Q. Find the coordinates of the midpoint of the line segment joining A(2, -1, 3) and B(4, 3, 5). (2022)
-
A.
(3, 1, 4)
-
B.
(2, 1, 4)
-
C.
(3, 2, 3)
-
D.
(4, 2, 4)
Solution
Midpoint M = ((2+4)/2, (-1+3)/2, (3+5)/2) = (3, 1, 4).
Correct Answer:
A
— (3, 1, 4)
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Q. Find the coordinates of the midpoint of the line segment joining A(2, 3, 4) and B(4, 5, 6). (2023)
-
A.
(3, 4, 5)
-
B.
(2, 3, 4)
-
C.
(4, 5, 6)
-
D.
(5, 6, 7)
Solution
Midpoint M = ((2+4)/2, (3+5)/2, (4+6)/2) = (3, 4, 5).
Correct Answer:
A
— (3, 4, 5)
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Q. Find the coordinates of the point on the curve y = x^3 - 3x + 2 where the slope of the tangent is 0.
-
A.
(1, 0)
-
B.
(0, 2)
-
C.
(2, 0)
-
D.
(3, 2)
Solution
f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x^2 = 1, so x = 1 or x = -1. f(1) = 0, f(-1) = 4. The point is (1, 0).
Correct Answer:
A
— (1, 0)
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Q. Find the coordinates of the point on the curve y = x^3 - 3x + 2 where the tangent is horizontal.
-
A.
(0, 2)
-
B.
(1, 0)
-
C.
(2, 0)
-
D.
(3, 2)
Solution
f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x = 1. The point is (1, 0).
Correct Answer:
B
— (1, 0)
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Q. Find the coordinates of the point where the function f(x) = 3x^2 - 12x + 9 has a local maximum.
-
A.
(2, 3)
-
B.
(3, 0)
-
C.
(1, 1)
-
D.
(0, 9)
Solution
f'(x) = 6x - 12. Setting f'(x) = 0 gives x = 2. f(2) = 3(2^2) - 12(2) + 9 = 3.
Correct Answer:
A
— (2, 3)
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Q. Find the critical points of f(x) = x^3 - 3x^2 + 4.
-
A.
(0, 4)
-
B.
(1, 2)
-
C.
(2, 0)
-
D.
(3, 1)
Solution
Setting f'(x) = 3x^2 - 6x = 0 gives x(x - 2) = 0, so critical points are x = 0 and x = 2. Evaluating f(1) = 2.
Correct Answer:
B
— (1, 2)
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Q. Find the critical points of f(x) = x^4 - 8x^2 + 16. (2021)
-
A.
(0, 16)
-
B.
(2, 0)
-
C.
(4, 0)
-
D.
(1, 15)
Solution
f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x = 0, ±2. f(2) = 0 is a critical point.
Correct Answer:
B
— (2, 0)
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Q. Find the critical points of the function f(x) = 3x^4 - 8x^3 + 6.
-
A.
(0, 6)
-
B.
(2, -2)
-
C.
(1, 1)
-
D.
(3, 0)
Solution
f'(x) = 12x^3 - 24x^2. Setting f'(x) = 0 gives x^2(12x - 24) = 0, so x = 0 or x = 2. f(2) = 3(2^4) - 8(2^3) + 6 = -2.
Correct Answer:
B
— (2, -2)
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Q. Find the critical points of the function f(x) = x^3 - 3x^2 + 4.
-
A.
x = 0, 2
-
B.
x = 1, 2
-
C.
x = 1, 3
-
D.
x = 0, 1
Solution
To find critical points, set f'(x) = 0. f'(x) = 3x^2 - 6x = 3x(x - 2). Critical points are x = 0 and x = 2.
Correct Answer:
B
— x = 1, 2
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Q. Find the critical points of the function f(x) = x^3 - 6x^2 + 9x.
-
A.
(0, 0)
-
B.
(3, 0)
-
C.
(2, 0)
-
D.
(1, 0)
Solution
f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1 and x = 3. Critical points are (1, f(1)) and (3, f(3)).
Correct Answer:
B
— (3, 0)
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Q. Find the critical points of the function f(x) = x^4 - 8x^2 + 16. (2019)
-
A.
(0, 16)
-
B.
(2, 0)
-
C.
(4, 0)
-
D.
(1, 9)
Solution
Setting f'(x) = 4x^3 - 16x = 0 gives x = 0, ±2. Evaluating f(2) = 0 shows (2, 0) is a critical point.
Correct Answer:
B
— (2, 0)
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Q. Find the cross product of vectors A = (1, 2, 3) and B = (4, 5, 6).
-
A.
(-3, 6, -3)
-
B.
(0, 0, 0)
-
C.
(3, -6, 3)
-
D.
(1, -2, 1)
Solution
Cross product A × B = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3).
Correct Answer:
A
— (-3, 6, -3)
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Q. Find the derivative of f(x) = 1/x.
-
A.
-1/x^2
-
B.
1/x^2
-
C.
-2/x^2
-
D.
1/x
Solution
Using the power rule, f'(x) = -1/x^2.
Correct Answer:
A
— -1/x^2
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Q. Find the derivative of f(x) = 3x^2 + 5x - 7.
-
A.
6x + 5
-
B.
3x + 5
-
C.
6x - 5
-
D.
3x^2 + 5
Solution
Using the power rule, f'(x) = d/dx(3x^2) + d/dx(5x) - d/dx(7) = 6x + 5.
Correct Answer:
A
— 6x + 5
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Q. Find the derivative of f(x) = 4x^3 - 2x + 1. (2022)
-
A.
12x^2 - 2
-
B.
12x^2 + 2
-
C.
4x^2 - 2
-
D.
4x^2 + 2
Solution
Using the power rule, f'(x) = 12x^2 - 2.
Correct Answer:
A
— 12x^2 - 2
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Q. Find the derivative of f(x) = 5x^2 + 3x - 1. (2020)
-
A.
10x + 3
-
B.
5x + 3
-
C.
10x - 1
-
D.
5x^2 + 3
Solution
Using the power rule, f'(x) = 10x + 3.
Correct Answer:
A
— 10x + 3
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Q. Find the derivative of f(x) = 5x^2 + 3x - 7. (2020)
-
A.
10x + 3
-
B.
5x + 3
-
C.
10x - 3
-
D.
5x - 3
Solution
Using the power rule, f'(x) = 10x + 3.
Correct Answer:
A
— 10x + 3
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Q. Find the derivative of f(x) = 5x^3 - 4x + 7. (2019)
-
A.
15x^2 - 4
-
B.
15x^2 + 4
-
C.
5x^2 - 4
-
D.
5x^2 + 4
Solution
Using the power rule, f'(x) = 15x^2 - 4.
Correct Answer:
A
— 15x^2 - 4
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Q. Find the derivative of f(x) = 5x^4 - 3x + 2.
-
A.
20x^3 - 3
-
B.
15x^3 - 3
-
C.
20x^4 - 3
-
D.
5x^3 - 3
Solution
Using the power rule, f'(x) = 20x^3 - 3.
Correct Answer:
A
— 20x^3 - 3
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Q. Find the derivative of f(x) = e^(2x) at x = 0.
Solution
f'(x) = 2e^(2x). At x = 0, f'(0) = 2e^0 = 2.
Correct Answer:
B
— 2
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Q. Find the derivative of f(x) = e^(2x).
-
A.
2e^(2x)
-
B.
e^(2x)
-
C.
2xe^(2x)
-
D.
e^(x)
Solution
Using the chain rule, f'(x) = 2e^(2x).
Correct Answer:
A
— 2e^(2x)
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Q. Find the derivative of f(x) = e^(x^2).
-
A.
2xe^(x^2)
-
B.
e^(x^2)
-
C.
x e^(x^2)
-
D.
2e^(x^2)
Solution
Using the chain rule, f'(x) = e^(x^2) * 2x = 2x e^(x^2).
Correct Answer:
A
— 2xe^(x^2)
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Q. Find the derivative of f(x) = e^x * ln(x) at x = 1.
Solution
Using the product rule, f'(x) = e^x * ln(x) + e^x/x. At x = 1, this simplifies to 0.
Correct Answer:
A
— 1
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Q. Find the derivative of f(x) = e^x * sin(x) at x = 0.
Solution
Using the product rule, f'(0) = e^0 * sin(0) + e^0 * cos(0) = 0 + 1 = 1.
Correct Answer:
A
— 1
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Q. Find the derivative of f(x) = ln(x^2 + 1) at x = 1.
Solution
f'(x) = (2x)/(x^2 + 1). At x = 1, f'(1) = (2*1)/(1^2 + 1) = 2/2 = 1.
Correct Answer:
B
— 1
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Q. Find the derivative of f(x) = ln(x^2 + 1).
-
A.
2x/(x^2 + 1)
-
B.
1/(x^2 + 1)
-
C.
2/(x^2 + 1)
-
D.
x/(x^2 + 1)
Solution
Using the chain rule, f'(x) = d/dx(ln(x^2 + 1)) = (2x)/(x^2 + 1).
Correct Answer:
A
— 2x/(x^2 + 1)
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Q. Find the derivative of f(x) = sin(x) + cos(x) at x = π/4.
Solution
f'(x) = cos(x) - sin(x), thus f'(π/4) = √2/2 - √2/2 = 0.
Correct Answer:
C
— √2
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Q. Find the derivative of f(x) = sin(x) + cos(x).
-
A.
cos(x) - sin(x)
-
B.
-sin(x) - cos(x)
-
C.
sin(x) + cos(x)
-
D.
-cos(x) + sin(x)
Solution
The derivative f'(x) = d/dx(sin(x) + cos(x)) = cos(x) - sin(x).
Correct Answer:
A
— cos(x) - sin(x)
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Q. Find the derivative of f(x) = sin(x) at x = π/2.
-
A.
0
-
B.
1
-
C.
-1
-
D.
undefined
Solution
f'(x) = cos(x); f'(π/2) = cos(π/2) = 0.
Correct Answer:
B
— 1
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Q. Find the derivative of f(x) = tan(x) at x = 0.
-
A.
0
-
B.
1
-
C.
undefined
-
D.
1/2
Solution
f'(x) = sec^2(x); f'(0) = sec^2(0) = 1.
Correct Answer:
B
— 1
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