In differential equations, which represents their rates of change
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.
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Curve of function i.e. y=h(x) is called
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.
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Interval used to indicate interval of all integers between a and b is called
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.
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Inverse operation of logarithm is
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.
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Interval which does not include its endpoints, and is indicated with parentheses is known as
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.
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Laplace transform of function f(t)=cos(wt) is
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.
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The interval \( [a, +\infty) = \{\, x \mid x \geq a \,\} \) is
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.
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A formula or rule for computing derivative of composition of two or more functions is known as
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.
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If a number has no zero immediately after decimal point, characteristic is
Answer: B
For a number less than 1, the characteristic is negative: it is one more than the number of zeros immediately after the decimal point, taken with a minus sign.
With no such zeros, as in 0.5 or 0.842, the characteristic is \( -1 \), usually written \( \bar{1} \).
\[ \log 0.842 = \bar{1}.9253 = -1 + 0.9253 \]
The mantissa stays positive; only the characteristic carries the sign.
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