Single-Variable Calculus
Key Takeaways
- Derivatives measure instantaneous rate of change: f'(x) = lim(h→0) [f(x+h) - f(x)] / h.
- The power rule d/dx[xⁿ] = nxⁿ⁻¹ is the most-used differentiation rule on the FE exam.
- The chain rule d/dx[f(g(x))] = f'(g(x))·g'(x) is critical for composite functions.
- Integration is the reverse of differentiation: ∫f'(x)dx = f(x) + C.
- Definite integrals compute area under curves, volumes of revolution, and accumulated quantities.
- Know integration techniques: substitution, integration by parts, and partial fractions.
Single-Variable Calculus
Calculus is the mathematical foundation for nearly every engineering discipline. The FE exam tests your ability to differentiate, integrate, and apply these concepts to real-world engineering problems.
Limits and Continuity
A function f(x) is continuous at x = a if:
- f(a) exists
- lim(x→a) f(x) exists
- lim(x→a) f(x) = f(a)
L'Hôpital's Rule: If lim(x→a) f(x)/g(x) gives 0/0 or ∞/∞, then:
Differentiation Rules
| Rule | Formula | Example |
|---|---|---|
| Constant | d/dx[c] = 0 | d/dx[5] = 0 |
| Power | d/dx[xⁿ] = nxⁿ⁻¹ | d/dx[x³] = 3x² |
| Constant Multiple | d/dx[cf(x)] = cf'(x) | d/dx[3x²] = 6x |
| Sum/Difference | d/dx[f ± g] = f' ± g' | d/dx[x² + x] = 2x + 1 |
| Product | d/dx[fg] = f'g + fg' | d/dx[x·sin x] = sin x + x cos x |
| Quotient | d/dx[f/g] = (f'g - fg')/g² | d/dx[sin x/x] = (x cos x - sin x)/x² |
| Chain | d/dx[f(g(x))] = f'(g(x))·g'(x) | d/dx[sin(3x)] = 3cos(3x) |
Common Derivatives
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | -sin x |
| tan x | sec²x |
| eˣ | eˣ |
| ln x | 1/x |
| aˣ | aˣ ln a |
| arcsin x | 1/√(1-x²) |
| arctan x | 1/(1+x²) |
Applications of Derivatives
Critical Points and Extrema
- Critical points: Where f'(x) = 0 or f'(x) is undefined
- First Derivative Test: f' changes from + to - → local maximum; from - to + → local minimum
- Second Derivative Test: At critical point x = c:
- f''(c) > 0 → local minimum (concave up)
- f''(c) < 0 → local maximum (concave down)
- f''(c) = 0 → test is inconclusive
Inflection Points
Points where concavity changes: f''(x) = 0 and f'' changes sign.
Related Rates
When quantities change with respect to time, differentiate the relationship implicitly with respect to t:
- Write the equation relating the variables
- Differentiate both sides with respect to t
- Substitute known values and solve for the unknown rate
Integration
Basic Integration Rules
| Function | Integral |
|---|---|
| xⁿ (n ≠ -1) | xⁿ⁺¹/(n+1) + C |
| 1/x | ln |
| eˣ | eˣ + C |
| sin x | -cos x + C |
| cos x | sin x + C |
| sec²x | tan x + C |
| 1/(1+x²) | arctan x + C |
| 1/√(1-x²) | arcsin x + C |
Integration Techniques
Substitution (u-substitution):
Integration by Parts:
Use the LIATE rule for choosing u: Logarithmic → Inverse trig → Algebraic → Trigonometric → Exponential
Partial Fractions: Decompose rational functions before integrating:
Applications of Integrals
Area Between Curves
where f(x) ≥ g(x) on [a, b]
Volume of Revolution
Disk method (about x-axis):
Shell method (about y-axis):
Arc Length
Series and Sequences
Taylor Series about x = a:
Maclaurin Series (Taylor about x = 0):
| Function | 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 - x⁴/4 + ... |
| 1/(1-x) | 1 + x + x² + x³ + ... ( |
What is the derivative of f(x) = x³ sin(x)?
Evaluate the integral ∫₀¹ 2x dx.
What is the Maclaurin series expansion of eˣ?