Q.1 If y = x^3, what is the slope of the tangent at x = 2?
12
6
8
4
Explanation - The slope of the tangent is given by dy/dx = 3x^2. At x = 2, slope = 3*(2^2) = 12.
Correct answer is: 12
Q.2 The function y = x^2 + 3x + 2 has a minimum value at which point?
x = -3/2
x = 0
x = -1
x = 2
Explanation - For a quadratic ax^2+bx+c, the vertex occurs at x = -b/(2a). Here a=1, b=3 → x = -3/2.
Correct answer is: x = -3/2
Q.3 If f(x) = sin(x), what is the maximum slope of the tangent?
1
0
-1
Does not exist
Explanation - Derivative f'(x) = cos(x). The maximum value of cos(x) is 1.
Correct answer is: 1
Q.4 The normal to y = x^2 at (1,1) has slope:
-2
1/2
-1/2
2
Explanation - Tangent slope is dy/dx = 2x = 2 at x=1. Normal slope is negative reciprocal = -1/2.
Correct answer is: -1/2
Q.5 At what point is the tangent to y = x^2 + 4x + 5 parallel to the line y = 2x + 3?
(-3, 2)
(-2, 1)
(-1, 0)
(-4, 5)
Explanation - Derivative dy/dx = 2x + 4. For parallel slope = 2. Solve 2x+4=2 → x=-1. Substituting, y=2. Point (-1,2).
Correct answer is: (-3, 2)
Q.6 If f(x) = x^4 - 4x^3, find the point of inflection.
x=3
x=2
x=1
x=0
Explanation - f''(x)=12x^2 - 24x=12x(x-2). Change of sign at x=0,2. Inflection occurs at x=2.
Correct answer is: x=2
Q.7 A particle moves with s(t)=t^3-6t^2+9t. At t=2, velocity is:
-3
0
1
3
Explanation - Velocity is derivative: v(t)=3t^2-12t+9. At t=2, v=3(4)-24+9=0.
Correct answer is: 0
Q.8 The function f(x)=ln(x) is concave downward for:
x>0
x>1
0<x<1
All x
Explanation - f''(x) = -1/x^2, which is negative for all x>0 → concave down.
Correct answer is: x>0
Q.9 Maximum area of a rectangle inscribed in a semicircle of radius r occurs when the rectangle is:
Square
Half-square
Base = 2r
Height = r
Explanation - Maximization shows maximum area when rectangle is a square inscribed in the semicircle.
Correct answer is: Square
Q.10 If y = e^x, slope of tangent at x=0 is:
0
1
e
e^0
Explanation - Derivative of e^x is e^x. At x=0, slope = e^0 = 1.
Correct answer is: 1
Q.11 If f(x) = |x|, derivative at x=0 is:
0
1
-1
Does not exist
Explanation - Left derivative = -1, right derivative = 1, hence derivative at 0 does not exist.
Correct answer is: Does not exist
Q.12 If f(x) = 1/x, slope of tangent at x=1 is:
0
-1
1
2
Explanation - f'(x) = -1/x^2. At x=1, slope = -1.
Correct answer is: -1
Q.13 For y = x^3 - 6x^2 + 9x, minimum occurs at:
x=3
x=2
x=1
x=0
Explanation - y' = 3x^2 -12x+9=3(x-1)(x-3). Critical points: x=1,3. y''=6x-12. At x=3, y''=6>0 → minimum.
Correct answer is: x=3
Q.14 If radius of a circle increases at 2 cm/s, rate of increase of area when radius=3 cm is:
6π
12π
18π
24π
Explanation - A=πr^2. dA/dt=2πr(dr/dt)=2π*3*2=12π.
Correct answer is: 12π
Q.15 If y = cos(x), maximum slope of tangent is:
1
0
-1
Does not exist
Explanation - Derivative is -sin(x). Max value is 1.
Correct answer is: 1
Q.16 If f(x)=x^2, the rate of change of f at x=5 is:
5
10
25
2
Explanation - Derivative f'(x)=2x. At x=5, slope=10.
Correct answer is: 10
Q.17 Find the slope of the tangent to y=tan(x) at x=π/4.
1
2
0
√2
Explanation - Derivative dy/dx=sec^2(x). At x=π/4, slope=sec^2(π/4)=2.
Correct answer is: 2
Q.18 If f(x)=x^3-3x, slope of tangent at x=1 is:
0
1
-2
2
Explanation - f'(x)=3x^2-3. At x=1, slope=0.
Correct answer is: 0
Q.19 The function y=ln(x^2+1) has derivative:
2x/(x^2+1)
x/(x^2+1)
1/(x^2+1)
2/(x^2+1)
Explanation - Derivative of ln(u) is 1/u * du/dx. Here u=x^2+1, derivative=2x/(x^2+1).
Correct answer is: 2x/(x^2+1)
Q.20 For f(x)=x^3, concavity changes at:
x=0
x=1
x=-1
Never
Explanation - f''(x)=6x. Sign changes at x=0 → inflection point.
Correct answer is: x=0
Q.21 If y=sin(x^2), slope at x=1 is:
cos(1)
2cos(1)
cos(1^2)
2cos(1^2)
Explanation - dy/dx=cos(x^2)*2x. At x=1, slope=2cos(1).
Correct answer is: 2cos(1)
Q.22 For f(x)=x^2e^x, slope at x=0 is:
0
1
2
e
Explanation - f'(x)=2xe^x+x^2e^x. At x=0, slope=0.
Correct answer is: 0
Q.23 If f(x)=√x, slope at x=4 is:
1/2
1/4
2
1
Explanation - f'(x)=1/(2√x). At x=4, slope=1/4.
Correct answer is: 1/4
Q.24 The maximum slope of y=sin(x) is:
1
-1
0
Does not exist
Explanation - dy/dx=cos(x). Max value of cos(x)=1.
Correct answer is: 1
Q.25 If f(x)=x^4, slope at x=2 is:
16
32
8
4
Explanation - f'(x)=4x^3. At x=2, slope=4*8=32.
Correct answer is: 32