Calculus
  • Demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity. Know the definition of convergence and divergence of a function as the domain variable approaches either a number or infinity.

  • Prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions.

  • Use graphical calculators to verify and estimate limits.

  • Prove and use special limits, such as the limits of (sin(x))/x and (1−cos(x))/x as x tends to 0.

  • Demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function. 

  • Demonstrate an understanding and the application of the intermediate value theorem and the extreme value theorem.

  • Demonstrate an understanding of the formal definition of the derivative of a function at a point and the notion of differentiability.

  • Demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function.  

  • Demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of changes can use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function.

  • Understand the relation between differentiability and continuity.

  • Derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions.

  • Know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.

  • Find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of problems in physics, chemistry, economics, and so forth.

  • Compute derivatives of higher orders.

  • Know and can apply Rolle’s Theorem, the mean value theorem, and L’Hôpital’s rule.

  • Inflexion points, and intervals in which the function is increasing and decreasing.

  • Know Newton’s method for approximating the zeros of a function.

  • Use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts.

  • Use differentiation to solve related rate problems in a variety of pure and applied contexts.

  • Know the definition of the definite integral by using Riemann sums to approximate integrals.

  • Apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals. 

  • Knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives.

  • Use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work.

  • Compute, by hand, the integrals of a wide variety of functions by using techniques of integration, such as substitution, integration by parts, and trigonometric substitution. 

  • Know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals. 

  • Compute, by hand, the integrals of rational functions by combining the techniques in standard with the algebraic techniques of partial fractions and completing the square.

  • Compute the integrals of trigonometric functions by using the techniques noted above.

  • Understand the algorithms involved in Simpson’s rule and Newton’s method. Calculators or computers or both to approximate integrals numerically.

  • Understand improper integrals as limits of definite integrals.

  • Understand and can compute the radius (interval) of the convergence of power series.

  • Differentiate and integrate the terms of a power series in order to form new series from known ones.

  • Calculate Taylor polynomials and Taylor series of basic functions, including the remainder term.

  • Know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems.