Logarithms Questions and Answers

Answer: A

A deterministic model is one in which the present state is fixed entirely by the parameters and the earlier states: run it twice from the same start and it gives the same answer twice.

A stochastic or probabilistic model includes randomness, so repeated runs can differ.

Answer: D

Volume changes at the difference between what flows in and what flows out.

\[ \frac{dV}{dt} = \text{rate in} - \text{rate out} \]

The two rates are equal and constant, so the difference is zero:

\[ \frac{dV}{dt} = 0 \]

The volume is therefore constant - the tank is in steady state, neither filling nor draining.

Answer: D

The value prescribed at the starting point of a problem is the initial condition, also called the seed value.

\[ y(t_{0}) = y_{0} \]

It is what picks a single solution out of the general family, each member of which differs only by the arbitrary constant.

Answer: A

Boyle's law states that at constant temperature the pressure and volume of a gas are inversely proportional:

\[ pV = \text{constant} \quad\Longrightarrow\quad V = \frac{\text{constant}}{p} \]

Halving the pressure therefore doubles the volume. The approximation holds best at low pressure, where the gas is close to ideal.

Answer: D

A linear accelerator drives particles forward with oscillating electric fields, and an electric field can only exert force on something carrying charge.

\[ F = qE \]

So only charged particles - electrons, protons, ions - can be accelerated. A neutral particle has \( q = 0 \) and feels no force at all.

Answer: A

Napier's logarithms are the natural logarithms, taken to base \( e \) and written \( \ln x \).

Common logarithms, to base 10, came slightly later and are named after Briggs. The mantissa is only the fractional part of a logarithm, and the antilogarithm is the inverse operation, so neither names a system of logarithms at all.

Answer: D

A function has a Laplace transform only if the integral

\[ \int_{0}^{\infty} e^{-st}f(t)\,dt \]

converges. That requires \( f \) to be piecewise continuous and of exponential order.

\( \tan t \) has infinite discontinuities at \( t = \tfrac{\pi}{2}, \tfrac{3\pi}{2}, \ldots \), where it blows up faster than any exponential can damp. The integral diverges, so no Laplace transform exists.

Answer: D

The value of a variable at the designated starting time \( t = 0 \) is the seed value - also called the initial value.

It is what an initial condition supplies, and it is what selects one particular solution from the general family.

Answer: A

The Laplace transform is defined by an integral:

\[ F(s) = \int_{0}^{\infty} e^{-st}f(t)\,dt \]

The function of \( t \) is multiplied by \( e^{-st} \) and integrated over all positive time, producing a new function of \( s \). Its power comes from turning differentiation in \( t \) into multiplication in \( s \).

Answer: A

An ordinary differential equation involves one independent variable, so every derivative in it is an ordinary derivative:

\[ \frac{d^{2}y}{dx^{2}} + 3\frac{dy}{dx} + 2y \;=\; 0 \]

A partial differential equation has two or more independent variables and therefore partial derivatives, as in \( \partial u/\partial t = \partial^{2}u/\partial x^{2} \). Linear and non linear describe how the unknown appears, which is a separate classification from how many independent variables there are.