Calculus focuses on understanding change, motion, and rates of growth, forming the foundation for advanced mathematics, science, and engineering. Here are the key learning goals:
1. Limits and Continuity
- Understand the concept of a limit and its notation.
- Calculate limits algebraically, graphically, and numerically.
- Evaluate one-sided, two-sided, and infinite limits.
- Identify and analyze discontinuities (removable, jump, and infinite).
- Apply the Intermediate Value Theorem.
2. Derivatives and Differentiation
- Understand the derivative as the instantaneous rate of change and the slope of a tangent line.
- Differentiate functions using the definition of a derivative.
- Apply differentiation rules (power, product, quotient, and chain rules).
- Differentiate trigonometric, exponential, logarithmic, and implicit functions.
- Use higher-order derivatives to analyze motion (velocity, acceleration, jerk).
3. Applications of Derivatives
- Find critical points and determine extrema using the First and Second Derivative Tests.
- Use the Mean Value Theorem to analyze function behavior.
- Apply derivatives to curve sketching (concavity, inflection points, asymptotes).
- Solve optimization problems in real-world applications.
- Apply related rates to problems involving multiple changing variables.
4. Integrals and Integration
- Understand the integral as the antiderivative and the area under a curve.
- Apply the Fundamental Theorem of Calculus.
- Evaluate definite and indefinite integrals.
- Use integration techniques (substitution, integration by parts, partial fractions, and trigonometric substitution).
- Approximate area under a curve using Riemann sums and trapezoidal rule.
5. Applications of Integration
- Calculate areas between curves.
- Determine volumes of solids using disk, washer, and shell methods.
- Solve problems involving work, fluid force, and center of mass.
- Apply integration to motion (position, velocity, acceleration).
6. Differential Equations
- Solve basic separable differential equations.
- Use slope fields to visualize solutions.
- Apply exponential growth and decay models.
- Solve logistic growth problems.
7. Sequences and Series (for Calculus II and Beyond)
- Understand sequences and series, including convergence and divergence.
- Apply tests for convergence (n-th term, integral, ratio, root, alternating series tests).
- Work with power series and Taylor/Maclaurin series.
- Use approximations and error bounds for series.