NDA Calculus: Functions, Limits, Continuity and Derivatives

Owned by the Differential Calculus portion of the UPSC NDA Mathematics syllabus, with only a short bridge to the related integral-calculus route. Full integration methods are owned by /nda/mathematics/integration, and differential equations are owned by /nda/mathematics/differential-equations.

Real-valued functions; domain, range and graphs; composite, one-one, onto and inverse functions; limits; continuity; derivatives and their geometrical / physical interpretation; derivatives of sums, products, quotients and composite functions; second-order derivatives; increasing and decreasing functions; applications of derivatives to maxima and minima. Integration is linked but not duplicated here.

Algebraic manipulation, functions, trigonometric identities, basic coordinate geometry.

• Function, domain, range and graph
• Limit and continuity
• Derivative as rate of change / slope
• Antiderivative
• Definite integral

• d(x^n)/dx = n x^(n-1)
• d(sin x)/dx=cos x; d(cos x)/dx=-sin x; d(e^x)/dx=e^x; d(ln x)/dx=1/x for x>0
• Product: (uv)'=u'v+uv'; quotient: (u/v)'=(u'v-uv')/v^2, v!=0
• Chain rule: d[f(g(x))]/dx=f'(g(x))g'(x)
• At a differentiable interior extremum, f'(x)=0 is a candidate condition, not a complete classification test.

• Continuity at a point requires the limit to exist and equal the function value.
• Differentiability implies continuity at that point, but continuity alone does not guarantee differentiability.
• Quotient rule requires a non-zero denominator.
• A stationary point is not automatically a maximum or minimum; classify by sign change, second derivative where valid, or endpoint comparison on a closed interval.
• Inverse-function work requires the stated one-one / onto conditions on the relevant domain and codomain.

1. For f(x)=x^3-3x, f'(x)=3x^2-3=3(x^2-1), so stationary points are x=+/-1. Classify using sign change or second derivative rather than assuming both are extrema of the same type.
2. For lim x->2 (x^2-4)/(x-2), factor to (x-2)(x+2)/(x-2), giving limit 4 after cancellation for x != 2.
3. For f(x)=x^2-4x+7, f'(x)=2x-4=0 gives x=2. Since f''(x)=2>0, x=2 is a local minimum.

• Substituting into an indeterminate limit before simplifying.
• Forgetting chain-rule inner derivative.
• Treating f'(x)=0 as sufficient for maximum.
• Dropping +C in an indefinite integral.
• Confusing signed definite integral with geometric area.

• Knowledge Gap: The concept, definition or theorem needed to solve the item is not understood.
• Recall Gap: The concept was learned but a formula, condition or relation cannot be retrieved accurately.
• Execution Error: The correct method is known but algebra, arithmetic, sign, substitution, diagram or step execution fails.
• Decision / Selection Error: The candidate chooses an inefficient method, continues a time-heavy path, or attempts a low-confidence item without adequate basis.
• Needs Review: Evidence is insufficient or contradictory; do not mislabel the error until another worked item is checked.
• Time pressure note: Time pressure is treated as a contributing condition, not automatically as the root cause.

Classify the item as function / mapping, limit, continuity, direct differentiation or application of derivative. Simplify algebra before taking a limit or derivative. For maxima / minima, write the candidate condition and then classify. If the item is dominated by integration or a differential equation, move to the dedicated frozen route rather than duplicating methods here.

Negative marking is a real exam condition, but this page does not promise a universal attempt threshold. Use method certainty and option evidence before committing to a low-confidence answer.

• Required reviewer: NDA Mathematics faculty with demonstrable subject competence.