Logarithms Questions and Answers

Answer: A

Faà di Bruno's formula generalises the chain rule to higher derivatives, giving \( \tfrac{d^{n}}{dx^{n}}f(g(x)) \) as a sum over the partitions of \( n \).

The first case is the ordinary chain rule; the second is

\[ (f \circ g)'' = f''(g)\,(g')^{2} + f'(g)\,g'' \]

Leibniz's formula is the analogous generalisation for the product rule, not the chain rule.

Answer: B

Scientific notation writes a number as a figure between 1 and 10 multiplied by a power of ten, \( a \times 10^{n} \) with \( 1 \le a < 10 \).

\[ 0.000581 \;=\; 5.81 \times 10^{-4}, \qquad 1563.4 \;=\; 1.5634 \times 10^{3} \]

It is exactly this form that gives a logarithm its two parts: the exponent becomes the characteristic and the leading figure supplies the mantissa.

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: A

The graph of a function \( y = h(x) \) that satisfies a differential equation is its solution curve.

The equation prescribes the slope at every point, and a solution curve is one whose tangent matches that prescription everywhere. It is also called an integral curve.

Answer: C

An integer interval holds only the whole numbers lying between its endpoints, rather than every real number there:

\[ \{\, x \in \mathbb{Z} : a \le x \le b \,\} \]

So between 2 and 5 it contains 2, 3, 4 and 5, whereas the closed interval \( [2,5] \) contains every real number in that range. Open and closed describe whether the endpoints are included; this option describes which kind of number is included.

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: 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: C

The standard transform pair is

\[ \mathcal{L}\{\cos \omega t\} = \frac{s}{s^{2}+\omega^{2}} \]

while its companion is

\[ \mathcal{L}\{\sin \omega t\} = \frac{\omega}{s^{2}+\omega^{2}} \]

The denominators are identical; only the numerator distinguishes them - \( s \) for cosine, \( \omega \) for sine.

Answer: D

Read the two features separately.

The square bracket at \( a \) means the endpoint is included (\( x \geq a \)), so the interval is closed at that end. It extends to \( +\infty \) with no upper endpoint, so it is unbounded.

Hence it is closed and unbounded. Infinity is never itself included, which is why that side is always written with a parenthesis.

Answer: C

The chain rule differentiates a composition of functions:

\[ \frac{d}{dx}f\big(g(x)\big) = f'\big(g(x)\big) \cdot g'(x) \]

The product rule handles a product of functions, \( (uv)' = u'v + uv' \), which is a different situation entirely.