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 change.                         s 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.                                                    

  • Inflection 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.