Calculus

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.