Differential Calculus Questions and Answers

Answer: A

The form must be recognised before anything is done to it, since a determinate power needs no such treatment. Logarithms then turn the power into \( g \ln f \), a product that becomes a quotient. That quotient is evaluated, and only at the end is the exponential taken to recover the limit itself rather than its logarithm

Answer: D

Each differentiation multiplies by the constant \( a \) and leaves the exponential alone, so after \( n \) steps

\[ \frac{d^{n}}{dx^{n}} e^{ax} \;=\; a^{n} e^{ax}. \]

This is why the exponential is the natural basis for linear differential equations: substituting it turns each derivative into a power of \( a \) and the equation into a polynomial

Answer: C

Repeated use of the product rule generates binomial coefficients, exactly as in the binomial theorem:

\[ (uv)^{(n)} \;=\; \sum_{r=0}^{n} \binom{n}{r} u^{(n-r)} v^{(r)}. \]

For \( n = 1 \) this is \( u'v + uv' \). The two-term forms offered omit every mixed derivative between the extremes, which for \( n = 2 \) loses the \( 2u'v' \) term

Answer: B

The derivatives run in a cycle of four:

\[ \sin x \;\to\; \cos x \;\to\; -\sin x \;\to\; -\cos x \;\to\; \sin x. \]

So the fourth derivative returns the original function, and in general \( y^{(n)} = \sin(x + n\pi/2) \). Any derivative whose order is a multiple of four reproduces the function unchanged

Answer: A

Curvature \( \kappa \) measures how fast the tangent direction turns per unit arc length, and the radius of curvature is \( \rho = 1/\kappa \). It is the radius of the circle that matches the curve in position, tangent and second derivative at the point, so a straight line has zero curvature and infinite radius

Answer: D

The standard Cartesian formula is

\[ \rho \;=\; \frac{\left(1 + y'^{2}\right)^{3/2}}{|y''|}. \]

The inverted form is the curvature itself. The exponent must be \( 3/2 \), not \( 1/2 \): the arc length contributes one power of \( (1+y'^{2})^{1/2} \) and the turning of the tangent the remaining one

Answer: C

At the origin \( y' = 2x = 0 \) and \( y'' = 2 \), so the formula collapses to the reciprocal of the second derivative:

\[ \rho \;=\; \frac{(1 + 0)^{3/2}}{|2|} \;=\; \frac{1}{2}. \]

The parabola is therefore locally the circle \( x^{2} + (y - 1/2)^{2} = 1/4 \), whose centre sits on the axis at the height of the focus

Answer: B

A circle turns at a constant rate, and its curvature is the reciprocal of its radius:

\[ \kappa \;=\; \frac{1}{\rho} \;=\; \frac{1}{5} \;=\; 0.2. \]

This is the one curve whose curvature is the same at every point, which is what makes it the natural yardstick: any curve's curvature at a point is that of the circle which best fits it there

Answer: A

Each point of a curve has a centre of curvature, lying on the normal at a distance \( \rho \) on the concave side. As the point moves along the curve the centre traces the evolute, and the original curve is then one involute of it. Inflexion points, where the curvature vanishes, send the centre off to infinity

Answer: D

The denominator vanishes at \( x = 3 \), giving a vertical asymptote. For the horizontal one, divide through by \( x \):

\[ \lim_{x \to \pm\infty} \frac{2x+1}{x-3} \;=\; \lim_{x \to \pm\infty} \frac{2 + 1/x}{1 - 3/x} \;=\; 2. \]

So \( y = 2 \) is approached at both ends, the ratio of the leading coefficients