The mean value of f(x) over [a, b] is defined as?
A
(1/(b - a)) ∫ₐᵇ f(x) dx
B
∫ₐᵇ f(x) dx
C
∫ₐᵇ f(x)² dx
D
(b - a) ∫ₐᵇ f(x) dx
Analysis & Theory
Mean value = (1/(b - a)) ∫ₐᵇ f(x) dx.
The RMS value of f(x) over [a, b] is defined as?
A
√(1/(b - a) ∫ₐᵇ f(x)² dx)
B
(1/(b - a)) ∫ₐᵇ f(x) dx
C
∫ₐᵇ |f(x)| dx
D
∫ₐᵇ f(x)² dx
Analysis & Theory
RMS value = √(1/(b - a) ∫ₐᵇ f(x)² dx).
The mean value of f(x) = x over [0, 1] is?
A
1/2
B
1/3
C
2/3
D
1
Analysis & Theory
Mean = (1/(1-0)) ∫₀¹ x dx = [x²/2]₀¹ = 1/2.
The RMS value of f(x) = x over [0, 1] is?
A
1/√3
B
1/2
C
√2/2
D
1/3
Analysis & Theory
RMS = √(∫₀¹ x² dx) = √(1/3) = 1/√3.
The mean value of sin(x) over [0, π] is?
A
2/π
B
1
C
0
D
1/2
Analysis & Theory
Mean = (1/π) ∫₀^π sin(x) dx = (1/π)(2) = 2/π.
The RMS value of sin(x) over [0, π] is?
A
1/√2
B
√2/2
C
1
D
2/π
Analysis & Theory
RMS = √((1/π) ∫₀^π sin²(x) dx) = √(1/2) = 1/√2.
The mean value of cos(x) over [0, π/2] is?
A
2/π
B
1
C
0
D
1/2
Analysis & Theory
Mean = (2/π) ∫₀^(π/2) cos(x) dx = (2/π)(1) = 2/π.
The RMS value of cos(x) over [0, π/2] is?
A
1/√2
B
√2/2
C
1
D
2/π
Analysis & Theory
RMS = √((2/π) ∫₀^(π/2) cos²(x) dx) = √(1/2) = 1/√2.
The mean value of f(x) = e^x over [0, 1] is?
A
e - 1
B
(e - 1)
C
(e - 1)/(1)
D
(e - 1)
Analysis & Theory
Mean = (1/1) ∫₀¹ e^x dx = e - 1.
The RMS value of f(x) = e^x over [0, 1] is?
A
√((e² - 1)/2)
B
√(e - 1)
C
√e
D
e/√2
Analysis & Theory
RMS = √((1/1) ∫₀¹ e^(2x) dx) = √((e² - 1)/2).