Aimed to provide a rigorous understanding of the foundational concepts in calculus.
Course Structure
Limits and Continuity
Introduction and estimation of limits from graphs and tables.
One-sided limits and their graphical behavior.
Techniques for approximating limits: tables, sums, differences, products, and quotients.
Composite functions and direct substitution methods.
Understanding limits of trigonometric and piecewise functions.
Strategy and techniques for finding limits, including the Squeeze Theorem.
Discontinuities and continuity over intervals, including removable discontinuities.
Limits at infinity and the Intermediate Value Theorem.
Derivatives: Definition and Basic Rules
Understanding derivatives as the slope of a curve.
Secant lines and average rate of change.
Differentiability and estimation of derivatives.
Application of the power rule for different powers.
Fundamental derivative rules involving constant, sum, difference, and constant multiple.
Differentiation of polynomials and trigonometric functions.
Product and quotient rules, and derivatives of more complex trigonometric functions.
Derivatives: Chain Rule and Advanced Topics
Identifying composite functions and applying the chain rule.
Implicit differentiation and differentiation of inverse functions.
Advanced differentiation strategies using multiple rules.
Second derivatives and logarithmic differentiation.
Applications of Derivatives
Interpreting derivatives in various contexts, especially motion.
Solving related rates problems.
Approximation using local linearity.
Application of L'Hôpital's rule for indeterminate forms.
Analyzing Functions
Applying the Mean Value Theorem and finding critical points.
Identifying intervals of increase/decrease and local/absolute extrema.
Understanding concavity and inflection points using the second derivative test.
Sketching curves and solving optimization problems.
Integrals
Introduction to accumulation of change and Riemann sums.
Summation notation and defining integrals.
Fundamental Theorem of Calculus and interpreting accumulation functions.
Techniques for finding definite and indefinite integrals, including substitution.
Differential Equations
Writing and verifying solutions to differential equations.
Sketching and reasoning with slope fields.
Solving separable differential equations and applying exponential models.
Applications of Integrals
Calculating average values and solving motion problems through integrals.
Determining areas between curves and volumes using cross-sectional methods (disk and washer methods).
Additional Resources
Course challenges and quizzes are available for testing knowledge and skills learned.
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Conclusion
This course serves as a comprehensive introduction to the foundational concepts of calculus, with applications spanning across various mathematical problems.