Logarithms Questions and Answers

Answer: D

A continuous model treats its quantities as varying smoothly over the whole domain, with a value defined at every point - a velocity field, a temperature distribution, an electric field.

A discrete model would define values only at separated points or time steps. The distinction is what separates differential equations from difference equations.

Answer: A

Transforming \( y'' - y = 0 \) with \( y(0) = 12 \) and \( y'(0) = 0 \):

\[ s^{2}Y - s \cdot 12 - 0 - Y = 0 \]

\[ Y(s^{2} - 1) = 12s \quad\Longrightarrow\quad Y = \frac{12s}{s^{2}-1} \]

Since \( \mathcal{L}\{\cosh t\} = \dfrac{s}{s^{2}-1} \):

\[ y = 12\cosh t \]

The hyperbolic cosine appears because the denominator is \( s^{2} - 1 \); an ordinary cosine would need \( s^{2} + 1 \).

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: A

A curve can be parametrised in infinitely many ways - changing the parameter traces the same set of points at a different rate.

\[ (\cos t, \sin t) \quad\text{and}\quad (\cos 2t, \sin 2t) \]

both describe the unit circle. So parametric representations are non-unique: the curve determines the shape, not the parametrisation.

Answer: D

An open interval excludes its endpoints and is written with parentheses:

\[ (a,b) = \{\, x \mid a < x < b \,\} \]

The closed interval \( [a,b] \) uses square brackets and includes both ends.

Answer: A

An integral curve is a curve whose tangent at every point matches the direction the differential equation prescribes there - that is, a particular solution drawn in the plane.

The family of all such curves fills the direction field, and an initial condition picks out one of them.

Answer: B

In a differential equation the rates of change are the derivatives.

\[ \frac{dy}{dt} \ \text{is the rate at which } y \text{ changes with } t \]

The functions themselves record the quantities; only their derivatives record how fast those quantities move.

Answer: D

Logarithm and exponential are inverse operations:

\[ \log_{a}(a^{x}) = x, \qquad a^{\log_{a} x} = x \]

A logarithm asks "to what power?", and exponentiation supplies that power - each undoes the other. Taking antilogs is exactly this exponentiation.

Answer: B

A partial differential equation involves partial derivatives with respect to two or more independent variables.

\[ \frac{\partial u}{\partial t} = c^{2}\frac{\partial^{2} u}{\partial x^{2}} \]

An ordinary differential equation has just one independent variable and so no partial derivatives at all.

Answer: B

Cross-multiplying the proportion clears both denominators:

\[ \frac{a}{b} = \frac{c}{d} \quad\Longrightarrow\quad ad = bc \]

Multiply both sides by \( bd \): the left gives \( ad \) and the right \( bc \). This holds provided \( b \) and \( d \) are non-zero.