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Section Topics and Learning Outcomes

Topics
Understanding the Derivative: How do we measure velocity?; The notion of a limit; The derivative of a function at a point; The derivative function; Interpreting, estimating, and using the derivative; The second derivative; Limits, Continuity, and Differentiability; The tangent line approximation.
Computing Derivatives: Elementary derivative rules; The sine and cosine functions; The product and quotient rules; Derivatives of other trigonometry functions; The chain rule; Derivatives of inverse functions; Derivatives of functions given implicitly
Using Derivatives: Related rates; Using derivatives to evaluate limits; Using derivatives to identify extreme values; Global optimization; Applied optimization;
The Definite Integral: Determining distance traveled from velocity; Riemann sums; The definite integral; The Fundamental Theorem of Calculus.
Evaluating Integrals: Constructing accurate graphs of antiderivatives; The Second Fundamental Theorem of Calculus; Integration by substitution.
Learning Outcomes
  • Calculate limits using graphical, algebraic and numerical methods and the Squeeze Theorem.
  • Determine continuity of a function at points, identify various types of discontinuities and identify intervals of continuity for functions.
  • Calculate derivatives from the definition and using differentiation rules and implicit differentiation.
  • Use differentiation to identify tangent lines, to solve geometric problems and to solve problems involving rates of change.
  • Solve optimization and related-rates problems by analytic methods.
  • Graph a function, showing all relevant information such as relative extrema, inflection points, and asymptotes, with analytic methods and without the aid of graphing technology.
  • Approximate roots of equations by using the bisection method and Newton’s method.
  • Calculate integrals by using the Fundamental Theorem of Calculus, u-substitution, and observations about symmetry of graphs.
  • Use integration to find the area under a curve, to solve problems involving net change, and to solve other geometric problems.
  • State and apply named theorems of calculus (the Intermediate Value Theorem, the Mean Value Theorem, and the Fundamental Theorem of Calculus).
  • Identify whether a derivative or an integral (or neither) is more appropriate for use in a given problem.
  • Communicate their knowledge of the basic principles of Calculus I, both orally (e.g. class discussions) and in writing (e.g. written assessments).
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