ETS 1.0 Calculus

1 Limits of Functions

• • Epsilon-delta definition
• • Limit theorems and properties
• • One-sided limits
• • Infinities and indeterminate forms

2 Continuity

• • Definition and characterization
• • Intermediate value theorem
• • Properties of continuous functions

3 Topological Concepts

• • Open and closed sets
• • Interior and limit points
• • Bounded, connected, and compact sets
• • Completeness of \( \mathbb{R} \)

1 Derivatives & Differentiation Rules

• • Definition: \( f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \)
• • Product, quotient, chain rules
• • Derivatives of standard functions

2 Rolle's & MVT

• • Rolle's theorem statement & proof
• • Mean value theorem
• • Cauchy's mean value theorem

3 L'Hôpital's Rule & Applications

• • Indeterminate forms: \( \frac{0}{0}, \frac{\infty}{\infty} \), etc.
• • L'Hôpital's rule application
• • Monotonicity and extrema

1 Taylor's Theorem & Expansion

• • Taylor's theorem with remainder
• • Lagrange's form of remainder
• • Taylor polynomial approximation

2 Taylor & Maclaurin Series

• • Series expansions
• • Common series: \( e^x, \sin x, \cos x, \ln x \)
• • Domain of convergence

3 Power Series Operations

• • Term-wise differentiation
• • Term-wise integration
• • Radius of convergence preservation

1 Definite Integrals

• • Riemann sums and upper/lower sums
• • Definition of Riemann integrability
• • Riemann integral properties

2 Fundamental Theorem of Calculus

• • FTC Part 1 & Part 2
• • Connection: derivatives ↔ integrals
• • Antiderivatives

3 Integration Techniques & Applications

• • Substitution and integration by parts
• • Computing areas and volumes
• • Arc length and surface area

1 Limits & Continuity in \( \mathbb{R}^2 \)

• • Multivariable limits: \( \lim_{(x,y) \to (a,b)} f(x,y) \)
• • Continuity for multivariable functions
• • Path independence and limits

2 Partial Derivatives

• • Definition: \( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \)
• • Higher-order partial derivatives
• • Schwarz's theorem (mixed partials equality)

3 Differentiability & Gradient

• • Total derivative and Jacobian
• • Gradient vector and directional derivatives
• • Tangent plane to surfaces

1 Maxima & Minima in \( \mathbb{R}^2 \)

• • Critical points: \( \nabla f = 0 \)
• • Hessian matrix and 2nd derivative test
• • Saddle points and classification

2 Lagrange Multipliers

• • Method of Lagrange multipliers
• • Constrained optimization: \( \nabla f = \lambda \nabla g \)
• • Multiple constraints

3 Double & Triple Integrals

• • Double integrals: \( \iint_R f(x,y) \, dA \)
• • Change of order and Fubini's theorem
• • Triple integrals and volume calculations
• • Polar, cylindrical, spherical coordinates
• • Homogeneous functions & Euler's theorem