Applied Calculus — Unit 1–3 Important Questions (Part A): quick revision, 16 bites

The Derivative: Calculus' Core Definition

The derivative is a limit — nothing more.

Takeaway. Every derivative rule (power, product, chain) is built from this single limit definition.

Power Rule: Derivative of xⁿ

Bring the power down, subtract one.

Takeaway. For f(x) = x², f'(x) = 2x; for f(x) = x⁴ + x², f'(x) = 4x³ + 2x.

Worked: Derivative of x⁴ + x²

Apply the power rule term by term.

Takeaway. Differentiation is linear — handle each term separately, then sum.

Conditions for Maxima and Minima

First derivative zero, second derivative decides.

Takeaway. Stationary point + sign of f'' is the standard 2-mark answer for maxima/minima conditions.

Partial Derivatives: One Variable at a Time

Freeze the others, differentiate one.

Takeaway. Partial differentiation = ordinary differentiation with other variables held constant.

Euler's Theorem for Homogeneous Functions

Degree times the function — that's Euler.

Holds **only** when f is homogeneous of degree n — meaning f(tx, ty) = tⁿ f(x, y).

Takeaway. Euler's theorem is a one-line identity for homogeneous functions of any degree.

Chain Rule for Multivariable Functions

Outer times inner — every layer.

Takeaway. Chain rule links how f changes to how each input changes.

Total Derivative vs Partial Derivative

Total derivative counts every path.

Takeaway. Partial derivative freezes variables; total derivative lets them all vary.

Maxima/Minima for Two Variables

Three conditions decide the surface's peaks.

Takeaway. The discriminant D = f_xx·f_yy − (f_xy)² is the two-variable analogue of the second-derivative test.

Lagrange's Multiplier Method

Optimize under a constraint in 4 steps.

Takeaway. λ absorbs the constraint so you can optimize f as if it were unconstrained.

Fundamental Theorem of Calculus

Differentiation and integration undo each other.

Takeaway. Definite integrals are evaluated by finding an antiderivative and subtracting endpoint values.

Indefinite Integral

Antiderivative plus a constant.

Takeaway. Always add +C — the constant is what makes it 'indefinite'.

Improper Integrals

Infinite limits or infinite integrands.

Takeaway. Improper integrals are always defined via a limit — never plug in ∞ directly.

Reduction Formula for ∫ sinⁿx dx

Peel off two powers at a time.

Takeaway. Reduction formulas lower the power by 2 each step until a base case (n = 0 or 1) is reached.

Don't Confuse Partial and Total Derivatives

Freeze vs. follow — know the difference.

Takeaway. The word 'partial' is a promise — only one variable moves.

2-Mark Answer Strategy

Definition + formula = full marks.

Takeaway. For 2-mark questions, a crisp definition plus the formula is the safest full-marks pattern.