First Principles AI

Question 1

Consider a data set at monthly sales figures. A polynomial regression model $P(x) = x^3 + 2x^2 + x - 3$ is fitted to this data. If the observed sales in the 5th month are 240 units, find the percentage error.

Solution Theory

Question 2

A retailer company has developed a polynomial regression model to predict weekly product demand: $D(w) = w^3 - 2w^2 + 5w - 4$ , where $D(w)$ represents predicted demand (in units) and $w$ is the week number. Use the remainder theorem to predict demand for 3rd week. If the observed demand is 22 units, calculate the prediction error.

Solution Theory

Question 3

A digital signal processing system has a transfer function with a numerator $B(z) = z^2 - z - 2$ . Use the factor theorem to find the zeros of the system.

Solution Theory

Question 4

A signal process system has a transfer function $H(z) = \dfrac{z^2 + 3z + 2}{z^2 - 0.2z + 0.9}$ . Find the zero(s) of the transfer function by using factor theorem.

Solution Theory

Question 5

A signal process system has a transfer function $H(z) = \dfrac{z^2 - 0.5z - 0.5}{z^2 + 1}$ . Find the zero(s) of the transfer function by using factor theorem.

Solution Theory

Question 6

A signal processing system has a transfer function with a denominator $A(z) = z^2 - 0.3z - 0.4$ . Use factor theorem to find the poles of the system and determine if the system is stable.

Solution Theory

Question 7

The denominator of signal processing system's transfer function equal to $A(z) = z^2 + 1.2z + 0.35$ . Use factor theorem to determine the location of the corresponding poles and assess the stability of the system.

Solution Theory

Questions