91Ó°ÊÓ

Given an operator \( \hat{A} \) and a function \( f \), we need to evaluate the expression \( g = \hat{A} f \). The operator, \( \hat{A} \), and the function, \( f \), are both provided as possible options. Our task is to determine what the result \( g \) is for each combination.

We'll evaluate each combination of operators and functions. Since this involves some calculus operations, it is crucial to carefully follow the operations described by each operator. We'll proceed with each operator option \( \hat{A} \) on each function option \( f \).

When \( \hat{A} \) is the SQRT operator:- For \( f = x^4 \), \( \hat{A} f = \sqrt{x^4} = x^2 \), assuming non-negative values of \( x \).- For \( f = e^{-ax} \), \( \hat{A}f = \sqrt{e^{-ax}} = e^{-ax/2} \).- For \( f = x^3 - 2x + 3 \), \( \hat{A}f = \sqrt{x^3 - 2x + 3} \).- For \( f = x^3y^2z^4 \), \( \hat{A}f = \sqrt{x^3y^2z^4} \).

When \( \hat{A} = \frac{d^3}{dx^3} + x^3 \):- For \( f = x^4 \), calculate the third derivative and add \( x^3 \): \[ \frac{d^3}{dx^3}x^4 = 24x, \quad \therefore \hat{A}f = 24x + x^3 = x^3 + 24x \]- For \( f = e^{-ax} \), calculate: \[ \frac{d^3}{dx^3}e^{-ax} = -a^3 e^{-ax}, \quad \therefore \hat{A}f = -a^3 e^{-ax} + x^3 \]- For \( f = x^3 - 2x + 3 \), \[ \frac{d^3}{dx^3}(x^3 - 2x + 3) = 6, \quad \therefore \hat{A}f = 6 + x^3 \]- For \( f = x^3y^2z^4 \), not meaningful as \( f \) is not univariate.

When \( \hat{A} = \int_0^1 dx \):- For \( f = x^4 \), integrate: \[ \int_0^1 x^4 \, dx = \left[ \frac{x^5}{5} \right]_0^1 = \frac{1}{5} \]- For \( f = e^{-ax} \), integrate: \[ \int_0^1 e^{-ax} \, dx = \left[ -\frac{1}{a}e^{-ax} \right]_0^1 = \frac{1}{a}(1 - e^{-a}) \]- For \( f = x^3 - 2x + 3 \), integrate: \[ \int_0^1 (x^3 - 2x + 3) \, dx = \left[ \frac{x^4}{4} - x^2 + 3x \right]_0^1 = \frac{11}{4} \]- For \( f = x^3y^2z^4 \), assume it is a constant, \[ \int_0^1 x^3y^2z^4 \, dx = \left[ \frac{x^4}{4}y^2z^4 \right]_0^1 = \frac{1}{4}y^2z^4 \]

When \( \hat{A} = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \):- For \( f = x^4 \), second derivative w.r.t \( x \): \[ \frac{\partial^2}{\partial x^2}x^4 = 12x^2 \] and others are zero.- For \( f = e^{-ax} \), second derivative w.r.t \( x \): \[ \frac{\partial^2}{\partial x^2}e^{-ax} = a^2 e^{-ax} \] and others are zero.- For \( f = x^3 - 2x + 3 \), second derivative w.r.t \( x \): \[ \frac{\partial^2}{\partial x^2}(x^3 - 2x + 3) = 6 \] and others are zero.- For \( f = x^3y^2z^4 \), calculate each: \[ \frac{\partial^2}{\partial x^2} = 6xy^2z^4, \quad \frac{\partial^2}{\partial y^2} = 2x^3z^4, \quad \frac{\partial^2}{\partial z^2} = 12x^3y^2z^2 \] sum results. Total: \[ \hat{A}f = 6xy^2z^4 + 2x^3z^4 + 12x^3y^2z^2 \]

We have determined the action of each operator on each function, resulting in a set of specific answers described in each Option. Review the corresponding combination for a specific need.