After completing this course, students should have developed a clear understanding of the fundamental concepts of single variable calculus and a range of skills allowing them to work effectively with the concepts.
The basic concepts are:
1. Derivatives as rates of change, computed as a limit of ratios
2. Integrals as a "sum," computed as a limit of Riemann sums
After completing this course, students should demonstrate competency in the following skills:
- Use both the limit definition and rules of differentiation to differentiate functions.
- Sketch the graph of a function using asymptotes, critical points, the derivative test for increasing/decreasing functions, and concavity.
- Apply differentiation to solve applied max/min problems.
- Apply differentiation to solve related rates problems.
- Evaluate integrals both by using Riemann sums and by using the Fundamental Theorem of Calculus.
- Apply integration to compute arc lengths, volumes of revolution and surface areas of revolution.
- Evaluate integrals using advanced techniques of integration, such as inverse substitution, partial fractions and integration by parts.
- Use L'Hospital's rule to evaluate certain indefinite forms.
- Determine convergence/divergence of improper integrals and evaluate convergent improper integrals.
- Determine the convergence/divergence of an infinite series and find the Taylor series expansion of a function near a point.