Single-Variable Calculus
Key Takeaways
- The power rule d/dx[xⁿ] = nxⁿ⁻¹ and chain rule d/dx[f(g(x))] = f′(g(x))·g′(x) are the most-used FE differentiation tools.
- Critical points occur where f′(x) = 0; the second-derivative test classifies them (f″ > 0 minimum, f″ < 0 maximum).
- Integration reverses differentiation: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1, and ∫(1/x) dx = ln|x| + C.
- L'Hôpital's Rule resolves 0/0 and ∞/∞ limits by differentiating numerator and denominator separately.
- Disk method gives volume of revolution V = π∫[f(x)]² dx; arc length L = ∫√(1+[f′(x)]²) dx.
- The Maclaurin series of eˣ is 1 + x + x²/2! + x³/3! + …; sin x and cos x have alternating-sign series.
Calculus is the analytical engine behind statics, dynamics, fluid mechanics, and engineering economics. On the FE you must differentiate, integrate, and find extrema quickly, then plug into Handbook formulas (centroids, moments of inertia, work integrals) that are themselves integrals.
Limits and Continuity
f(x) is continuous at x = a when f(a) exists, lim(x→a) f(x) exists, and the two are equal. L'Hôpital's Rule: if lim f(x)/g(x) gives the indeterminate forms 0/0 or ∞/∞, then lim f(x)/g(x) = lim f′(x)/g′(x).
Worked example — L'Hôpital. Evaluate lim(x→0) sin x / x. Direct substitution gives 0/0, so differentiate top and bottom: lim cos x / 1 = cos 0 = 1. This limit (= 1) is worth memorizing; it underlies the derivative of sine.
Differentiation Rules
| Rule | Formula | Example |
|---|---|---|
| Power | d/dx[xⁿ] = nxⁿ⁻¹ | d/dx[x³] = 3x² |
| Product | (fg)′ = f′g + fg′ | d/dx[x sin x] = sin x + x cos x |
| Quotient | (f/g)′ = (f′g − fg′)/g² | — |
| Chain | d/dx[f(g(x))] = f′(g)·g′ | d/dx[sin 3x] = 3 cos 3x |
Common Derivatives
| f(x) | f′(x) | f(x) | f′(x) |
|---|---|---|---|
| sin x | cos x | eˣ | eˣ |
| cos x | −sin x | ln x | 1/x |
| tan x | sec²x | arctan x | 1/(1+x²) |
Worked example — product + chain. Differentiate f(x) = x³ sin x. By the product rule, f′(x) = (3x²)(sin x) + (x³)(cos x) = 3x² sin x + x³ cos x. A frequent trap is treating it as a simple product of derivatives (3x²·cos x), which drops two terms.
Applications of Derivatives — Maxima and Minima
The FE loves optimization (minimize cost, maximize area). Procedure: (1) write the quantity as a function of one variable, (2) set f′(x) = 0 to find critical points, (3) classify with the second-derivative test — f″ > 0 gives a local minimum (concave up), f″ < 0 gives a local maximum (concave down), f″ = 0 is inconclusive. Inflection points are where f″(x) = 0 and concavity changes.
Worked example — maxima/minima. A rectangular pen of fixed perimeter P = 40 m is built against a wall (only three sides fenced). Maximize area. Let x = the two sides perpendicular to the wall; then the parallel side is 40 − 2x, and A(x) = x(40 − 2x) = 40x − 2x². Set A′(x) = 40 − 4x = 0 → x = 10 m. Since A″ = −4 < 0, this is a maximum. The dimensions are 10 m × 20 m and A = 200 m². Checking endpoints (x = 0 or x = 20 give A = 0) confirms the interior critical point is the true maximum.
Integration
Integration accumulates quantities and reverses differentiation.
| f(x) | ∫f(x) dx |
|---|---|
| xⁿ (n ≠ −1) | xⁿ⁺¹/(n+1) + C |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| sin x | −cos x + C |
| cos x | sin x + C |
Techniques
- u-substitution: ∫f(g(x))·g′(x) dx = ∫f(u) du with u = g(x).
- Integration by parts: ∫u dv = uv − ∫v du. Choose u by LIATE (Logarithmic, Inverse-trig, Algebraic, Trig, Exponential).
- Partial fractions: split a rational function such as 1/[(x−1)(x+2)] into A/(x−1) + B/(x+2) before integrating.
Worked example — definite integral. Evaluate ∫₀¹ 2x dx = [x²]₀¹ = 1² − 0² = 1. Always evaluate the antiderivative at the upper limit minus the lower limit; forgetting to subtract the lower bound is the classic error.
Applications of Integrals
| Quantity | Formula |
|---|---|
| Area between curves | A = ∫ₐᵇ [f(x) − g(x)] dx, f ≥ g |
| Volume (disk, about x-axis) | V = π∫ₐᵇ [f(x)]² dx |
| Volume (shell, about y-axis) | V = 2π∫ₐᵇ x·f(x) dx |
| Arc length | L = ∫ₐᵇ √(1 + [f′(x)]²) dx |
| Average value | f̄ = (1/(b−a)) ∫ₐᵇ f(x) dx |
Worked example — volume of revolution. Rotate y = √x for 0 ≤ x ≤ 4 about the x-axis. Disk method: V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π[x²/2]₀⁴ = π(16/2) = 8π ≈ 25.1. Squaring √x to get x is the step most often missed.
Series and Sequences
The Taylor series about x = a is f(x) = Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ. The Maclaurin series is the special case a = 0.
| Function | Maclaurin series |
|---|---|
| eˣ | 1 + x + x²/2! + x³/3! + … |
| sin x | x − x³/3! + x⁵/5! − … |
| cos x | 1 − x²/2! + x⁴/4! − … |
| ln(1+x) | x − x²/2 + x³/3 − … |
| 1/(1−x) | 1 + x + x² + x³ + … (|x| < 1) |
Worked example — series approximation. Estimate e^0.1 using the first three terms: 1 + 0.1 + 0.1²/2 = 1 + 0.1 + 0.005 = 1.105. The true value is 1.10517, so three terms already give four-figure accuracy — illustrating why low-order Taylor terms dominate near the expansion point. Recognize that the alternating series with x³/3! belongs to sin x and the one starting 1 − x²/2! belongs to cos x; mixing them up is a common distractor on the FE.
What is the derivative of f(x) = x³ sin(x)?
A rectangular pen uses a wall for one side and 40 m of fence for the other three sides. What length of the two equal perpendicular sides maximizes the enclosed area?
Evaluate the integral ∫₀¹ 2x dx.
What is the volume generated by rotating y = √x for 0 ≤ x ≤ 4 about the x-axis (disk method)?