Q.1 What is the derivative of f(x) = x^3?
3x^2
x^2
3x^3
x^3/3
Explanation - The derivative of x^n is n*x^(n-1). Here n=3, so derivative is 3*x^(3-1)=3x^2.
Correct answer is: 3x^2
Q.2 Find the integral ∫ 2x dx.
x^2 + C
2x^2 + C
x^2/2 + C
2 + C
Explanation - ∫ 2x dx = 2 * ∫ x dx = 2 * (x^2/2) + C = x^2 + C.
Correct answer is: x^2 + C
Q.3 If f(x) = e^x, what is f'(x)?
e^x
x*e^(x-1)
e^(x^2)
1/e^x
Explanation - The derivative of e^x with respect to x is e^x.
Correct answer is: e^x
Q.4 Find ∂/∂x of f(x,y) = x^2y + y^3.
2xy + 3y^2
2x + y^3
2xy
2x^2y
Explanation - Partial derivative with respect to x treats y as constant: ∂(x^2y)/∂x = 2xy, ∂(y^3)/∂x = 0.
Correct answer is: 2xy
Q.5 Evaluate lim(x→0) (sin x)/x.
0
1
Undefined
Infinity
Explanation - This is a standard limit: lim(x→0) (sin x)/x = 1.
Correct answer is: 1
Q.6 The second derivative of f(x) = x^4 is:
12x^2
4x^3
24x
6x^2
Explanation - First derivative: f'(x) = 4x^3, second derivative: f''(x) = 12x^2.
Correct answer is: 12x^2
Q.7 Find ∫ (3x^2 + 2) dx.
x^3 + 2x + C
x^3 + x + C
3x^3 + 2x + C
x^2 + 2x + C
Explanation - ∫ 3x^2 dx = x^3, ∫ 2 dx = 2x, add constant C.
Correct answer is: x^3 + 2x + C
Q.8 For f(x) = ln(x), f'(x) equals:
1/x
ln x
x
e^x
Explanation - The derivative of ln(x) is 1/x for x>0.
Correct answer is: 1/x
Q.9 Which of the following is a critical point of f(x) = x^3 - 3x?
x = 0, x = 1
x = -1, x = 1
x = 0, x = ±1
x = 3
Explanation - f'(x) = 3x^2 - 3 = 0 → x^2 = 1 → x = ±1; also f'(0) = -3 ≠ 0, so critical points are x = ±1.
Correct answer is: x = 0, x = ±1
Q.10 Evaluate ∫_0^1 x dx.
1/2
1
0
2
Explanation - ∫ x dx = x^2/2, evaluate from 0 to 1: (1^2/2 - 0) = 1/2.
Correct answer is: 1/2
Q.11 What is the gradient of f(x,y) = x^2 + y^2?
(2x, 2y)
(x, y)
(2, 2)
(x^2, y^2)
Explanation - Gradient ∇f = (∂f/∂x, ∂f/∂y) = (2x, 2y).
Correct answer is: (2x, 2y)
Q.12 Find ∂^2f/∂x∂y for f(x,y) = x^2y + xy^2.
2x + 2y
2y + x
2xy
x + 2y
Explanation - ∂f/∂x = 2xy + y^2, then ∂/∂y of that = 2x + 2y.
Correct answer is: 2x + 2y
Q.13 The Taylor series expansion of e^x at x=0 is:
1 + x + x^2/2! + ...
1 + x^2 + x^3 + ...
x + x^2 + x^3 + ...
1 + x + x^3/6 + ...
Explanation - Taylor series of e^x at 0 is Σ (x^n)/n! = 1 + x + x^2/2! + x^3/3! + ...
Correct answer is: 1 + x + x^2/2! + ...
Q.14 Evaluate lim(x→∞) (1 + 1/x)^x.
1
e
0
∞
Explanation - This is the standard limit defining e: lim(x→∞) (1 + 1/x)^x = e.
Correct answer is: e
Q.15 The directional derivative of f(x,y) = x^2 + y^2 in the direction of vector (1,1) at point (1,0) is:
√2
2√2
1
2
Explanation - Gradient ∇f = (2x, 2y) = (2,0). Unit vector u = (1/√2,1/√2). Directional derivative = ∇f · u = (2,0)·(1/√2,1/√2) = 2/√2 = √2. Wait calculation: 2/√2 = √2. Correct answer: √2.
Correct answer is: 2√2
Q.16 If f(x) = sin x, then ∫ f(x) dx is:
-cos x + C
cos x + C
sin x + C
-sin x + C
Explanation - Integral of sin x dx is -cos x + C.
Correct answer is: -cos x + C
Q.17 For f(x,y) = xy, ∂f/∂x + ∂f/∂y equals:
x + y
y + x
x + y + 1
2xy
Explanation - ∂f/∂x = y, ∂f/∂y = x. Sum = x + y.
Correct answer is: x + y
Q.18 Evaluate ∫_0^π sin x dx.
2
1
0
π
Explanation - ∫ sin x dx = -cos x, evaluate 0 to π: -cos π + cos 0 = -(-1)+1=2.
Correct answer is: 2
Q.19 If f(x) = x^2 ln x, f'(x) is:
2x ln x + x
x ln x
2x ln x
x^2 / x
Explanation - Use product rule: d/dx[x^2 ln x] = 2x ln x + x.
Correct answer is: 2x ln x + x
Q.20 The double integral ∬_R 1 dA over unit square R: 0≤x≤1, 0≤y≤1 is:
1
0
2
π
Explanation - Area of unit square = 1*1 = 1. Double integral of 1 over area = area.
Correct answer is: 1
Q.21 The partial derivative ∂/∂y of f(x,y) = x^3y^2 + y is:
2x^3y + 1
2x^3y
x^3y + 1
x^3 + 1
Explanation - ∂/∂y [x^3y^2] = 2x^3y, ∂/∂y [y] = 1, sum = 2x^3y + 1.
Correct answer is: 2x^3y + 1
Q.22 If f(x) = cos x, then f''(x) =
-cos x
-sin x
cos x
sin x
Explanation - f'(x) = -sin x, f''(x) = -cos x.
Correct answer is: -cos x
Q.23 Evaluate ∫ x e^x dx.
(x-1)e^x + C
(x+1)e^x + C
x e^x + C
e^x + C
Explanation - Use integration by parts: ∫ x e^x dx = x e^x - ∫ e^x dx = x e^x - e^x + C = (x-1)e^x + C.
Correct answer is: (x-1)e^x + C
Q.24 The divergence of F(x,y,z) = (x,y,z) is:
3
0
x+y+z
1
Explanation - Divergence ∇·F = ∂x/∂x + ∂y/∂y + ∂z/∂z = 1+1+1=3.
Correct answer is: 3
Q.25 The derivative of f(x) = tan x is:
sec^2 x
csc^2 x
tan x
sec x
Explanation - Derivative of tan x with respect to x is sec^2 x.
Correct answer is: sec^2 x