Differential Calculus Questions and Answers

Answer: A

Only \( 0/0 \) and \( \infty/\infty \) are ready for L'Hopital's rule, so a quotient is differentiated at once. A product is turned into a quotient by inverting one factor, and a power is turned into a product by taking logarithms, after which the same rule finishes the work

Answer: D

The first derivatives are set to zero to obtain the equations, which are then solved for the candidate points. Only at those points are the second derivatives worth computing, and the discriminant formed from them together with the sign of \( f_{xx} \) gives the classification. Evaluating second derivatives before knowing where to evaluate them wastes the work

Answer: C

The first is the standard implication, differentiability being the stronger property. The second is the extreme value theorem, which needs both closedness and continuity. The third is false: at a corner the one-sided derivatives exist but disagree, as they do for \( |x| \) at the origin, so no single derivative exists

Answer: B

Clairaut's theorem gives the first, and Euler's theorem \( x f_{x} + y f_{y} = n f \) the second. The third reverses the truth: a vanishing Jacobian marks functional *dependence*, a relation \( \phi(u,v) = 0 \) tying the two together, and independence is what a non-zero Jacobian certifies

Answer: A

For \( y = ax^{2} \) the slope vanishes at the origin and \( y'' = 2a \), so

\[ \rho \;=\; \frac{1}{2a}, \]

giving \( 1/8 \), \( 1/4 \) and \( 1/2 \) for \( a = 4, 2, 1 \). The straight line has zero curvature and so infinite radius, placing it last

Answer: D

The theorem is an existence result: it promises some \( c \) with \( f'(c) \) equal to the average slope, without saying where. Finding \( c \) means solving an equation, which for most functions cannot be done in closed form. The two hypotheses and the existence conclusion are all stated correctly

Answer: C

The difference quotient \( [f(x)-f(a)]/(x-a) \) can only tend to a finite limit if the numerator tends to zero, which is continuity at \( a \). So differentiability is strictly the stronger condition, and a discontinuity of any kind, removable or not, rules the derivative out

Answer: B

The condition is necessary for an interior extremum of a differentiable function but not sufficient. The function \( y = x^{3} \) has \( y' = 0 \) at the origin yet rises through it, the point being a horizontal inflexion. Deciding which case holds needs the second derivative, or the first non-vanishing one

Answer: A

Take logarithms and differentiate, which converts a product into a sum of relative errors:

\[ \ln y = 3 \ln x \quad \Longrightarrow \quad \frac{dy}{y} \;=\; 3\,\frac{dx}{x}. \]

So a one per cent error in \( x \) becomes three per cent in \( y \). The exponent multiplies the relative error, and the value \( x = 2 \) does not enter at all

Answer: D

The differential is the linear part of the increment, \( dy = f'(x)\,dx \), and it approximates the true increment \( \Delta y \) with an error of order \( dx^{2} \). The bare difference of two values of \\( f \\) is that exact increment rather than its linear approximation, and the version keeping a term in \\( dx^{2} \\) carries a second-order term the differential deliberately omits