?
-
A.
1
-
B.
0
-
C.
∞
-
D.
Undefined
Solution
Using L'Hôpital's Rule, we differentiate the numerator and denominator to find lim (x -> 0) (1/(1 + x)) = 1.
Correct Answer: A — 1
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Q. What is the limit: lim (x -> 1) (x^2 - 1)/(x - 1)? (2019)
-
A.
0
-
B.
1
-
C.
2
-
D.
Undefined
Solution
Factoring gives (x - 1)(x + 1)/(x - 1), which simplifies to x + 1. Thus, lim (x -> 1) (x + 1) = 2.
Correct Answer: C — 2
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Q. What is the maximum value of f(x) = -2x^2 + 10x - 12? (2022)
Solution
The maximum occurs at x = -b/(2a) = -10/(-4) = 2. f(2) = -2(2^2) + 10(2) - 12 = 6.
Correct Answer: C — 6
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Q. What is the maximum value of f(x) = -3x^2 + 12x - 5? (2019)
Solution
The vertex is at x = -12/(2*(-3)) = 2. The maximum value is f(2) = -3(2^2) + 12(2) - 5 = 7.
Correct Answer: C — 7
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Q. What is the maximum value of the function f(x) = -2x^2 + 8x - 5? (2019)
Solution
The vertex occurs at x = -b/(2a) = 2. f(2) = -2(2^2) + 8(2) - 5 = 9.
Correct Answer: B — 9
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Q. What is the minimum value of f(x) = 2x^2 - 8x + 10? (2021)
Solution
The minimum occurs at x = -b/(2a) = 8/(2*2) = 2. f(2) = 2(2^2) - 8(2) + 10 = 2.
Correct Answer: A — 1
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Q. What is the minimum value of the function f(x) = 2x^2 + 4x + 1? (2023)
Solution
The vertex is at x = -4/(2*2) = -1. The minimum value is f(-1) = 2(-1)^2 + 4(-1) + 1 = -1.
Correct Answer: B — 1
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Q. What is the second derivative of f(x) = 2x^4 - 3x^3 + x?
-
A.
24x^2 - 18x + 1
-
B.
24x^2 - 9
-
C.
12x^2 - 9
-
D.
8x^3 - 9
Solution
First derivative f'(x) = 8x^3 - 9x^2 + 1. Second derivative f''(x) = 24x^2 - 18x.
Correct Answer: A — 24x^2 - 18x + 1
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Q. What is the second derivative of f(x) = 2x^4 - 4x^3 + 3?
-
A.
24x^2 - 24x
-
B.
12x^2 - 12
-
C.
24x^3 - 12x^2
-
D.
8x^2 - 12
Solution
First derivative f'(x) = 8x^3 - 12x^2. Second derivative f''(x) = 24x^2 - 24x.
Correct Answer: A — 24x^2 - 24x
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Q. What is the second derivative of the function f(x) = 2x^4 - 4x^3 + 3?
-
A.
24x^2 - 24x
-
B.
24x^2 - 12
-
C.
12x^2 - 12
-
D.
8x^2 - 12
Solution
First derivative f'(x) = 8x^3 - 12x^2. Second derivative f''(x) = 24x^2 - 24x.
Correct Answer: A — 24x^2 - 24x
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Q. What is the slope of the tangent line to the curve y = x^2 + 2x at x = 1?
Solution
The derivative y' = 2x + 2. At x = 1, y' = 2(1) + 2 = 4.
Correct Answer: A — 3
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Q. What is the slope of the tangent line to the curve y = x^2 at the point (1,1)? (2023)
Solution
The derivative y' = 2x; At x = 1, slope = 2(1) = 2.
Correct Answer: B — 2
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Q. What is the slope of the tangent line to the curve y = x^2 at the point (2, 4)?
Solution
The derivative y' = 2x. At x = 2, the slope is y'(2) = 2(2) = 4.
Correct Answer: A — 2
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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 functions is continuous at all points?
-
A.
f(x) = 1/x
-
B.
f(x) = x^3
-
C.
f(x) = sqrt(x)
-
D.
f(x) = tan(x)
Solution
f(x) = x^3 is a polynomial function, which is continuous everywhere.
Correct Answer: B — f(x) = x^3
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Q. Which of the following functions is continuous at x = 0?
-
A.
f(x) = 1/x
-
B.
f(x) = e^x
-
C.
f(x) = tan(x)
-
D.
f(x) = 1/(x^2 + 1)
Solution
The function f(x) = e^x is continuous everywhere, including at x = 0.
Correct Answer: B — f(x) = e^x
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Q. Which of the following functions is continuous on the interval [0, 1]?
-
A.
f(x) = 1/x
-
B.
f(x) = x^3
-
C.
f(x) = sqrt(x)
-
D.
f(x) = 1/(x-1)
Solution
f(x) = x^3 is a polynomial function and is continuous on the interval [0, 1].
Correct Answer: B — f(x) = x^3
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Q. Which of the following statements is true about the function f(x) = 1/(x-1)? (2022)
-
A.
Continuous at x = 1
-
B.
Continuous everywhere
-
C.
Not continuous at x = 1
-
D.
Continuous at x = 0
Solution
The function f(x) = 1/(x-1) is not continuous at x = 1 because it is undefined there.
Correct Answer: C — Not continuous at x = 1
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Q. Which of the following statements is true about the function f(x) = 1/(x-3)?
-
A.
Continuous at x = 3
-
B.
Continuous everywhere
-
C.
Not continuous at x = 3
-
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 about the function f(x) = |x|?
-
A.
Continuous everywhere
-
B.
Discontinuous at x = 0
-
C.
Continuous only at x = 1
-
D.
Discontinuous everywhere
Solution
The function f(x) = |x| is continuous everywhere, including at x = 0.
Correct Answer: A — Continuous everywhere
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Q. Which of the following statements is true regarding the function f(x) = 1/(x-3)?
-
A.
Continuous at x = 3
-
B.
Discontinuous at x = 3
-
C.
Continuous everywhere
-
D.
Discontinuous everywhere
Solution
The function f(x) = 1/(x-3) is discontinuous at x = 3 because it is undefined at that point.
Correct Answer: B — Discontinuous at x = 3
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Q. Which of the following statements is true regarding the function f(x) = |x|?
-
A.
Continuous everywhere
-
B.
Discontinuous at x = 0
-
C.
Continuous only for x > 0
-
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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