The basic objective of Calculus is to relate small-scale (differential) quantities to large-scale (integrated) quantities. This is accomplished by means of the Fundamental Theorem of Calculus. Students should demonstrate an understanding of the integral as a cumulative sum, of the derivative as a rate of change, and of the inverse relationship between integration and differentiation.
Students completing 18.01 can:
- Use both the definition of derivative as a limit and the rules of differentiation to differentiate functions.
- Sketch the graph of a function using asymptotes, critical points, and the derivative test for increasing/decreasing and concavity properties.
- Set up max/min problems and use differentiation to solve them.
- Set up related rates problems and use differentiation to solve them.
- Evaluate integrals by using the Fundamental Theorem of Calculus.
- Apply integration to compute areas and volumes by slicing, volumes of revolution, arclength, and surface areas of revolution.
- Evaluate integrals using techniques of integration, such as substitution, inverse substitution, partial fractions and integration by parts.
- Set up and solve first order differential equations using separation of variables.
- Use L’Hôpital’s rule.
- Determine convergence/divergence of improper integrals, and evaluate convergent improper integrals.
- Estimate and compare series and integrals to determine convergence.
- Find the Taylor series expansion of a function near a point, with emphasis on the first two or three terms.