Logarithms Questions and Answers

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.

Answer: A

The logarithm to base 2 is the binary logarithm, written \( \log_{2} \) or \( \text{lb} \).

\[ \log_{2} 8 = 3 \quad\text{since}\quad 2^{3} = 8 \]

It is the natural measure in computing, where it counts bits and gives the depth of a balanced binary tree.

Answer: D

A logarithmic scale spaces values by their logarithm rather than by their size, so each fixed step multiplies the quantity by a constant factor:

\[ 1,\; 10,\; 100,\; 1000 \;\longrightarrow\; 0,\; 1,\; 2,\; 3 \]

That is what lets a single axis hold a range spanning many orders of magnitude. The decibel, the Richter scale and pH all work this way. Non linear scale is true of a logarithmic scale but describes a whole family rather than naming this one.

Answer: C

Operational calculus replaces the operations of calculus with algebraic ones. The Laplace transform is its best-known tool: differentiation in \( t \) becomes multiplication by \( s \).

\[ \mathcal{L}\{f'(t)\} = sF(s) - f(0) \]

A differential equation therefore turns into an algebraic one, which is solved and then transformed back.

Answer: B

A differential equation whose right-hand side does not contain the independent variable explicitly is called autonomous.

\[ \frac{dy}{dx} = f(y) \quad\text{is autonomous}; \qquad \frac{dy}{dx} = f(x,y) \quad\text{in general is not} \]

Autonomous equations are unchanged by a shift in \( x \), which is why their direction fields look the same in every vertical strip.

Answer: B

The general solution of a linear homogeneous equation is called the complementary function.

Every such equation has the trivial solution \( y = 0 \), which is why one solution is always "automatic"; the complementary function collects all of them.

For an inhomogeneous equation the full solution is the complementary function plus a particular integral.

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.