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.
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?