Q. What is the derivative of f(x) = 7x^2 + 2x - 5? (2023)
-
A.
14x + 2
-
B.
14x - 2
-
C.
7x + 2
-
D.
2x - 5
Solution
Using the power rule, f'(x) = 14x + 2.
Correct Answer:
A
— 14x + 2
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Q. What is the derivative of f(x) = e^x * ln(x)? (2022)
-
A.
e^x * ln(x)
-
B.
e^x/x
-
C.
e^x * (1 + ln(x))
-
D.
e^x * ln(x)/x
Solution
Using the product rule, f'(x) = e^x * ln(x) + e^x * (1/x) = e^x * (ln(x) + 1/x).
Correct Answer:
C
— e^x * (1 + ln(x))
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Q. What is the derivative of f(x) = e^x * x^2?
-
A.
e^x * (2x + 1)
-
B.
e^x * (x^2 + 2)
-
C.
e^x * 2x
-
D.
2e^x * x
Solution
Using the product rule, f'(x) = e^x * x^2 + e^x * 2x = e^x * (x^2 + 2x).
Correct Answer:
A
— e^x * (2x + 1)
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Q. What is the derivative of f(x) = ln(x)? (2019)
-
A.
1/x
-
B.
x
-
C.
e^x
-
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)
-
A.
2x * sin(x) + x^2 * cos(x)
-
B.
2x * cos(x) + x^2 * sin(x)
-
C.
2x * sin(x) - x^2 * cos(x)
-
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 * e^x?
-
A.
3x^2 * e^x + x^3 * e^x
-
B.
x^3 * e^x
-
C.
3x^2 * e^x
-
D.
x^3 * e^(x+1)
Solution
Using the product rule, f'(x) = 3x^2 * e^x + x^3 * e^x.
Correct Answer:
A
— 3x^2 * e^x + x^3 * e^x
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Q. What is the derivative of f(x) = x^3 * ln(x)? (2023)
-
A.
3x^2 * ln(x) + x^2
-
B.
3x^2 * ln(x) + x^3/x
-
C.
3x^2 * ln(x) + 3x^2
-
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 f(x) = x^3 - 4x + 7? (2021)
-
A.
3x^2 - 4
-
B.
3x^2 + 4
-
C.
x^2 - 4
-
D.
3x^2 - 7
Solution
Using the power rule, f'(x) = 3x^2 - 4.
Correct Answer:
A
— 3x^2 - 4
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Q. What is the derivative of f(x) = x^3 - 6x^2 + 9x?
-
A.
3x^2 - 12x + 9
-
B.
3x^2 - 6x + 9
-
C.
6x - 12
-
D.
3x^2 - 9
Solution
The derivative f'(x) = 3x^2 - 12x + 9.
Correct Answer:
A
— 3x^2 - 12x + 9
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Q. What is the derivative of f(x) = x^4 - 6x^2 + 9? (2022)
-
A.
4x^3 - 12x
-
B.
4x^3 + 12x
-
C.
2x^3 - 6x
-
D.
2x^3 + 6x
Solution
Using the power rule, f'(x) = 4x^3 - 12x.
Correct Answer:
A
— 4x^3 - 12x
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Q. What is the derivative of the function f(x) = 3x^2 + 5x - 7? (2021)
-
A.
3x + 5
-
B.
6x + 5
-
C.
6x - 5
-
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 first derivative of f(x) = ln(x)? (2019)
-
A.
1/x
-
B.
x
-
C.
ln(x)
-
D.
e^x
Solution
The derivative f'(x) = 1/x.
Correct Answer:
A
— 1/x
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Q. What is the first derivative of f(x) = tan(x)? (2021)
-
A.
sec^2(x)
-
B.
cos^2(x)
-
C.
sin^2(x)
-
D.
csc^2(x)
Solution
The derivative of f(x) = tan(x) is f'(x) = sec^2(x).
Correct Answer:
A
— sec^2(x)
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Q. What is the limit: lim (x -> 0) (1 - cos(x))/(x^2)? (2022)
-
A.
0
-
B.
1/2
-
C.
1
-
D.
Undefined
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (1 - cos(x))/(x^2) = lim (x -> 0) (2sin^2(x/2))/(x^2) = 1/2.
Correct Answer:
B
— 1/2
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Q. What is the limit: lim (x -> 0) (cos(x) - 1)/x^2? (2019)
-
A.
0
-
B.
-1/2
-
C.
1
-
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)
-
A.
1
-
B.
0
-
C.
e
-
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 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 -> 0) (tan(3x)/x)?
-
A.
3
-
B.
0
-
C.
1
-
D.
Infinity
Solution
Using the limit lim (x -> 0) (tan(x)/x) = 1, we have lim (x -> 0) (tan(3x)/x) = 3 * lim (x -> 0) (tan(3x)/(3x)) = 3.
Correct Answer:
A
— 3
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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 f(x) = 4x^2 - 16x + 15? (2022)
Solution
The minimum occurs at x = -b/(2a) = 16/(2*4) = 2. f(2) = 4(2^2) - 16(2) + 15 = 1.
Correct Answer:
B
— 2
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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 minimum value of the function f(x) = 3x^2 - 12x + 7? (2019)
Solution
The minimum occurs at x = -b/(2a) = 12/(2*3) = 2. f(2) = 3(2^2) - 12(2) + 7 = 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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