Question

Find the dot product fg on the interval [0, 1] for the functions

f(x) = x, g(x) = x².
O f . g = x⁴
O f . g = -1
O f . g = -1/4
O f . g = -1/2
O f . g = 1
O f . g = 1/2
O f . g = 1/4
O f . g = 0
O f . g = x³
O None of the options displayed.

Answer 1

The dot product of the functions f(x) = x and g(x) = x² over the interval [0, 1] is found by integrating their product x³, which results in 1/4.

Step-by-step explanation:

The task is to find the dot product of the functions f(x) = x and g(x) = x² over the interval [0, 1]. A dot product in this context involves multiplying the two functions together and integrating over the specified interval.

The product of f(x) and g(x) is x × x² = x³. The integral of x³ from 0 to 1 is computed by evaluating the antiderivative of x³, which is x⁴/4, at the two endpoints of the interval and subtracting the lower bound result from the upper bound result. This calculation results in:

∫ f(x)g(x) dx = ∫ x³ dx = [ x⁴/4 ] ˇ₀₁ = (1⁴/4) - (0⁴/4) = 1/4 - 0 = 1/4

Therefore, the correct option for the dot product f · g on the interval [0, 1] is 1/4.