Q. Find the coordinates of the point on the curve y = x^3 - 3x + 2 where the tangent is horizontal.
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A.
(0, 2)
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B.
(1, 0)
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C.
(2, 0)
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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.
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A.
(2, 3)
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B.
(3, 0)
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C.
(1, 1)
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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.
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A.
(0, 4)
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B.
(1, 2)
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C.
(2, 0)
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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 the function f(x) = 3x^4 - 8x^3 + 6.
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A.
(0, 6)
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B.
(2, -2)
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C.
(1, 1)
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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 - 6x^2 + 9x.
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A.
(0, 0)
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B.
(3, 0)
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C.
(2, 0)
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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 cross product of vectors A = (1, 2, 3) and B = (4, 5, 6).
-
A.
(-3, 6, -3)
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B.
(0, 0, 0)
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C.
(3, -6, 3)
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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.
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A.
-1/x^2
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B.
1/x^2
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C.
-2/x^2
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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.
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A.
6x + 5
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B.
3x + 5
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C.
6x - 5
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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) = 5x^4 - 3x + 2.
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A.
20x^3 - 3
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B.
15x^3 - 3
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C.
20x^4 - 3
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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).
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A.
2e^(2x)
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B.
e^(2x)
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C.
2xe^(2x)
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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).
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A.
2xe^(x^2)
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B.
e^(x^2)
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C.
x e^(x^2)
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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)
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B.
1/(x^2 + 1)
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C.
2/(x^2 + 1)
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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) 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
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D.
1/2
Solution
f'(x) = sec^2(x); f'(0) = sec^2(0) = 1.
Correct Answer: B — 1
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Q. Find the derivative of f(x) = tan(x) at x = π/4.
Solution
f'(x) = sec^2(x). At x = π/4, f'(π/4) = sec^2(π/4) = 2.
Correct Answer: A — 1
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Q. Find the derivative of f(x) = tan(x).
-
A.
sec^2(x)
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B.
csc^2(x)
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C.
sin^2(x)
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D.
cos^2(x)
Solution
The derivative f'(x) = d/dx(tan(x)) = sec^2(x).
Correct Answer: A — sec^2(x)
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Q. Find the derivative of f(x) = x^2 * e^x.
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A.
e^x(x^2 + 2x)
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B.
e^x(x^2 - 2x)
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C.
2xe^x
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D.
x^2e^x
Solution
Using the product rule: f'(x) = x^2 * e^x + 2x * e^x = e^x(x^2 + 2x).
Correct Answer: A — e^x(x^2 + 2x)
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Q. Find the derivative of f(x) = x^2 sin(1/x) at x = 0.
-
A.
0
-
B.
1
-
C.
undefined
-
D.
does not exist
Solution
Using the limit definition of the derivative, we find that f'(0) = 0, hence it is differentiable at x = 0.
Correct Answer: A — 0
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Q. Find the derivative of f(x) = x^3 - 3x^2 + 4 at x = 2.
Solution
f'(x) = 3x^2 - 6x. At x = 2, f'(2) = 3(2^2) - 6(2) = 12 - 12 = 0.
Correct Answer: B — 8
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Q. Find the derivative of f(x) = x^3 - 3x^2 + 4x - 5.
-
A.
3x^2 - 6x + 4
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B.
3x^2 - 3x + 4
-
C.
3x^2 - 6x + 5
-
D.
3x^2 + 6x - 4
Solution
Using the power rule, f'(x) = 3x^2 - 6x + 4.
Correct Answer: A — 3x^2 - 6x + 4
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Q. Find the derivative of f(x) = x^3 - 4x^2 + 6x.
-
A.
3x^2 - 8x + 6
-
B.
3x^2 - 4x + 6
-
C.
3x^2 - 8x
-
D.
x^2 - 4x + 6
Solution
Using the power rule, f'(x) = 3x^2 - 8x + 6.
Correct Answer: A — 3x^2 - 8x + 6
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Q. Find the determinant of the matrix \( D = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix} \).
Solution
The determinant of an upper triangular matrix is the product of its diagonal elements, which is 1.
Correct Answer: B — 1
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Q. Find the determinant of the matrix \( I = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \).
Solution
The determinant is calculated as \( 3*4 - 2*1 = 12 - 2 = 10 \).
Correct Answer: A — 10
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Q. Find the determinant of the matrix \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \).
Solution
The determinant of the identity matrix is always 1.
Correct Answer: B — 1
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix} \).
Solution
The determinant is calculated as \( 2*4 - 1*3 = 8 - 3 = 5 \).
Correct Answer: A — 5
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