CELE Engineering Mathematics — Differential CalculusRevision Notes

Formulas

Exam Tips

• Always attempt direct substitution first; only reach for L'Hôpital if an indeterminate form results.
• Factor and cancel common factors before differentiating — it is often faster than L'Hôpital.
• Memorize lim (sin x)/x = 1 and lim (1−cos x)/x = 0; they appear in disguised forms every exam.
• For limits at infinity of rational functions: divide every term by the highest power of x in the denominator.

Key Points

• The limit lim[x→a] f(x) = L means f(x) approaches L as x approaches a, regardless of f(a) itself.
• For a limit to exist, the left-hand limit and right-hand limit must be equal: lim[x→a⁻] f(x) = lim[x→a⁺] f(x).
• f is continuous at x = a if: (1) f(a) exists, (2) lim[x→a] f(x) exists, and (3) lim[x→a] f(x) = f(a).
• Indeterminate forms requiring further work: 0/0, ∞/∞, 0·∞, ∞−∞, 0⁰, 1^∞, ∞⁰.
• L'Hôpital's Rule: If the limit gives 0/0 or ∞/∞, differentiate numerator and denominator separately, then re-evaluate.
• Squeeze Theorem: If g(x) ≤ f(x) ≤ h(x) near a and lim g = lim h = L, then lim f = L.
• Standard limits to memorize: lim[x→0] (sin x)/x = 1; lim[x→0] (1 − cos x)/x = 0; lim[x→∞] (1 + 1/n)^n = e.

Definitions

Section Title

1. Limits and Continuity

Common Mistakes

• Applying L'Hôpital's Rule when the form is NOT indeterminate — always check the form first.
• Differentiating the whole fraction as a quotient instead of differentiating numerator and denominator separately.
• Using degrees instead of radians when evaluating trigonometric limits.
• Confusing 'limit does not exist' with 'limit equals infinity' — infinity is not a real number.

Formulas

Exam Tips

• Board-exam composite functions like y = (2x+1)^10 are solved in seconds with the chain rule — never expand.
• Logarithmic differentiation simplifies products and quotients raised to powers: take ln of both sides first.
• For d/dx [aˣ] = aˣ ln a — do NOT write aˣ alone; the ln a factor is always present.
• When differentiating implicitly, group all dy/dx terms on the left, then factor and divide.

Key Points

• The derivative f'(x) gives the instantaneous rate of change and the slope of the tangent line at any point.
• Power Rule: d/dx [xⁿ] = nxⁿ⁻¹ — the single most frequently used rule on board exams.
• Product Rule: (uv)' = u'v + uv' — use whenever two functions are multiplied.
• Quotient Rule: (u/v)' = (u'v − uv')/v² — use whenever a function is divided by another.
• Chain Rule: d/dx [f(g(x))] = f'(g(x)) · g'(x) — the most commonly missed rule, especially in composite trig/exponential functions.
• Implicit differentiation: differentiate both sides w.r.t. x, treating y as a function of x; collect dy/dx terms.
• Higher-order derivatives: y'', y''' are successive derivatives; used in classifying critical points and curvature.
• Partial derivatives: ∂f/∂x treats all other variables as constants; ∂f/∂y treats x as constant.

Definitions

Section Title

2. Differentiation Rules and Techniques

Common Mistakes

• Forgetting to apply the chain rule on composite functions, e.g., writing d/dx [sin(x²)] = cos(x²) instead of 2x cos(x²).
• Sign errors in the quotient rule — memorize: 'lo d-hi minus hi d-lo, over lo-squared'.
• Differentiating constants as zero but failing to recognize that e³ is a constant (not e^(3x)).
• In implicit differentiation, forgetting to multiply by dy/dx when differentiating y-terms.
• Treating y' as a separate variable rather than dy/dx when collecting terms.

Formulas

Exam Tips

• Board-exam optimization problems always have a constraint equation — identify it first and use it to eliminate one variable.
• For symmetric shapes (square, cube, equilateral), the optimal solution is almost always the symmetric case — use this as a quick check.
• In fencing/perimeter problems along a river (one side free), the optimal rectangle has width = half the length.
• After solving x from f'(x) = 0, substitute back to find the optimized quantity — partial credit is lost if you stop at x.
• If the constraint gives two solutions, check both — one may be outside the physical domain.

Key Points

• A critical point occurs where f'(x) = 0 or f'(x) is undefined.
• First Derivative Test: f' changes + to − at x₀ → local maximum; f' changes − to + → local minimum.
• Second Derivative Test: f''(x₀) < 0 → local max; f''(x₀) > 0 → local min; f''(x₀) = 0 → inconclusive.
• Absolute (global) extrema on a closed interval [a, b]: evaluate f at all critical points AND endpoints; the largest/smallest value is the answer.
• Inflection point: f'' changes sign; curvature changes from concave up to concave down (or vice versa).
• Optimization strategy: identify the quantity to optimize, write it as a function of ONE variable using the constraint, differentiate, set equal to zero, solve, then verify.
• Common optimization shapes on board exams: rectangle of maximum area for fixed perimeter (square), cylinder of maximum volume, open box from a flat sheet.

Definitions

Section Title

3. Maxima, Minima, and Optimization

Common Mistakes

• Forgetting to check endpoints when the problem specifies a closed interval or physical constraint (e.g., x > 0).
• Setting up the area/volume formula incorrectly before differentiating — always sketch the problem.
• Not verifying whether a critical point is a max or min — use either the first or second derivative test.
• Stopping after finding x without computing the maximum value of the quantity (the question often asks for the value, not x).
• Using the second-derivative test when f'' = 0 — switch to the first-derivative test in that case.

Formulas

Exam Tips

• Always draw a diagram and assign variable names before writing any equation.
• In cone tank problems, write r in terms of h using r/h = R/H before differentiating — eliminates one variable.
• Label all given rates with their signs: filling = positive dV/dt; draining = negative dV/dt.
• Finish by checking units — if dV/dt is in m³/s and r is in m, then dr/dt should be in m/s.
• Ladder problems: once x and dx/dt are given, find y from the Pythagorean theorem before substituting.

Key Points

• Related rates problems involve two or more quantities that both change with time; you differentiate their relationship with respect to t.
• Step 1: Draw and label a diagram identifying all time-varying quantities.
• Step 2: Write an equation relating the variables (geometric formula, Pythagorean theorem, similar triangles, etc.).
• Step 3: Differentiate both sides with respect to t using the chain rule.
• Step 4: Substitute the given numerical values (including the given rates) AFTER differentiating — NEVER before.
• Step 5: Solve for the unknown rate.
• Common geometric relations: sphere V = (4/3)πr³; circle A = πr²; cone V = (1/3)πr²h; right triangle: x² + y² = L².
• Rates are derivatives: dV/dt = rate of volume change; dr/dt = rate of radius change; dx/dt = speed of a moving point.

Definitions

Section Title

4. Related Rates

Common Mistakes

• Substituting numerical values BEFORE differentiating — the most common and costly error in related-rates problems.
• Getting the sign of rates wrong — decreasing quantities have negative rates; always assign signs consistently.
• Not using the similar-triangle ratio in cone problems, leading to two unknowns after differentiation.
• Confusing 'the shadow lengthening' rate with 'the tip of the shadow moving' rate in lamppost problems — these are different quantities.
• Forgetting the chain rule factor (dr/dt or dy/dt) when differentiating geometric formulas with respect to t.

Formulas

Exam Tips

• For radius of curvature at the origin or vertex of a parabola, y'=0, so R simplifies to 1/|y''| — very fast to compute.
• Normal line slope is −1/f'(x₀); if f'(x₀)=0 (horizontal tangent), the normal is vertical (undefined slope — write x = x₀).
• Curve analysis: always find critical points AND inflection points before sketching — board multiple-choice options differ by concavity.
• MVT problems: set f'(x) = average rate, solve for x, and verify x is inside the open interval.

Key Points

• Tangent line at (x₀, y₀): slope m = f'(x₀); equation: y − y₀ = m(x − x₀).
• Normal line: perpendicular to tangent; slope = −1/m.
• Radius of curvature R = [1 + (y')²]^(3/2) / |y''|; large R means nearly flat, small R means sharply curved.
• Center of curvature (evolute): the center of the osculating circle at a point.
• Curve sketching sequence: domain → intercepts → symmetry → asymptotes → critical points → inflection points → sketch.
• Rolle's Theorem: If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then ∃ c ∈ (a,b) where f'(c) = 0.
• Mean Value Theorem (MVT): ∃ c ∈ (a,b) such that f'(c) = [f(b)−f(a)]/(b−a) — the derivative equals the average rate of change.

Definitions

Section Title

5. Tangent Lines, Curvature, and Other Applications

Common Mistakes

• Confusing tangent slope with normal slope — they are negative reciprocals, not equal.
• Applying R = [1+(y')²]^(3/2)/|y''| with y' and y'' evaluated at different x values.
• Declaring an inflection point at x where y''=0 without verifying a sign change in y''.
• Mixing up Rolle's Theorem (requires f(a)=f(b)) with the general MVT.