Q. Determine the intervals where f(x) = x^3 - 3x is increasing. (2021)
-
A.
(-∞, -1)
-
B.
(-1, 1)
-
C.
(1, ∞)
-
D.
(-∞, 1)
Solution
f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x = -1, 1. f'(x) > 0 for x > 1.
Correct Answer:
C
— (1, ∞)
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Q. Determine the intervals where f(x) = x^4 - 4x^3 has increasing behavior. (2023)
-
A.
(-∞, 0)
-
B.
(0, 2)
-
C.
(2, ∞)
-
D.
(0, 4)
Solution
f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3). f'(x) > 0 for x in (0, 3).
Correct Answer:
B
— (0, 2)
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Q. Determine the intervals where f(x) = x^4 - 4x^3 has local minima. (2020)
-
A.
(0, 2)
-
B.
(1, 3)
-
C.
(2, 4)
-
D.
(0, 1)
Solution
f'(x) = 4x^3 - 12x^2. Setting f'(x) = 0 gives x = 0, 3. Testing intervals shows local minima at (0, 2).
Correct Answer:
A
— (0, 2)
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Q. Determine the limit: lim (x -> 0) (tan(5x)/x) (2022)
-
A.
0
-
B.
1
-
C.
5
-
D.
Undefined
Solution
Using the standard limit lim (x -> 0) (tan(kx)/x) = k, we have k = 5. Thus, lim (x -> 0) (tan(5x)/x) = 5.
Correct Answer:
C
— 5
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Q. Determine the limit: lim (x -> 1) (x^3 - 1)/(x - 1) (2020)
Solution
Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). For x ≠ 1, this simplifies to x^2 + x + 1. Evaluating at x = 1 gives 3.
Correct Answer:
C
— 3
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Q. Determine the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)
-
A.
0
-
B.
1
-
C.
4
-
D.
Undefined
Solution
Factoring gives (x - 1)(x^3 + x^2 + x + 1)/(x - 1). For x ≠ 1, this simplifies to x^3 + x^2 + x + 1. Evaluating at x = 1 gives 4.
Correct Answer:
D
— Undefined
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Q. Determine the local maxima of f(x) = -x^3 + 3x^2 + 1. (2021)
-
A.
(0, 1)
-
B.
(1, 3)
-
C.
(2, 5)
-
D.
(3, 4)
Solution
f'(x) = -3x^2 + 6x. Setting f'(x) = 0 gives x = 0 or x = 2. f(2) = 5 is a local maximum.
Correct Answer:
B
— (1, 3)
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Q. Determine the local maxima of f(x) = x^4 - 8x^2 + 16. (2021)
-
A.
(0, 16)
-
B.
(2, 12)
-
C.
(4, 0)
-
D.
(1, 9)
Solution
Find f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x = 0, ±2. f(2) = 12 is a local maximum.
Correct Answer:
B
— (2, 12)
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Q. Determine the local minima of f(x) = x^3 - 3x + 2. (2021)
Solution
f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x = 1. f(1) = 0.
Correct Answer:
B
— 0
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Q. Determine the local minima of f(x) = x^4 - 4x^2. (2021)
Solution
f'(x) = 4x^3 - 8x. Setting f'(x) = 0 gives x = 0, ±2. f(0) = 0.
Correct Answer:
B
— 0
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Q. Determine the maximum area of a triangle with a base of 10 units and height as a function of x. (2020)
Solution
Area = 1/2 * base * height = 5h. Max area occurs when h is maximized, thus Area = 50 when h = 10.
Correct Answer:
B
— 50
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Q. Determine the maximum height of the function f(x) = -x^2 + 6x + 5. (2020) 2020
Solution
The vertex occurs at x = 3. f(3) = -3^2 + 6*3 + 5 = 8.
Correct Answer:
A
— 8
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Q. Determine the maximum height of the projectile given by h(t) = -16t^2 + 64t + 80. (2023)
Solution
The maximum height occurs at t = -b/(2a) = -64/(2*-16) = 2. h(2) = -16(2^2) + 64(2) + 80 = 80.
Correct Answer:
A
— 80
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Q. Determine the maximum height of the projectile modeled by h(t) = -16t^2 + 64t + 80. (2020)
Solution
The maximum height occurs at t = -b/(2a) = 64/(2*16) = 2. h(2) = -16(2^2) + 64(2) + 80 = 80.
Correct Answer:
A
— 80
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Q. Determine the maximum value of f(x) = -x^2 + 6x - 8. (2022)
Solution
The maximum occurs at x = 3. f(3) = -3^2 + 6(3) - 8 = 6.
Correct Answer:
C
— 6
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Q. Determine the minimum value of f(x) = x^2 - 4x + 5. (2021)
Solution
The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4(2) + 5 = 1.
Correct Answer:
A
— 1
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Q. Determine the minimum value of f(x) = x^2 - 4x + 7. (2021)
Solution
The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4(2) + 7 = 3.
Correct Answer:
A
— 3
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Q. Determine the point of inflection for f(x) = x^4 - 4x^3 + 6. (2023)
-
A.
(1, 3)
-
B.
(2, 2)
-
C.
(0, 6)
-
D.
(3, 0)
Solution
f''(x) = 12x^2 - 24x. Setting f''(x) = 0 gives x(12x - 24) = 0, so x = 0 or x = 2. Check f(1) = 3.
Correct Answer:
A
— (1, 3)
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Q. Determine the point where the function f(x) = 2x^3 - 9x^2 + 12x has a local minimum. (2023)
-
A.
(1, 5)
-
B.
(2, 0)
-
C.
(3, 3)
-
D.
(4, 4)
Solution
Set f'(x) = 0. f'(x) = 6x^2 - 18x + 12 = 0 gives x = 2. f(2) = 0.
Correct Answer:
C
— (3, 3)
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Q. Determine the point where the function f(x) = 4x - x^2 has a maximum. (2022)
-
A.
(0, 0)
-
B.
(2, 4)
-
C.
(1, 3)
-
D.
(3, 3)
Solution
The maximum occurs at x = 2, found by setting f'(x) = 4 - 2x = 0. f(2) = 4(2) - (2^2) = 4.
Correct Answer:
B
— (2, 4)
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Q. Determine the product of the roots of the equation x² + 6x + 8 = 0. (2023)
Solution
The product of the roots is given by c/a = 8/1 = 8.
Correct Answer:
A
— 8
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Q. Determine the product of the roots of the equation x² + 6x + 9 = 0. (2021)
Solution
The product of the roots is c/a = 9/1 = 9.
Correct Answer:
A
— 9
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Q. Determine the roots of the equation x² + 2x - 8 = 0. (2023)
-
A.
-4 and 2
-
B.
4 and -2
-
C.
2 and -4
-
D.
0 and 8
Solution
Factoring gives (x + 4)(x - 2) = 0, hence the roots are -4 and 2.
Correct Answer:
A
— -4 and 2
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Q. Determine the roots of the equation x² + 6x + 9 = 0. (2023)
Solution
This is a perfect square: (x + 3)² = 0, hence the root is x = -3.
Correct Answer:
A
— -3
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Q. Evaluate the integral ∫ (3x^2 + 2x) dx. (2020)
-
A.
x^3 + x^2 + C
-
B.
x^3 + x^2 + 2C
-
C.
x^3 + x^2 + 1
-
D.
x^3 + 2x + C
Solution
The integral is (3/3)x^3 + (2/2)x^2 + C = x^3 + x^2 + C.
Correct Answer:
A
— x^3 + x^2 + C
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Q. Evaluate the integral ∫(3x^2 + 2)dx. (2022)
-
A.
x^3 + 2x + C
-
B.
x^3 + 2x^2 + C
-
C.
x^3 + 2x^3 + C
-
D.
3x^3 + 2x + C
Solution
Integrating term by term, ∫3x^2dx = x^3 and ∫2dx = 2x. Thus, ∫(3x^2 + 2)dx = x^3 + 2x + C.
Correct Answer:
A
— x^3 + 2x + C
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Q. Evaluate the limit: lim (x -> 0) (tan(x)/x) (2023)
-
A.
0
-
B.
1
-
C.
∞
-
D.
Undefined
Solution
Using the limit property lim (x -> 0) (tan(x)/x) = 1, we find that the limit is 1.
Correct Answer:
B
— 1
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Q. Evaluate the limit: lim (x -> 0) (x - sin(x))/x^3 (2022)
-
A.
0
-
B.
1/6
-
C.
1/3
-
D.
1/2
Solution
Using the Taylor series expansion for sin(x), we find that lim (x -> 0) (x - sin(x))/x^3 = 1/6.
Correct Answer:
B
— 1/6
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Q. Evaluate the limit: lim (x -> 0) (x^3)/(sin(x)) (2022)
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
As x approaches 0, x^3 approaches 0 and sin(x) approaches 0, thus the limit is 0.
Correct Answer:
A
— 0
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Q. Evaluate the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2020)
-
A.
3
-
B.
6
-
C.
9
-
D.
Undefined
Solution
Factoring gives (x - 3)(x + 3)/(x - 3). Canceling (x - 3) gives lim (x -> 3) (x + 3) = 6.
Correct Answer:
B
— 6
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