Logarithms Questions and Answers

Answer: B

Writing derivatives with dashes - \( y' \), \( y'' \), \( y''' \) - is prime notation, due to Lagrange.

Leibniz notation is the alternative, \( \tfrac{dy}{dx} \), \( \tfrac{d^{2}y}{dx^{2}} \), which has the advantage of naming the independent variable explicitly.

Answer: C

A constant factor passes straight through differentiation.

\[ \frac{\partial}{\partial x}(\epsilon x) = \epsilon \cdot \frac{\partial x}{\partial x} = \epsilon \cdot 1 = \epsilon \]

Since \( \epsilon \) does not depend on \( x \), it is not itself differentiated.

Answer: B

Separation of variables is also called the Fourier method, after Fourier's use of it on the heat equation.

The idea is to look for a solution built as a product of single-variable functions:

\[ u(x,t) \;=\; X(x)\,T(t) \]

Substituting and dividing through leaves each side depending on one variable only, so both must equal a constant. That turns one partial differential equation into two ordinary ones, whose solutions are then recombined as a Fourier series.

Answer: B

A first-order equation is separable when it can be rearranged so that each variable sits with its own differential:

\[ \frac{dy}{dx} = f(x)g(y) \quad\Longrightarrow\quad \frac{dy}{g(y)} = f(x)\,dx \]

Each side can then be integrated on its own. Separability is precisely the property that the variables can be pulled apart in this way.

Answer: C

A floating model rests on neither theory nor observation - it simply assumes a structure that seems plausible.

A deductive model is derived from theory and an inductive one built up from data; the floating model has neither footing, which is why its conclusions carry little weight on their own.

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

An interval is closed when both endpoints are finite and both belong to it, written with square brackets:

\[ [a,b] \;=\; \{\, x : a \le x \le b \,\} \]

• Closed \( [a,b] \) - both endpoints included.
• Open \( (a,b) \) - neither endpoint included.
• Infinite - at least one end unbounded, as in \( [a,\infty) \).

Answer: D

The characteristic is one less than the count of digits before the decimal point.

\[ \text{characteristic} = 1 - 1 = 0 \]

A one-digit number lies between 1 and 10, so its logarithm lies between \( \log 1 = 0 \) and \( \log 10 = 1 \) - the whole part must therefore be 0. For instance \( \log 7 = 0.8451 \).

Answer: B

\( \sin y = 1 \) is non-linear, because the unknown sits inside a sine rather than appearing to the first power on its own.

A linear equation admits the unknown only as \( ay + b \); anything else - a power, a product of unknowns, or a transcendental function of the unknown - makes it non-linear:

\[ 3y + 2 = 0 \;\;\text{linear}, \qquad \sin y = 1,\; y^{2} = 4,\; e^{y} = 3 \;\;\text{non-linear} \]

The practical consequence is that superposition fails: two solutions of a non-linear equation cannot be added to give a third.

Answer: A

The mean Earth–Sun distance is about \( 1.5 \times 10^{8} \) km - 150,000,000 km.

This distance is itself the definition of the astronomical unit (AU), the standard yardstick for distances within the solar system. Light covers it in about 8 minutes 20 seconds.